diff --git "a/data/sensemath_v2_d16.json" "b/data/sensemath_v2_d16.json" new file mode 100644--- /dev/null +++ "b/data/sensemath_v2_d16.json" @@ -0,0 +1,28602 @@ +[ + { + "family_id": "magnitude_estimation_001202", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100000000000000 * 9800000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 9800000000000000?", + "options": [ + "98980000000000000000000000000000", + "98980000000000000000000000000020", + "98979999999999999999999999999990", + "98980000000000000000000000000010" + ], + "answer": "98980000000000000000000000000000", + "correct_value": 98980000000000000000000000000000, + "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 9800000000000000≈10000000000000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10100000000000000 * 5001825339182240", + "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 5001825339182240?", + "options": [ + "50518435925740623999999999999990", + "50518435925740624000000000000010", + "50518435925740624000000000000000", + "50518435925740624000000000000020" + ], + "answer": "50518435925740624000000000000000", + "correct_value": 50518435925740624000000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5155500612349868 * 6723920597574282", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5155500612349868 * 6723920597574282?", + "options": [ + "34665176758186101218092702894786", + "34665176758186101218092702894796", + "34665176758186101218092702894776", + "34665176758186101218092702894766" + ], + "answer": "34665176758186101218092702894776", + "correct_value": 34665176758186101218092702894776, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001203", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10200000000000000 * 9900000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10200000000000000 * 9900000000000000?", + "options": [ + "100979999999999999999999999999990", + "100980000000000000000000000000020", + "100980000000000000000000000000010", + "100980000000000000000000000000000" + ], + "answer": "100980000000000000000000000000000", + "correct_value": 100980000000000000000000000000000, + "explanation": "Round to 10000000000000000: 10200000000000000≈10000000000000000, 9900000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200000000000000 * 5152184862388030", + "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200000000000000 * 5152184862388030?", + "options": [ + "52552285596357906000000000000000", + "52552285596357905999999999999990", + "52552285596357906000000000000010", + "52552285596357906000000000000020" + ], + "answer": "52552285596357906000000000000000", + "correct_value": 52552285596357906000000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5552630215968205 * 6748424206165293", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5552630215968205 * 6748424206165293?", + "options": [ + "37471504157324653254658262509055", + "37471504157324653254658262509065", + "37471504157324653254658262509085", + "37471504157324653254658262509075" + ], + "answer": "37471504157324653254658262509065", + "correct_value": 37471504157324653254658262509065, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001204", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9900000000000000 * 10100000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 9900000000000000 * 10100000000000000?", + "options": [ + "99990000000000000000000000000000", + "99990000000000000000000000000020", + "99990000000000000000000000000010", + "99989999999999999999999999999990" + ], + "answer": "99990000000000000000000000000000", + "correct_value": 99990000000000000000000000000000, + "explanation": "Round to 10000000000000000: 9900000000000000≈10000000000000000, 10100000000000000≈10000000000000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900000000000000 * 4842111086770157", + "ns_shortcut": "Round 9900000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900000000000000 * 4842111086770157?", + "options": [ + "47936899759024554300000000000010", + "47936899759024554299999999999990", + "47936899759024554300000000000020", + "47936899759024554300000000000000" + ], + "answer": "47936899759024554300000000000000", + "correct_value": 47936899759024554300000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5356067257309484 * 5112654749188299", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5356067257309484 * 5112654749188299?", + "options": [ + "27383722700055280443823634527726", + "27383722700055280443823634527716", + "27383722700055280443823634527706", + "27383722700055280443823634527736" + ], + "answer": "27383722700055280443823634527716", + "correct_value": 27383722700055280443823634527716, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001205", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800000000000000 * 10100000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 10100000000000000?", + "options": [ + "98980000000000000000000000000020", + "98979999999999999999999999999990", + "98980000000000000000000000000000", + "98980000000000000000000000000010" + ], + "answer": "98980000000000000000000000000000", + "correct_value": 98980000000000000000000000000000, + "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 10100000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200000000000000 * 5120910310714107", + "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200000000000000 * 5120910310714107?", + "options": [ + "52233285169283891400000000000010", + "52233285169283891400000000000020", + "52233285169283891400000000000000", + "52233285169283891399999999999990" + ], + "answer": "52233285169283891400000000000000", + "correct_value": 52233285169283891400000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5812476740312957 * 4132091761466112", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5812476740312957 * 4132091761466112?", + "options": [ + "24017687252360571339628030013204", + "24017687252360571339628030013194", + "24017687252360571339628030013174", + "24017687252360571339628030013184" + ], + "answer": "24017687252360571339628030013184", + "correct_value": 24017687252360571339628030013184, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_001206", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100000000000000 * 9800000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 9800000000000000?", + "options": [ + "98980000000000000000000000000020", + "98980000000000000000000000000000", + "98980000000000000000000000000010", + "98979999999999999999999999999990" + ], + "answer": "98980000000000000000000000000000", + "correct_value": 98980000000000000000000000000000, + "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 9800000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900000000000000 * 5103161774968639", + "ns_shortcut": "Round 9900000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900000000000000 * 5103161774968639?", + "options": [ + "50521301572189526100000000000010", + "50521301572189526099999999999990", + "50521301572189526100000000000000", + "50521301572189526100000000000020" + ], + "answer": "50521301572189526100000000000000", + "correct_value": 50521301572189526100000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4152024030649896 * 5280123037233386", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4152024030649896 * 5280123037233386?", + "options": [ + "21923197735381134231671408627876", + "21923197735381134231671408627866", + "21923197735381134231671408627846", + "21923197735381134231671408627856" + ], + "answer": "21923197735381134231671408627856", + "correct_value": 21923197735381134231671408627856, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_001207", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800000000000000 * 9800000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 9800000000000000?", + "options": [ + "96040000000000000000000000000000", + "96039999999999999999999999999990", + "96040000000000000000000000000020", + "96040000000000000000000000000010" + ], + "answer": "96040000000000000000000000000000", + "correct_value": 96040000000000000000000000000000, + "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 9800000000000000≈10000000000000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10100000000000000 * 5066956740204353", + "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 5066956740204353?", + "options": [ + "51176263076063965300000000000010", + "51176263076063965299999999999990", + "51176263076063965300000000000000", + "51176263076063965300000000000020" + ], + "answer": "51176263076063965300000000000000", + "correct_value": 51176263076063965300000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6760244877354416 * 5958069320222046", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 6760244877354416 * 5958069320222046?", + "options": [ + "40278007600953594070154158655136", + "40278007600953594070154158655146", + "40278007600953594070154158655126", + "40278007600953594070154158655156" + ], + "answer": "40278007600953594070154158655136", + "correct_value": 40278007600953594070154158655136, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001208", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100000000000000 * 9900000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 9900000000000000?", + "options": [ + "99990000000000000000000000000020", + "99990000000000000000000000000000", + "99989999999999999999999999999990", + "99990000000000000000000000000010" + ], + "answer": "99990000000000000000000000000000", + "correct_value": 99990000000000000000000000000000, + "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 9900000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9800000000000000 * 5073883247960602", + "ns_shortcut": "Round 9800000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 5073883247960602?", + "options": [ + "49724055830013899600000000000000", + "49724055830013899600000000000010", + "49724055830013899599999999999990", + "49724055830013899600000000000020" + ], + "answer": "49724055830013899600000000000000", + "correct_value": 49724055830013899600000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5516961977124086 * 5045411496661417", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5516961977124086 * 5045411496661417?", + "options": [ + "27835343386025764962916137589852", + "27835343386025764962916137589872", + "27835343386025764962916137589882", + "27835343386025764962916137589862" + ], + "answer": "27835343386025764962916137589862", + "correct_value": 27835343386025764962916137589862, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_001209", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10200000000000000 * 10200000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 10200000000000000 * 10200000000000000?", + "options": [ + "104039999999999999999999999999990", + "104040000000000000000000000000010", + "104040000000000000000000000000020", + "104040000000000000000000000000000" + ], + "answer": "104040000000000000000000000000000", + "correct_value": 104040000000000000000000000000000, + "explanation": "Round to 10000000000000000: 10200000000000000≈10000000000000000, 10200000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10100000000000000 * 4854921440655604", + "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 4854921440655604?", + "options": [ + "49034706550621600400000000000020", + "49034706550621600400000000000000", + "49034706550621600399999999999990", + "49034706550621600400000000000010" + ], + "answer": "49034706550621600400000000000000", + "correct_value": 49034706550621600400000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6404689492378734 * 4383592137668171", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 6404689492378734 * 4383592137668171?", + "options": [ + "28075546502997367571226749075514", + "28075546502997367571226749075524", + "28075546502997367571226749075534", + "28075546502997367571226749075504" + ], + "answer": "28075546502997367571226749075514", + "correct_value": 28075546502997367571226749075514, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001210", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9900000000000000 * 9800000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 9900000000000000 * 9800000000000000?", + "options": [ + "97020000000000000000000000000000", + "97020000000000000000000000000010", + "97020000000000000000000000000020", + "97019999999999999999999999999990" + ], + "answer": "97020000000000000000000000000000", + "correct_value": 97020000000000000000000000000000, + "explanation": "Round to 10000000000000000: 9900000000000000≈10000000000000000, 9800000000000000≈10000000000000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10100000000000000 * 5129201688292491", + "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 5129201688292491?", + "options": [ + "51804937051754159100000000000000", + "51804937051754159100000000000010", + "51804937051754159100000000000020", + "51804937051754159099999999999990" + ], + "answer": "51804937051754159100000000000000", + "correct_value": 51804937051754159100000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6891108841929046 * 4030300335533585", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 6891108841929046 * 4030300335533585?", + "options": [ + "27773238277825088451451120009900", + "27773238277825088451451120009920", + "27773238277825088451451120009910", + "27773238277825088451451120009930" + ], + "answer": "27773238277825088451451120009910", + "correct_value": 27773238277825088451451120009910, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001211", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10200000000000000 * 10100000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10200000000000000 * 10100000000000000?", + "options": [ + "103020000000000000000000000000010", + "103019999999999999999999999999990", + "103020000000000000000000000000020", + "103020000000000000000000000000000" + ], + "answer": "103020000000000000000000000000000", + "correct_value": 103020000000000000000000000000000, + "explanation": "Round to 10000000000000000: 10200000000000000≈10000000000000000, 10100000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9800000000000000 * 4906744557477605", + "ns_shortcut": "Round 9800000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 4906744557477605?", + "options": [ + "48086096663280529000000000000020", + "48086096663280528999999999999990", + "48086096663280529000000000000000", + "48086096663280529000000000000010" + ], + "answer": "48086096663280529000000000000000", + "correct_value": 48086096663280529000000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5822628859159496 * 4057342524693544", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5822628859159496 * 4057342524693544?", + "options": [ + "23624399675775679328704817493814", + "23624399675775679328704817493844", + "23624399675775679328704817493834", + "23624399675775679328704817493824" + ], + "answer": "23624399675775679328704817493824", + "correct_value": 23624399675775679328704817493824, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_001212", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10200000000000000 * 9900000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10200000000000000 * 9900000000000000?", + "options": [ + "100980000000000000000000000000020", + "100979999999999999999999999999990", + "100980000000000000000000000000000", + "100980000000000000000000000000010" + ], + "answer": "100980000000000000000000000000000", + "correct_value": 100980000000000000000000000000000, + "explanation": "Round to 10000000000000000: 10200000000000000≈10000000000000000, 9900000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200000000000000 * 4993047752579264", + "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200000000000000 * 4993047752579264?", + "options": [ + "50929087076308492800000000000010", + "50929087076308492800000000000000", + "50929087076308492800000000000020", + "50929087076308492799999999999990" + ], + "answer": "50929087076308492800000000000000", + "correct_value": 50929087076308492800000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4202610641184814 * 6629331028845125", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4202610641184814 * 6629331028845125?", + "options": [ + "27860497125761193450740107931760", + "27860497125761193450740107931740", + "27860497125761193450740107931770", + "27860497125761193450740107931750" + ], + "answer": "27860497125761193450740107931750", + "correct_value": 27860497125761193450740107931750, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_001213", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10200000000000000 * 9900000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10200000000000000 * 9900000000000000?", + "options": [ + "100980000000000000000000000000020", + "100980000000000000000000000000000", + "100979999999999999999999999999990", + "100980000000000000000000000000010" + ], + "answer": "100980000000000000000000000000000", + "correct_value": 100980000000000000000000000000000, + "explanation": "Round to 10000000000000000: 10200000000000000≈10000000000000000, 9900000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200000000000000 * 4806801361212255", + "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200000000000000 * 4806801361212255?", + "options": [ + "49029373884365001000000000000020", + "49029373884365000999999999999990", + "49029373884365001000000000000000", + "49029373884365001000000000000010" + ], + "answer": "49029373884365001000000000000000", + "correct_value": 49029373884365001000000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5290781332893331 * 6742394815088882", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5290781332893331 * 6742394815088882?", + "options": [ + "35672536626669039108908990045932", + "35672536626669039108908990045962", + "35672536626669039108908990045952", + "35672536626669039108908990045942" + ], + "answer": "35672536626669039108908990045942", + "correct_value": 35672536626669039108908990045942, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_001214", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10200000000000000 * 9800000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 10200000000000000 * 9800000000000000?", + "options": [ + "99960000000000000000000000000000", + "99960000000000000000000000000010", + "99960000000000000000000000000020", + "99959999999999999999999999999990" + ], + "answer": "99960000000000000000000000000000", + "correct_value": 99960000000000000000000000000000, + "explanation": "Round to 10000000000000000: 10200000000000000≈10000000000000000, 9800000000000000≈10000000000000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900000000000000 * 4897509021467765", + "ns_shortcut": "Round 9900000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900000000000000 * 4897509021467765?", + "options": [ + "48485339312530873499999999999990", + "48485339312530873500000000000020", + "48485339312530873500000000000000", + "48485339312530873500000000000010" + ], + "answer": "48485339312530873500000000000000", + "correct_value": 48485339312530873500000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5862411509202670 * 4343246576561364", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5862411509202670 * 4343246576561364?", + "options": [ + "25461898717738435742009967641900", + "25461898717738435742009967641870", + "25461898717738435742009967641890", + "25461898717738435742009967641880" + ], + "answer": "25461898717738435742009967641880", + "correct_value": 25461898717738435742009967641880, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_001215", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100000000000000 * 9800000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 9800000000000000?", + "options": [ + "98980000000000000000000000000010", + "98980000000000000000000000000000", + "98980000000000000000000000000020", + "98979999999999999999999999999990" + ], + "answer": "98980000000000000000000000000000", + "correct_value": 98980000000000000000000000000000, + "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 9800000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200000000000000 * 5092523396627801", + "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200000000000000 * 5092523396627801?", + "options": [ + "51943738645603570200000000000010", + "51943738645603570200000000000020", + "51943738645603570200000000000000", + "51943738645603570199999999999990" + ], + "answer": "51943738645603570200000000000000", + "correct_value": 51943738645603570200000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5727248378323872 * 4166809038465592", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5727248378323872 * 4166809038465592?", + "options": [ + "23864350308337314167715704212214", + "23864350308337314167715704212234", + "23864350308337314167715704212244", + "23864350308337314167715704212224" + ], + "answer": "23864350308337314167715704212224", + "correct_value": 23864350308337314167715704212224, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_001216", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800000000000000 * 10100000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 10100000000000000?", + "options": [ + "98980000000000000000000000000010", + "98980000000000000000000000000000", + "98979999999999999999999999999990", + "98980000000000000000000000000020" + ], + "answer": "98980000000000000000000000000000", + "correct_value": 98980000000000000000000000000000, + "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 10100000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10100000000000000 * 5053602910249821", + "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 5053602910249821?", + "options": [ + "51041389393523192100000000000000", + "51041389393523192099999999999990", + "51041389393523192100000000000010", + "51041389393523192100000000000020" + ], + "answer": "51041389393523192100000000000000", + "correct_value": 51041389393523192100000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5647170076319609 * 6712414650260605", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5647170076319609 * 6712414650260605?", + "options": [ + "37906147152801042291611121703465", + "37906147152801042291611121703435", + "37906147152801042291611121703455", + "37906147152801042291611121703445" + ], + "answer": "37906147152801042291611121703445", + "correct_value": 37906147152801042291611121703445, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_001217", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10200000000000000 * 10200000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 10200000000000000 * 10200000000000000?", + "options": [ + "104040000000000000000000000000020", + "104040000000000000000000000000010", + "104039999999999999999999999999990", + "104040000000000000000000000000000" + ], + "answer": "104040000000000000000000000000000", + "correct_value": 104040000000000000000000000000000, + "explanation": "Round to 10000000000000000: 10200000000000000≈10000000000000000, 10200000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200000000000000 * 4822535086594521", + "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200000000000000 * 4822535086594521?", + "options": [ + "49189857883264114200000000000010", + "49189857883264114200000000000020", + "49189857883264114199999999999990", + "49189857883264114200000000000000" + ], + "answer": "49189857883264114200000000000000", + "correct_value": 49189857883264114200000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4527955898884954 * 5244305882721060", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4527955898884954 * 5244305882721060?", + "options": [ + "23745985757223889383949412931250", + "23745985757223889383949412931260", + "23745985757223889383949412931230", + "23745985757223889383949412931240" + ], + "answer": "23745985757223889383949412931240", + "correct_value": 23745985757223889383949412931240, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_001218", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9900000000000000 * 9800000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 9900000000000000 * 9800000000000000?", + "options": [ + "97020000000000000000000000000010", + "97019999999999999999999999999990", + "97020000000000000000000000000020", + "97020000000000000000000000000000" + ], + "answer": "97020000000000000000000000000000", + "correct_value": 97020000000000000000000000000000, + "explanation": "Round to 10000000000000000: 9900000000000000≈10000000000000000, 9800000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200000000000000 * 5044852726630192", + "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200000000000000 * 5044852726630192?", + "options": [ + "51457497811627958400000000000010", + "51457497811627958399999999999990", + "51457497811627958400000000000000", + "51457497811627958400000000000020" + ], + "answer": "51457497811627958400000000000000", + "correct_value": 51457497811627958400000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6313687022006600 * 6449904410416486", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 6313687022006600 * 6449904410416486?", + "options": [ + "40722677769229698642153440807600", + "40722677769229698642153440807610", + "40722677769229698642153440807620", + "40722677769229698642153440807590" + ], + "answer": "40722677769229698642153440807600", + "correct_value": 40722677769229698642153440807600, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001219", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800000000000000 * 10200000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 10200000000000000?", + "options": [ + "99960000000000000000000000000020", + "99960000000000000000000000000010", + "99959999999999999999999999999990", + "99960000000000000000000000000000" + ], + "answer": "99960000000000000000000000000000", + "correct_value": 99960000000000000000000000000000, + "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 10200000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9800000000000000 * 4980221007934411", + "ns_shortcut": "Round 9800000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 4980221007934411?", + "options": [ + "48806165877757227800000000000000", + "48806165877757227799999999999990", + "48806165877757227800000000000010", + "48806165877757227800000000000020" + ], + "answer": "48806165877757227800000000000000", + "correct_value": 48806165877757227800000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4930653271891821 * 4917490342333151", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4930653271891821 * 4917490342333151?", + "options": [ + "24246439845921381904476814057971", + "24246439845921381904476814057991", + "24246439845921381904476814057961", + "24246439845921381904476814057981" + ], + "answer": "24246439845921381904476814057971", + "correct_value": 24246439845921381904476814057971, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001220", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9900000000000000 * 9900000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 9900000000000000 * 9900000000000000?", + "options": [ + "98010000000000000000000000000020", + "98010000000000000000000000000010", + "98009999999999999999999999999990", + "98010000000000000000000000000000" + ], + "answer": "98010000000000000000000000000000", + "correct_value": 98010000000000000000000000000000, + "explanation": "Round to 10000000000000000: 9900000000000000≈10000000000000000, 9900000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200000000000000 * 4800975125696054", + "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200000000000000 * 4800975125696054?", + "options": [ + "48969946282099750800000000000010", + "48969946282099750800000000000020", + "48969946282099750799999999999990", + "48969946282099750800000000000000" + ], + "answer": "48969946282099750800000000000000", + "correct_value": 48969946282099750800000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5945567960139418 * 4191725138415944", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5945567960139418 * 4191725138415944?", + "options": [ + "24922186680676803734901914080602", + "24922186680676803734901914080612", + "24922186680676803734901914080592", + "24922186680676803734901914080582" + ], + "answer": "24922186680676803734901914080592", + "correct_value": 24922186680676803734901914080592, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001221", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800000000000000 * 9800000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 9800000000000000?", + "options": [ + "96039999999999999999999999999990", + "96040000000000000000000000000010", + "96040000000000000000000000000020", + "96040000000000000000000000000000" + ], + "answer": "96040000000000000000000000000000", + "correct_value": 96040000000000000000000000000000, + "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 9800000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10100000000000000 * 4945815364780484", + "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 4945815364780484?", + "options": [ + "49952735184282888400000000000010", + "49952735184282888399999999999990", + "49952735184282888400000000000020", + "49952735184282888400000000000000" + ], + "answer": "49952735184282888400000000000000", + "correct_value": 49952735184282888400000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5755494055002924 * 4451531537784221", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5755494055002924 * 4451531537784221?", + "options": [ + "25620763301375108116522636062194", + "25620763301375108116522636062224", + "25620763301375108116522636062214", + "25620763301375108116522636062204" + ], + "answer": "25620763301375108116522636062204", + "correct_value": 25620763301375108116522636062204, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_001222", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100000000000000 * 9800000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 9800000000000000?", + "options": [ + "98979999999999999999999999999990", + "98980000000000000000000000000010", + "98980000000000000000000000000000", + "98980000000000000000000000000020" + ], + "answer": "98980000000000000000000000000000", + "correct_value": 98980000000000000000000000000000, + "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 9800000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900000000000000 * 5030392877535389", + "ns_shortcut": "Round 9900000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900000000000000 * 5030392877535389?", + "options": [ + "49800889487600351100000000000010", + "49800889487600351100000000000020", + "49800889487600351099999999999990", + "49800889487600351100000000000000" + ], + "answer": "49800889487600351100000000000000", + "correct_value": 49800889487600351100000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5329769258417834 * 6048338598006816", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5329769258417834 * 6048338598006816?", + "options": [ + "32236249124158749501022107956534", + "32236249124158749501022107956564", + "32236249124158749501022107956544", + "32236249124158749501022107956554" + ], + "answer": "32236249124158749501022107956544", + "correct_value": 32236249124158749501022107956544, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001223", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100000000000000 * 9900000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 9900000000000000?", + "options": [ + "99990000000000000000000000000010", + "99990000000000000000000000000000", + "99990000000000000000000000000020", + "99989999999999999999999999999990" + ], + "answer": "99990000000000000000000000000000", + "correct_value": 99990000000000000000000000000000, + "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 9900000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10100000000000000 * 4923131470611700", + "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 4923131470611700?", + "options": [ + "49723627853178170000000000000000", + "49723627853178170000000000000020", + "49723627853178169999999999999990", + "49723627853178170000000000000010" + ], + "answer": "49723627853178170000000000000000", + "correct_value": 49723627853178170000000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4619638249980770 * 6884881938188582", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4619638249980770 * 6884881938188582?", + "options": [ + "31805663948257712840790133568160", + "31805663948257712840790133568130", + "31805663948257712840790133568140", + "31805663948257712840790133568150" + ], + "answer": "31805663948257712840790133568140", + "correct_value": 31805663948257712840790133568140, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001224", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800000000000000 * 10100000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 10100000000000000?", + "options": [ + "98980000000000000000000000000010", + "98980000000000000000000000000000", + "98980000000000000000000000000020", + "98979999999999999999999999999990" + ], + "answer": "98980000000000000000000000000000", + "correct_value": 98980000000000000000000000000000, + "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 10100000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10100000000000000 * 5104882698796544", + "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 5104882698796544?", + "options": [ + "51559315257845094400000000000020", + "51559315257845094400000000000010", + "51559315257845094399999999999990", + "51559315257845094400000000000000" + ], + "answer": "51559315257845094400000000000000", + "correct_value": 51559315257845094400000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5745083024855275 * 5838254583182977", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5745083024855275 * 5838254583182977?", + "options": [ + "33541257300628030237114268653695", + "33541257300628030237114268653675", + "33541257300628030237114268653665", + "33541257300628030237114268653685" + ], + "answer": "33541257300628030237114268653675", + "correct_value": 33541257300628030237114268653675, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001225", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9900000000000000 * 10100000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 9900000000000000 * 10100000000000000?", + "options": [ + "99990000000000000000000000000020", + "99990000000000000000000000000010", + "99990000000000000000000000000000", + "99989999999999999999999999999990" + ], + "answer": "99990000000000000000000000000000", + "correct_value": 99990000000000000000000000000000, + "explanation": "Round to 10000000000000000: 9900000000000000≈10000000000000000, 10100000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200000000000000 * 4924573409349543", + "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200000000000000 * 4924573409349543?", + "options": [ + "50230648775365338599999999999990", + "50230648775365338600000000000020", + "50230648775365338600000000000010", + "50230648775365338600000000000000" + ], + "answer": "50230648775365338600000000000000", + "correct_value": 50230648775365338600000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5356516177011934 * 4584296011508358", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5356516177011934 * 4584296011508358?", + "options": [ + "24555855745855806786308706744372", + "24555855745855806786308706744362", + "24555855745855806786308706744382", + "24555855745855806786308706744392" + ], + "answer": "24555855745855806786308706744372", + "correct_value": 24555855745855806786308706744372, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001226", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800000000000000 * 9900000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 9900000000000000?", + "options": [ + "97020000000000000000000000000000", + "97020000000000000000000000000010", + "97020000000000000000000000000020", + "97019999999999999999999999999990" + ], + "answer": "97020000000000000000000000000000", + "correct_value": 97020000000000000000000000000000, + "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 9900000000000000≈10000000000000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9800000000000000 * 4886334506404108", + "ns_shortcut": "Round 9800000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 4886334506404108?", + "options": [ + "47886078162760258399999999999990", + "47886078162760258400000000000010", + "47886078162760258400000000000020", + "47886078162760258400000000000000" + ], + "answer": "47886078162760258400000000000000", + "correct_value": 47886078162760258400000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5782567748443046 * 5954438778181177", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5782567748443046 * 5954438778181177?", + "options": [ + "34431945638789090503797453745162", + "34431945638789090503797453745132", + "34431945638789090503797453745152", + "34431945638789090503797453745142" + ], + "answer": "34431945638789090503797453745142", + "correct_value": 34431945638789090503797453745142, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_001227", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9900000000000000 * 9800000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 9900000000000000 * 9800000000000000?", + "options": [ + "97020000000000000000000000000020", + "97020000000000000000000000000000", + "97020000000000000000000000000010", + "97019999999999999999999999999990" + ], + "answer": "97020000000000000000000000000000", + "correct_value": 97020000000000000000000000000000, + "explanation": "Round to 10000000000000000: 9900000000000000≈10000000000000000, 9800000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10100000000000000 * 5164167204012696", + "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 5164167204012696?", + "options": [ + "52158088760528229600000000000020", + "52158088760528229600000000000000", + "52158088760528229600000000000010", + "52158088760528229599999999999990" + ], + "answer": "52158088760528229600000000000000", + "correct_value": 52158088760528229600000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4532621438566095 * 4742685695159983", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4532621438566095 * 4742685695159983?", + "options": [ + "21496798858262882444117144576395", + "21496798858262882444117144576375", + "21496798858262882444117144576405", + "21496798858262882444117144576385" + ], + "answer": "21496798858262882444117144576385", + "correct_value": 21496798858262882444117144576385, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_001228", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800000000000000 * 9900000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 9900000000000000?", + "options": [ + "97020000000000000000000000000010", + "97019999999999999999999999999990", + "97020000000000000000000000000000", + "97020000000000000000000000000020" + ], + "answer": "97020000000000000000000000000000", + "correct_value": 97020000000000000000000000000000, + "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 9900000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200000000000000 * 5126783907362554", + "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200000000000000 * 5126783907362554?", + "options": [ + "52293195855098050800000000000010", + "52293195855098050800000000000000", + "52293195855098050800000000000020", + "52293195855098050799999999999990" + ], + "answer": "52293195855098050800000000000000", + "correct_value": 52293195855098050800000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6105391114893111 * 6690701216482219", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 6105391114893111 * 6690701216482219?", + "options": [ + "40849347759515069075755617093319", + "40849347759515069075755617093299", + "40849347759515069075755617093309", + "40849347759515069075755617093329" + ], + "answer": "40849347759515069075755617093309", + "correct_value": 40849347759515069075755617093309, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001229", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9900000000000000 * 10200000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 9900000000000000 * 10200000000000000?", + "options": [ + "100980000000000000000000000000020", + "100980000000000000000000000000010", + "100979999999999999999999999999990", + "100980000000000000000000000000000" + ], + "answer": "100980000000000000000000000000000", + "correct_value": 100980000000000000000000000000000, + "explanation": "Round to 10000000000000000: 9900000000000000≈10000000000000000, 10200000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10100000000000000 * 5074043480065477", + "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 5074043480065477?", + "options": [ + "51247839148661317700000000000010", + "51247839148661317699999999999990", + "51247839148661317700000000000000", + "51247839148661317700000000000020" + ], + "answer": "51247839148661317700000000000000", + "correct_value": 51247839148661317700000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5329807492629888 * 5448721080235767", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5329807492629888 * 5448721080235767?", + "options": [ + "29040634438691008106753910804086", + "29040634438691008106753910804096", + "29040634438691008106753910804106", + "29040634438691008106753910804116" + ], + "answer": "29040634438691008106753910804096", + "correct_value": 29040634438691008106753910804096, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001230", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800000000000000 * 10100000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 10100000000000000?", + "options": [ + "98980000000000000000000000000000", + "98980000000000000000000000000010", + "98979999999999999999999999999990", + "98980000000000000000000000000020" + ], + "answer": "98980000000000000000000000000000", + "correct_value": 98980000000000000000000000000000, + "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 10100000000000000≈10000000000000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9800000000000000 * 4865539774409013", + "ns_shortcut": "Round 9800000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 4865539774409013?", + "options": [ + "47682289789208327400000000000020", + "47682289789208327399999999999990", + "47682289789208327400000000000010", + "47682289789208327400000000000000" + ], + "answer": "47682289789208327400000000000000", + "correct_value": 47682289789208327400000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6510477638092508 * 6840064374314141", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 6510477638092508 * 6840064374314141?", + "options": [ + "44532086152085437242818010555618", + "44532086152085437242818010555638", + "44532086152085437242818010555648", + "44532086152085437242818010555628" + ], + "answer": "44532086152085437242818010555628", + "correct_value": 44532086152085437242818010555628, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_001231", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100000000000000 * 10100000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 10100000000000000?", + "options": [ + "102010000000000000000000000000000", + "102010000000000000000000000000020", + "102009999999999999999999999999990", + "102010000000000000000000000000010" + ], + "answer": "102010000000000000000000000000000", + "correct_value": 102010000000000000000000000000000, + "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 10100000000000000≈10000000000000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9800000000000000 * 4871645584344437", + "ns_shortcut": "Round 9800000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 4871645584344437?", + "options": [ + "47742126726575482599999999999990", + "47742126726575482600000000000010", + "47742126726575482600000000000000", + "47742126726575482600000000000020" + ], + "answer": "47742126726575482600000000000000", + "correct_value": 47742126726575482600000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4555507416800086 * 6776608281358877", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4555507416800086 * 6776608281358877?", + "options": [ + "30870889286479248144331130463442", + "30870889286479248144331130463432", + "30870889286479248144331130463422", + "30870889286479248144331130463412" + ], + "answer": "30870889286479248144331130463422", + "correct_value": 30870889286479248144331130463422, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001232", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10200000000000000 * 10200000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 10200000000000000 * 10200000000000000?", + "options": [ + "104040000000000000000000000000020", + "104040000000000000000000000000010", + "104039999999999999999999999999990", + "104040000000000000000000000000000" + ], + "answer": "104040000000000000000000000000000", + "correct_value": 104040000000000000000000000000000, + "explanation": "Round to 10000000000000000: 10200000000000000≈10000000000000000, 10200000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900000000000000 * 5016077832363676", + "ns_shortcut": "Round 9900000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900000000000000 * 5016077832363676?", + "options": [ + "49659170540400392400000000000020", + "49659170540400392400000000000010", + "49659170540400392400000000000000", + "49659170540400392399999999999990" + ], + "answer": "49659170540400392400000000000000", + "correct_value": 49659170540400392400000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6016010535027170 * 5334562348016950", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 6016010535027170 * 5334562348016950?", + "options": [ + "32092783285429247617563870531510", + "32092783285429247617563870531500", + "32092783285429247617563870531520", + "32092783285429247617563870531490" + ], + "answer": "32092783285429247617563870531500", + "correct_value": 32092783285429247617563870531500, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001233", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10200000000000000 * 10200000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 10200000000000000 * 10200000000000000?", + "options": [ + "104040000000000000000000000000010", + "104040000000000000000000000000020", + "104039999999999999999999999999990", + "104040000000000000000000000000000" + ], + "answer": "104040000000000000000000000000000", + "correct_value": 104040000000000000000000000000000, + "explanation": "Round to 10000000000000000: 10200000000000000≈10000000000000000, 10200000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900000000000000 * 4883591223167229", + "ns_shortcut": "Round 9900000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900000000000000 * 4883591223167229?", + "options": [ + "48347553109355567100000000000010", + "48347553109355567099999999999990", + "48347553109355567100000000000000", + "48347553109355567100000000000020" + ], + "answer": "48347553109355567100000000000000", + "correct_value": 48347553109355567100000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4305574056969059 * 5037627339029452", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4305574056969059 * 5037627339029452?", + "options": [ + "21689877579603282862629853725658", + "21689877579603282862629853725688", + "21689877579603282862629853725678", + "21689877579603282862629853725668" + ], + "answer": "21689877579603282862629853725668", + "correct_value": 21689877579603282862629853725668, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_001234", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800000000000000 * 10100000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 10100000000000000?", + "options": [ + "98980000000000000000000000000020", + "98980000000000000000000000000010", + "98980000000000000000000000000000", + "98979999999999999999999999999990" + ], + "answer": "98980000000000000000000000000000", + "correct_value": 98980000000000000000000000000000, + "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 10100000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10100000000000000 * 4818275652129337", + "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 4818275652129337?", + "options": [ + "48664584086506303700000000000000", + "48664584086506303700000000000010", + "48664584086506303699999999999990", + "48664584086506303700000000000020" + ], + "answer": "48664584086506303700000000000000", + "correct_value": 48664584086506303700000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5361084824088603 * 4942101398729140", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5361084824088603 * 4942101398729140?", + "options": [ + "26495024807833850350802957991420", + "26495024807833850350802957991410", + "26495024807833850350802957991440", + "26495024807833850350802957991430" + ], + "answer": "26495024807833850350802957991420", + "correct_value": 26495024807833850350802957991420, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001235", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100000000000000 * 10200000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 10200000000000000?", + "options": [ + "103019999999999999999999999999990", + "103020000000000000000000000000010", + "103020000000000000000000000000000", + "103020000000000000000000000000020" + ], + "answer": "103020000000000000000000000000000", + "correct_value": 103020000000000000000000000000000, + "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 10200000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200000000000000 * 4953293039806411", + "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200000000000000 * 4953293039806411?", + "options": [ + "50523589006025392200000000000000", + "50523589006025392200000000000010", + "50523589006025392199999999999990", + "50523589006025392200000000000020" + ], + "answer": "50523589006025392200000000000000", + "correct_value": 50523589006025392200000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6105607364251042 * 4245060455227103", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 6105607364251042 * 4245060455227103?", + "options": [ + "25918672377125480835987614391326", + "25918672377125480835987614391336", + "25918672377125480835987614391316", + "25918672377125480835987614391346" + ], + "answer": "25918672377125480835987614391326", + "correct_value": 25918672377125480835987614391326, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001236", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800000000000000 * 9800000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 9800000000000000?", + "options": [ + "96040000000000000000000000000000", + "96040000000000000000000000000020", + "96040000000000000000000000000010", + "96039999999999999999999999999990" + ], + "answer": "96040000000000000000000000000000", + "correct_value": 96040000000000000000000000000000, + "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 9800000000000000≈10000000000000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9800000000000000 * 4867979432292460", + "ns_shortcut": "Round 9800000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 4867979432292460?", + "options": [ + "47706198436466108000000000000010", + "47706198436466108000000000000000", + "47706198436466108000000000000020", + "47706198436466107999999999999990" + ], + "answer": "47706198436466108000000000000000", + "correct_value": 47706198436466108000000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4554192021933899 * 5058578672725689", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4554192021933899 * 5058578672725689?", + "options": [ + "23037738633652304729407317231411", + "23037738633652304729407317231431", + "23037738633652304729407317231401", + "23037738633652304729407317231421" + ], + "answer": "23037738633652304729407317231411", + "correct_value": 23037738633652304729407317231411, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001237", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100000000000000 * 10200000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 10200000000000000?", + "options": [ + "103020000000000000000000000000010", + "103020000000000000000000000000000", + "103020000000000000000000000000020", + "103019999999999999999999999999990" + ], + "answer": "103020000000000000000000000000000", + "correct_value": 103020000000000000000000000000000, + "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 10200000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9800000000000000 * 4950420671806159", + "ns_shortcut": "Round 9800000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 4950420671806159?", + "options": [ + "48514122583700358200000000000020", + "48514122583700358199999999999990", + "48514122583700358200000000000000", + "48514122583700358200000000000010" + ], + "answer": "48514122583700358200000000000000", + "correct_value": 48514122583700358200000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6153933374897656 * 6134924508923020", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 6153933374897656 * 6134924508923020?", + "options": [ + "37753916687938985369851282441110", + "37753916687938985369851282441130", + "37753916687938985369851282441140", + "37753916687938985369851282441120" + ], + "answer": "37753916687938985369851282441120", + "correct_value": 37753916687938985369851282441120, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_001238", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9900000000000000 * 9800000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 9900000000000000 * 9800000000000000?", + "options": [ + "97020000000000000000000000000010", + "97019999999999999999999999999990", + "97020000000000000000000000000020", + "97020000000000000000000000000000" + ], + "answer": "97020000000000000000000000000000", + "correct_value": 97020000000000000000000000000000, + "explanation": "Round to 10000000000000000: 9900000000000000≈10000000000000000, 9800000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9800000000000000 * 4892875286646204", + "ns_shortcut": "Round 9800000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 4892875286646204?", + "options": [ + "47950177809132799200000000000010", + "47950177809132799200000000000020", + "47950177809132799199999999999990", + "47950177809132799200000000000000" + ], + "answer": "47950177809132799200000000000000", + "correct_value": 47950177809132799200000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4977342321974566 * 4646971614284825", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4977342321974566 * 4646971614284825?", + "options": [ + "23129568484794328158825929760960", + "23129568484794328158825929760970", + "23129568484794328158825929760950", + "23129568484794328158825929760940" + ], + "answer": "23129568484794328158825929760950", + "correct_value": 23129568484794328158825929760950, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001239", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9900000000000000 * 10200000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 9900000000000000 * 10200000000000000?", + "options": [ + "100979999999999999999999999999990", + "100980000000000000000000000000000", + "100980000000000000000000000000010", + "100980000000000000000000000000020" + ], + "answer": "100980000000000000000000000000000", + "correct_value": 100980000000000000000000000000000, + "explanation": "Round to 10000000000000000: 9900000000000000≈10000000000000000, 10200000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900000000000000 * 4981853464827535", + "ns_shortcut": "Round 9900000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900000000000000 * 4981853464827535?", + "options": [ + "49320349301792596500000000000010", + "49320349301792596500000000000000", + "49320349301792596499999999999990", + "49320349301792596500000000000020" + ], + "answer": "49320349301792596500000000000000", + "correct_value": 49320349301792596500000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4737986033067062 * 5777948164299855", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4737986033067062 * 5777948164299855?", + "options": [ + "27375837702238182973719491876030", + "27375837702238182973719491876010", + "27375837702238182973719491876020", + "27375837702238182973719491876000" + ], + "answer": "27375837702238182973719491876010", + "correct_value": 27375837702238182973719491876010, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001240", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100000000000000 * 9900000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 9900000000000000?", + "options": [ + "99990000000000000000000000000020", + "99990000000000000000000000000000", + "99990000000000000000000000000010", + "99989999999999999999999999999990" + ], + "answer": "99990000000000000000000000000000", + "correct_value": 99990000000000000000000000000000, + "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 9900000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900000000000000 * 4823594971824780", + "ns_shortcut": "Round 9900000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900000000000000 * 4823594971824780?", + "options": [ + "47753590221065322000000000000020", + "47753590221065322000000000000000", + "47753590221065322000000000000010", + "47753590221065321999999999999990" + ], + "answer": "47753590221065322000000000000000", + "correct_value": 47753590221065322000000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4615464719270934 * 4097685519414294", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4615464719270934 * 4097685519414294?", + "options": [ + "18912722945524065829810378330616", + "18912722945524065829810378330586", + "18912722945524065829810378330606", + "18912722945524065829810378330596" + ], + "answer": "18912722945524065829810378330596", + "correct_value": 18912722945524065829810378330596, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_001241", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800000000000000 * 10200000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 10200000000000000?", + "options": [ + "99959999999999999999999999999990", + "99960000000000000000000000000020", + "99960000000000000000000000000000", + "99960000000000000000000000000010" + ], + "answer": "99960000000000000000000000000000", + "correct_value": 99960000000000000000000000000000, + "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 10200000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200000000000000 * 5000648950652713", + "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200000000000000 * 5000648950652713?", + "options": [ + "51006619296657672600000000000010", + "51006619296657672600000000000000", + "51006619296657672600000000000020", + "51006619296657672599999999999990" + ], + "answer": "51006619296657672600000000000000", + "correct_value": 51006619296657672600000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4317963557893896 * 6661580945381115", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4317963557893896 * 6661580945381115?", + "options": [ + "28764463760116022606778452174030", + "28764463760116022606778452174040", + "28764463760116022606778452174060", + "28764463760116022606778452174050" + ], + "answer": "28764463760116022606778452174040", + "correct_value": 28764463760116022606778452174040, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001242", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10200000000000000 * 10100000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10200000000000000 * 10100000000000000?", + "options": [ + "103019999999999999999999999999990", + "103020000000000000000000000000010", + "103020000000000000000000000000000", + "103020000000000000000000000000020" + ], + "answer": "103020000000000000000000000000000", + "correct_value": 103020000000000000000000000000000, + "explanation": "Round to 10000000000000000: 10200000000000000≈10000000000000000, 10100000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9800000000000000 * 4905882100252471", + "ns_shortcut": "Round 9800000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 4905882100252471?", + "options": [ + "48077644582474215800000000000000", + "48077644582474215799999999999990", + "48077644582474215800000000000010", + "48077644582474215800000000000020" + ], + "answer": "48077644582474215800000000000000", + "correct_value": 48077644582474215800000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5457400268454454 * 6559819562684101", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5457400268454454 * 6559819562684101?", + "options": [ + "35799561042404991836279108435864", + "35799561042404991836279108435874", + "35799561042404991836279108435844", + "35799561042404991836279108435854" + ], + "answer": "35799561042404991836279108435854", + "correct_value": 35799561042404991836279108435854, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_001243", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800000000000000 * 9800000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 9800000000000000?", + "options": [ + "96039999999999999999999999999990", + "96040000000000000000000000000010", + "96040000000000000000000000000020", + "96040000000000000000000000000000" + ], + "answer": "96040000000000000000000000000000", + "correct_value": 96040000000000000000000000000000, + "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 9800000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200000000000000 * 4839293691081513", + "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200000000000000 * 4839293691081513?", + "options": [ + "49360795649031432600000000000010", + "49360795649031432600000000000020", + "49360795649031432600000000000000", + "49360795649031432599999999999990" + ], + "answer": "49360795649031432600000000000000", + "correct_value": 49360795649031432600000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5947788105771199 * 6816630932267362", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5947788105771199 * 6816630932267362?", + "options": [ + "40543876380371855341662667307058", + "40543876380371855341662667307028", + "40543876380371855341662667307038", + "40543876380371855341662667307048" + ], + "answer": "40543876380371855341662667307038", + "correct_value": 40543876380371855341662667307038, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001244", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10200000000000000 * 10200000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 10200000000000000 * 10200000000000000?", + "options": [ + "104040000000000000000000000000010", + "104040000000000000000000000000020", + "104040000000000000000000000000000", + "104039999999999999999999999999990" + ], + "answer": "104040000000000000000000000000000", + "correct_value": 104040000000000000000000000000000, + "explanation": "Round to 10000000000000000: 10200000000000000≈10000000000000000, 10200000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900000000000000 * 5119816128141137", + "ns_shortcut": "Round 9900000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900000000000000 * 5119816128141137?", + "options": [ + "50686179668597256300000000000000", + "50686179668597256300000000000010", + "50686179668597256299999999999990", + "50686179668597256300000000000020" + ], + "answer": "50686179668597256300000000000000", + "correct_value": 50686179668597256300000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5345398198724517 * 6153362299073483", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5345398198724517 * 6153362299073483?", + "options": [ + "32892171749566748690621456682731", + "32892171749566748690621456682701", + "32892171749566748690621456682721", + "32892171749566748690621456682711" + ], + "answer": "32892171749566748690621456682711", + "correct_value": 32892171749566748690621456682711, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_001245", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9900000000000000 * 10100000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 9900000000000000 * 10100000000000000?", + "options": [ + "99990000000000000000000000000020", + "99990000000000000000000000000010", + "99989999999999999999999999999990", + "99990000000000000000000000000000" + ], + "answer": "99990000000000000000000000000000", + "correct_value": 99990000000000000000000000000000, + "explanation": "Round to 10000000000000000: 9900000000000000≈10000000000000000, 10100000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10100000000000000 * 4913911818799656", + "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 4913911818799656?", + "options": [ + "49630509369876525600000000000020", + "49630509369876525600000000000000", + "49630509369876525600000000000010", + "49630509369876525599999999999990" + ], + "answer": "49630509369876525600000000000000", + "correct_value": 49630509369876525600000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6838050545676369 * 4381969857009705", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 6838050545676369 * 4381969857009705?", + "options": [ + "29964131371862613915755022161135", + "29964131371862613915755022161155", + "29964131371862613915755022161165", + "29964131371862613915755022161145" + ], + "answer": "29964131371862613915755022161145", + "correct_value": 29964131371862613915755022161145, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_001246", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100000000000000 * 9800000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 9800000000000000?", + "options": [ + "98980000000000000000000000000000", + "98980000000000000000000000000020", + "98979999999999999999999999999990", + "98980000000000000000000000000010" + ], + "answer": "98980000000000000000000000000000", + "correct_value": 98980000000000000000000000000000, + "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 9800000000000000≈10000000000000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10100000000000000 * 5156959225765561", + "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 5156959225765561?", + "options": [ + "52085288180232166100000000000000", + "52085288180232166100000000000010", + "52085288180232166099999999999990", + "52085288180232166100000000000020" + ], + "answer": "52085288180232166100000000000000", + "correct_value": 52085288180232166100000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4752119667718159 * 6042125643626516", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4752119667718159 * 6042125643626516?", + "options": [ + "28712904105901806796391347104064", + "28712904105901806796391347104054", + "28712904105901806796391347104044", + "28712904105901806796391347104034" + ], + "answer": "28712904105901806796391347104044", + "correct_value": 28712904105901806796391347104044, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001247", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800000000000000 * 10200000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 10200000000000000?", + "options": [ + "99959999999999999999999999999990", + "99960000000000000000000000000020", + "99960000000000000000000000000000", + "99960000000000000000000000000010" + ], + "answer": "99960000000000000000000000000000", + "correct_value": 99960000000000000000000000000000, + "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 10200000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9800000000000000 * 4801815449158777", + "ns_shortcut": "Round 9800000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 4801815449158777?", + "options": [ + "47057791401756014600000000000010", + "47057791401756014600000000000020", + "47057791401756014600000000000000", + "47057791401756014599999999999990" + ], + "answer": "47057791401756014600000000000000", + "correct_value": 47057791401756014600000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5817385105311973 * 5181621934063465", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5817385105311973 * 5181621934063465?", + "options": [ + "30143490260578619555324406366455", + "30143490260578619555324406366445", + "30143490260578619555324406366465", + "30143490260578619555324406366435" + ], + "answer": "30143490260578619555324406366445", + "correct_value": 30143490260578619555324406366445, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001248", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100000000000000 * 10200000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 10200000000000000?", + "options": [ + "103020000000000000000000000000000", + "103019999999999999999999999999990", + "103020000000000000000000000000010", + "103020000000000000000000000000020" + ], + "answer": "103020000000000000000000000000000", + "correct_value": 103020000000000000000000000000000, + "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 10200000000000000≈10000000000000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900000000000000 * 5035269051252501", + "ns_shortcut": "Round 9900000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900000000000000 * 5035269051252501?", + "options": [ + "49849163607399759900000000000020", + "49849163607399759900000000000000", + "49849163607399759900000000000010", + "49849163607399759899999999999990" + ], + "answer": "49849163607399759900000000000000", + "correct_value": 49849163607399759900000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6586488355273688 * 6439527846202457", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 6586488355273688 * 6439527846202457?", + "options": [ + "42413875172473135500059693051426", + "42413875172473135500059693051416", + "42413875172473135500059693051406", + "42413875172473135500059693051436" + ], + "answer": "42413875172473135500059693051416", + "correct_value": 42413875172473135500059693051416, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001249", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100000000000000 * 10200000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 10200000000000000?", + "options": [ + "103019999999999999999999999999990", + "103020000000000000000000000000010", + "103020000000000000000000000000020", + "103020000000000000000000000000000" + ], + "answer": "103020000000000000000000000000000", + "correct_value": 103020000000000000000000000000000, + "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 10200000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9800000000000000 * 4971993418120652", + "ns_shortcut": "Round 9800000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 4971993418120652?", + "options": [ + "48725535497582389600000000000000", + "48725535497582389600000000000020", + "48725535497582389599999999999990", + "48725535497582389600000000000010" + ], + "answer": "48725535497582389600000000000000", + "correct_value": 48725535497582389600000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5965904767560386 * 4974928785303299", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5965904767560386 * 4974928785303299?", + "options": [ + "29679951358514351487207307513404", + "29679951358514351487207307513434", + "29679951358514351487207307513414", + "29679951358514351487207307513424" + ], + "answer": "29679951358514351487207307513414", + "correct_value": 29679951358514351487207307513414, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001250", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100000000000000 * 10200000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 10200000000000000?", + "options": [ + "103020000000000000000000000000010", + "103020000000000000000000000000020", + "103019999999999999999999999999990", + "103020000000000000000000000000000" + ], + "answer": "103020000000000000000000000000000", + "correct_value": 103020000000000000000000000000000, + "explanation": "Round to 10000000000000000: 10100000000000000≈10000000000000000, 10200000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200000000000000 * 5041863505507249", + "ns_shortcut": "Round 10200000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200000000000000 * 5041863505507249?", + "options": [ + "51427007756173939800000000000020", + "51427007756173939799999999999990", + "51427007756173939800000000000010", + "51427007756173939800000000000000" + ], + "answer": "51427007756173939800000000000000", + "correct_value": 51427007756173939800000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6773845913961462 * 6532756337663620", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 6773845913961462 * 6532756337663620?", + "options": [ + "44251884824788557279707799412430", + "44251884824788557279707799412450", + "44251884824788557279707799412440", + "44251884824788557279707799412460" + ], + "answer": "44251884824788557279707799412440", + "correct_value": 44251884824788557279707799412440, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_001251", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800000000000000 * 9800000000000000", + "ns_shortcut": "Round both to 10000000000000000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 9800000000000000 * 9800000000000000?", + "options": [ + "96040000000000000000000000000020", + "96040000000000000000000000000010", + "96039999999999999999999999999990", + "96040000000000000000000000000000" + ], + "answer": "96040000000000000000000000000000", + "correct_value": 96040000000000000000000000000000, + "explanation": "Round to 10000000000000000: 9800000000000000≈10000000000000000, 9800000000000000≈10000000000000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10100000000000000 * 5039428072712825", + "ns_shortcut": "Round 10100000000000000≈10000000000000000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 10100000000000000 * 5039428072712825?", + "options": [ + "50898223534399532500000000000010", + "50898223534399532499999999999990", + "50898223534399532500000000000020", + "50898223534399532500000000000000" + ], + "answer": "50898223534399532500000000000000", + "correct_value": 50898223534399532500000000000000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6664231850608989 * 6391661009179223", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 6664231850608989 * 6391661009179223?", + "options": [ + "42595510875667771521171595835547", + "42595510875667771521171595835537", + "42595510875667771521171595835557", + "42595510875667771521171595835567" + ], + "answer": "42595510875667771521171595835547", + "correct_value": 42595510875667771521171595835547, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001252", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999999999999999 * 9011938623869753", + "ns_shortcut": "Use (10000000000000000-1)*m = 10000000000000000*m - 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 9999999999999999 * 9011938623869753?", + "options": [ + "90119386238697520988061376130247", + "90119386238697520988061376130267", + "90119386238697520988061376130237", + "90119386238697520988061376130257" + ], + "answer": "90119386238697520988061376130247", + "correct_value": 90119386238697520988061376130247, + "explanation": "Distributive: (10000000000000000-1)*9011938623869753 = 10000000000000000*9011938623869753 - 1*9011938623869753." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999956 * 8644745744233812", + "ns_shortcut": "Use (10000000000000000-44)*m, correction is 44*8644745744233812.", + "shortcut_strength": 0.999999999999912, + "pure_math": { + "question": "What is 9999999999999956 * 8644745744233812?", + "options": [ + "86447457442337739631187253712292", + "86447457442337739631187253712262", + "86447457442337739631187253712282", + "86447457442337739631187253712272" + ], + "answer": "86447457442337739631187253712272", + "correct_value": 86447457442337739631187253712272, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4923371277053531 * 6368259785730583", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4923371277053531 * 6368259785730583?", + "options": [ + "31353307313881026117254434838593", + "31353307313881026117254434838583", + "31353307313881026117254434838573", + "31353307313881026117254434838563" + ], + "answer": "31353307313881026117254434838573", + "correct_value": 31353307313881026117254434838573, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001253", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000001 * 4494188112978724", + "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 10000000000000001 * 4494188112978724?", + "options": [ + "44941881129787244494188112978744", + "44941881129787244494188112978714", + "44941881129787244494188112978734", + "44941881129787244494188112978724" + ], + "answer": "44941881129787244494188112978724", + "correct_value": 44941881129787244494188112978724, + "explanation": "Distributive: (10000000000000000+1)*4494188112978724 = 10000000000000000*4494188112978724 + 1*4494188112978724." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10000000000000028 * 8515507261070221", + "ns_shortcut": "Use (10000000000000000+28)*m, correction is 28*8515507261070221.", + "shortcut_strength": 0.999999999999944, + "pure_math": { + "question": "What is 10000000000000028 * 8515507261070221?", + "options": [ + "85155072610702448434203309966198", + "85155072610702448434203309966178", + "85155072610702448434203309966208", + "85155072610702448434203309966188" + ], + "answer": "85155072610702448434203309966188", + "correct_value": 85155072610702448434203309966188, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5377865884745713 * 6589366975414545", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5377865884745713 * 6589366975414545?", + "options": [ + "35436731859151924928220086595575", + "35436731859151924928220086595585", + "35436731859151924928220086595595", + "35436731859151924928220086595605" + ], + "answer": "35436731859151924928220086595585", + "correct_value": 35436731859151924928220086595585, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001254", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000001 * 4858514932636808", + "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 10000000000000001 * 4858514932636808?", + "options": [ + "48585149326368084858514932636808", + "48585149326368084858514932636818", + "48585149326368084858514932636828", + "48585149326368084858514932636798" + ], + "answer": "48585149326368084858514932636808", + "correct_value": 48585149326368084858514932636808, + "explanation": "Distributive: (10000000000000000+1)*4858514932636808 = 10000000000000000*4858514932636808 + 1*4858514932636808." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10000000000000027 * 2733255024386121", + "ns_shortcut": "Use (10000000000000000+27)*m, correction is 27*2733255024386121.", + "shortcut_strength": 0.999999999999946, + "pure_math": { + "question": "What is 10000000000000027 * 2733255024386121?", + "options": [ + "27332550243861283797885658425267", + "27332550243861283797885658425277", + "27332550243861283797885658425257", + "27332550243861283797885658425287" + ], + "answer": "27332550243861283797885658425267", + "correct_value": 27332550243861283797885658425267, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4016698708572911 * 6796267905713954", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4016698708572911 * 6796267905713954?", + "options": [ + "27298560519996761491501919100104", + "27298560519996761491501919100114", + "27298560519996761491501919100094", + "27298560519996761491501919100084" + ], + "answer": "27298560519996761491501919100094", + "correct_value": 27298560519996761491501919100094, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001255", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000002 * 2074831903814983", + "ns_shortcut": "Use (10000000000000000+2)*m = 10000000000000000*m + 2*m.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "What is 10000000000000002 * 2074831903814983?", + "options": [ + "20748319038149834149663807629976", + "20748319038149834149663807629956", + "20748319038149834149663807629986", + "20748319038149834149663807629966" + ], + "answer": "20748319038149834149663807629966", + "correct_value": 20748319038149834149663807629966, + "explanation": "Distributive: (10000000000000000+2)*2074831903814983 = 10000000000000000*2074831903814983 + 2*2074831903814983." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999957 * 8254254351744488", + "ns_shortcut": "Use (10000000000000000-43)*m, correction is 43*8254254351744488.", + "shortcut_strength": 0.999999999999914, + "pure_math": { + "question": "What is 9999999999999957 * 8254254351744488?", + "options": [ + "82542543517444525067062874987006", + "82542543517444525067062874987026", + "82542543517444525067062874987016", + "82542543517444525067062874987036" + ], + "answer": "82542543517444525067062874987016", + "correct_value": 82542543517444525067062874987016, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4596314980696829 * 4445658723356322", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4596314980696829 * 4445658723356322?", + "options": [ + "20433647789228202608841222502938", + "20433647789228202608841222502958", + "20433647789228202608841222502948", + "20433647789228202608841222502928" + ], + "answer": "20433647789228202608841222502938", + "correct_value": 20433647789228202608841222502938, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001256", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000001 * 5049805118679571", + "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 10000000000000001 * 5049805118679571?", + "options": [ + "50498051186795715049805118679571", + "50498051186795715049805118679591", + "50498051186795715049805118679581", + "50498051186795715049805118679561" + ], + "answer": "50498051186795715049805118679571", + "correct_value": 50498051186795715049805118679571, + "explanation": "Distributive: (10000000000000000+1)*5049805118679571 = 10000000000000000*5049805118679571 + 1*5049805118679571." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999953 * 7683288106032672", + "ns_shortcut": "Use (10000000000000000-47)*m, correction is 47*7683288106032672.", + "shortcut_strength": 0.999999999999906, + "pure_math": { + "question": "What is 9999999999999953 * 7683288106032672?", + "options": [ + "76832881060326358885459016464406", + "76832881060326358885459016464436", + "76832881060326358885459016464416", + "76832881060326358885459016464426" + ], + "answer": "76832881060326358885459016464416", + "correct_value": 76832881060326358885459016464416, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5585558016500916 * 4775288695808696", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5585558016500916 * 4775288695808696?", + "options": [ + "26672652055980466057656844765536", + "26672652055980466057656844765526", + "26672652055980466057656844765546", + "26672652055980466057656844765556" + ], + "answer": "26672652055980466057656844765536", + "correct_value": 26672652055980466057656844765536, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001257", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000001 * 1061684344876105", + "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 10000000000000001 * 1061684344876105?", + "options": [ + "10616843448761051061684344876095", + "10616843448761051061684344876125", + "10616843448761051061684344876115", + "10616843448761051061684344876105" + ], + "answer": "10616843448761051061684344876105", + "correct_value": 10616843448761051061684344876105, + "explanation": "Distributive: (10000000000000000+1)*1061684344876105 = 10000000000000000*1061684344876105 + 1*1061684344876105." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10000000000000018 * 5039884088307084", + "ns_shortcut": "Use (10000000000000000+18)*m, correction is 18*5039884088307084.", + "shortcut_strength": 0.999999999999964, + "pure_math": { + "question": "What is 10000000000000018 * 5039884088307084?", + "options": [ + "50398840883070930717913589527522", + "50398840883070930717913589527512", + "50398840883070930717913589527502", + "50398840883070930717913589527532" + ], + "answer": "50398840883070930717913589527512", + "correct_value": 50398840883070930717913589527512, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5609646615275373 * 5867744144269920", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5609646615275373 * 5867744144269920?", + "options": [ + "32915971078205646682560840680160", + "32915971078205646682560840680180", + "32915971078205646682560840680170", + "32915971078205646682560840680150" + ], + "answer": "32915971078205646682560840680160", + "correct_value": 32915971078205646682560840680160, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001258", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000001 * 2386261281789032", + "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 10000000000000001 * 2386261281789032?", + "options": [ + "23862612817890322386261281789032", + "23862612817890322386261281789052", + "23862612817890322386261281789022", + "23862612817890322386261281789042" + ], + "answer": "23862612817890322386261281789032", + "correct_value": 23862612817890322386261281789032, + "explanation": "Distributive: (10000000000000000+1)*2386261281789032 = 10000000000000000*2386261281789032 + 1*2386261281789032." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999956 * 8439388374464827", + "ns_shortcut": "Use (10000000000000000-44)*m, correction is 44*8439388374464827.", + "shortcut_strength": 0.999999999999912, + "pure_math": { + "question": "What is 9999999999999956 * 8439388374464827?", + "options": [ + "84393883744647898666911523547622", + "84393883744647898666911523547612", + "84393883744647898666911523547602", + "84393883744647898666911523547632" + ], + "answer": "84393883744647898666911523547612", + "correct_value": 84393883744647898666911523547612, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4907883938296077 * 5830873817096810", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4907883938296077 * 5830873817096810?", + "options": [ + "28617251953160571217182352214380", + "28617251953160571217182352214390", + "28617251953160571217182352214370", + "28617251953160571217182352214360" + ], + "answer": "28617251953160571217182352214370", + "correct_value": 28617251953160571217182352214370, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001259", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000001 * 5232951492619627", + "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 10000000000000001 * 5232951492619627?", + "options": [ + "52329514926196275232951492619637", + "52329514926196275232951492619617", + "52329514926196275232951492619647", + "52329514926196275232951492619627" + ], + "answer": "52329514926196275232951492619627", + "correct_value": 52329514926196275232951492619627, + "explanation": "Distributive: (10000000000000000+1)*5232951492619627 = 10000000000000000*5232951492619627 + 1*5232951492619627." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10000000000000028 * 8115604350974979", + "ns_shortcut": "Use (10000000000000000+28)*m, correction is 28*8115604350974979.", + "shortcut_strength": 0.999999999999944, + "pure_math": { + "question": "What is 10000000000000028 * 8115604350974979?", + "options": [ + "81156043509750017236921827299432", + "81156043509750017236921827299412", + "81156043509750017236921827299422", + "81156043509750017236921827299402" + ], + "answer": "81156043509750017236921827299412", + "correct_value": 81156043509750017236921827299412, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5343421197924214 * 4693024188092021", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5343421197924214 * 4693024188092021?", + "options": [ + "25076804929021978654942416096494", + "25076804929021978654942416096484", + "25076804929021978654942416096514", + "25076804929021978654942416096504" + ], + "answer": "25076804929021978654942416096494", + "correct_value": 25076804929021978654942416096494, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001260", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999999999999998 * 1407322558019713", + "ns_shortcut": "Use (10000000000000000-2)*m = 10000000000000000*m - 2*m.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "What is 9999999999999998 * 1407322558019713?", + "options": [ + "14073225580197127185354883960584", + "14073225580197127185354883960574", + "14073225580197127185354883960564", + "14073225580197127185354883960594" + ], + "answer": "14073225580197127185354883960574", + "correct_value": 14073225580197127185354883960574, + "explanation": "Distributive: (10000000000000000-2)*1407322558019713 = 10000000000000000*1407322558019713 - 2*1407322558019713." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999966 * 2813108782065125", + "ns_shortcut": "Use (10000000000000000-34)*m, correction is 34*2813108782065125.", + "shortcut_strength": 0.999999999999932, + "pure_math": { + "question": "What is 9999999999999966 * 2813108782065125?", + "options": [ + "28131087820651154354301409785750", + "28131087820651154354301409785770", + "28131087820651154354301409785740", + "28131087820651154354301409785760" + ], + "answer": "28131087820651154354301409785750", + "correct_value": 28131087820651154354301409785750, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4311482825325165 * 5211984849428471", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4311482825325165 * 5211984849428471?", + "options": [ + "22471383164165818836074983772705", + "22471383164165818836074983772725", + "22471383164165818836074983772735", + "22471383164165818836074983772715" + ], + "answer": "22471383164165818836074983772715", + "correct_value": 22471383164165818836074983772715, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_001261", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999999999999998 * 6470993008833395", + "ns_shortcut": "Use (10000000000000000-2)*m = 10000000000000000*m - 2*m.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "What is 9999999999999998 * 6470993008833395?", + "options": [ + "64709930088333937058013982333210", + "64709930088333937058013982333220", + "64709930088333937058013982333230", + "64709930088333937058013982333200" + ], + "answer": "64709930088333937058013982333210", + "correct_value": 64709930088333937058013982333210, + "explanation": "Distributive: (10000000000000000-2)*6470993008833395 = 10000000000000000*6470993008833395 - 2*6470993008833395." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10000000000000018 * 5655073340204744", + "ns_shortcut": "Use (10000000000000000+18)*m, correction is 18*5655073340204744.", + "shortcut_strength": 0.999999999999964, + "pure_math": { + "question": "What is 10000000000000018 * 5655073340204744?", + "options": [ + "56550733402047541791320123685382", + "56550733402047541791320123685402", + "56550733402047541791320123685412", + "56550733402047541791320123685392" + ], + "answer": "56550733402047541791320123685392", + "correct_value": 56550733402047541791320123685392, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4688405905327996 * 5210233974431528", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4688405905327996 * 5210233974431528?", + "options": [ + "24427691733865330796050483457908", + "24427691733865330796050483457898", + "24427691733865330796050483457878", + "24427691733865330796050483457888" + ], + "answer": "24427691733865330796050483457888", + "correct_value": 24427691733865330796050483457888, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_001262", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000002 * 3434660663634979", + "ns_shortcut": "Use (10000000000000000+2)*m = 10000000000000000*m + 2*m.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "What is 10000000000000002 * 3434660663634979?", + "options": [ + "34346606636349796869321327269948", + "34346606636349796869321327269978", + "34346606636349796869321327269968", + "34346606636349796869321327269958" + ], + "answer": "34346606636349796869321327269958", + "correct_value": 34346606636349796869321327269958, + "explanation": "Distributive: (10000000000000000+2)*3434660663634979 = 10000000000000000*3434660663634979 + 2*3434660663634979." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999963 * 1752597361076109", + "ns_shortcut": "Use (10000000000000000-37)*m, correction is 37*1752597361076109.", + "shortcut_strength": 0.999999999999926, + "pure_math": { + "question": "What is 9999999999999963 * 1752597361076109?", + "options": [ + "17525973610761025153897640183967", + "17525973610761025153897640183987", + "17525973610761025153897640183957", + "17525973610761025153897640183977" + ], + "answer": "17525973610761025153897640183967", + "correct_value": 17525973610761025153897640183967, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4063775899240359 * 5449566808476649", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4063775899240359 * 5449566808476649?", + "options": [ + "22145818257587607539001089877001", + "22145818257587607539001089876991", + "22145818257587607539001089877011", + "22145818257587607539001089876981" + ], + "answer": "22145818257587607539001089876991", + "correct_value": 22145818257587607539001089876991, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001263", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000001 * 1169684179928312", + "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 10000000000000001 * 1169684179928312?", + "options": [ + "11696841799283121169684179928322", + "11696841799283121169684179928312", + "11696841799283121169684179928302", + "11696841799283121169684179928332" + ], + "answer": "11696841799283121169684179928312", + "correct_value": 11696841799283121169684179928312, + "explanation": "Distributive: (10000000000000000+1)*1169684179928312 = 10000000000000000*1169684179928312 + 1*1169684179928312." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999984 * 3784988998153142", + "ns_shortcut": "Use (10000000000000000-16)*m, correction is 16*3784988998153142.", + "shortcut_strength": 0.999999999999968, + "pure_math": { + "question": "What is 9999999999999984 * 3784988998153142?", + "options": [ + "37849889981531359440176029549738", + "37849889981531359440176029549748", + "37849889981531359440176029549718", + "37849889981531359440176029549728" + ], + "answer": "37849889981531359440176029549728", + "correct_value": 37849889981531359440176029549728, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6778683712221519 * 5217164744843272", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6778683712221519 * 5217164744843272?", + "options": [ + "35365509679845425016298600770158", + "35365509679845425016298600770168", + "35365509679845425016298600770188", + "35365509679845425016298600770178" + ], + "answer": "35365509679845425016298600770168", + "correct_value": 35365509679845425016298600770168, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001264", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999999999999999 * 1810859629489210", + "ns_shortcut": "Use (10000000000000000-1)*m = 10000000000000000*m - 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 9999999999999999 * 1810859629489210?", + "options": [ + "18108596294892098189140370510800", + "18108596294892098189140370510790", + "18108596294892098189140370510780", + "18108596294892098189140370510810" + ], + "answer": "18108596294892098189140370510790", + "correct_value": 18108596294892098189140370510790, + "explanation": "Distributive: (10000000000000000-1)*1810859629489210 = 10000000000000000*1810859629489210 - 1*1810859629489210." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999976 * 1637621934169284", + "ns_shortcut": "Use (10000000000000000-24)*m, correction is 24*1637621934169284.", + "shortcut_strength": 0.999999999999952, + "pure_math": { + "question": "What is 9999999999999976 * 1637621934169284?", + "options": [ + "16376219341692800697073579937194", + "16376219341692800697073579937184", + "16376219341692800697073579937174", + "16376219341692800697073579937204" + ], + "answer": "16376219341692800697073579937184", + "correct_value": 16376219341692800697073579937184, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5405931106176310 * 4184057188058838", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5405931106176310 * 4184057188058838?", + "options": [ + "22618724902947855225241481727800", + "22618724902947855225241481727770", + "22618724902947855225241481727780", + "22618724902947855225241481727790" + ], + "answer": "22618724902947855225241481727780", + "correct_value": 22618724902947855225241481727780, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001265", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000002 * 3573383218644172", + "ns_shortcut": "Use (10000000000000000+2)*m = 10000000000000000*m + 2*m.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "What is 10000000000000002 * 3573383218644172?", + "options": [ + "35733832186441727146766437288344", + "35733832186441727146766437288364", + "35733832186441727146766437288354", + "35733832186441727146766437288334" + ], + "answer": "35733832186441727146766437288344", + "correct_value": 35733832186441727146766437288344, + "explanation": "Distributive: (10000000000000000+2)*3573383218644172 = 10000000000000000*3573383218644172 + 2*3573383218644172." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999959 * 5851488878202253", + "ns_shortcut": "Use (10000000000000000-41)*m, correction is 41*5851488878202253.", + "shortcut_strength": 0.999999999999918, + "pure_math": { + "question": "What is 9999999999999959 * 5851488878202253?", + "options": [ + "58514888782022290088955993707637", + "58514888782022290088955993707647", + "58514888782022290088955993707617", + "58514888782022290088955993707627" + ], + "answer": "58514888782022290088955993707627", + "correct_value": 58514888782022290088955993707627, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6312530937697899 * 5915658060960045", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6312530937697899 * 5915658060960045?", + "options": [ + "37342774526652247828498119445465", + "37342774526652247828498119445455", + "37342774526652247828498119445445", + "37342774526652247828498119445475" + ], + "answer": "37342774526652247828498119445455", + "correct_value": 37342774526652247828498119445455, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001266", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999999999999999 * 5511347764236148", + "ns_shortcut": "Use (10000000000000000-1)*m = 10000000000000000*m - 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 9999999999999999 * 5511347764236148?", + "options": [ + "55113477642361474488652235763852", + "55113477642361474488652235763842", + "55113477642361474488652235763872", + "55113477642361474488652235763862" + ], + "answer": "55113477642361474488652235763852", + "correct_value": 55113477642361474488652235763852, + "explanation": "Distributive: (10000000000000000-1)*5511347764236148 = 10000000000000000*5511347764236148 - 1*5511347764236148." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999983 * 8737523485418743", + "ns_shortcut": "Use (10000000000000000-17)*m, correction is 17*8737523485418743.", + "shortcut_strength": 0.999999999999966, + "pure_math": { + "question": "What is 9999999999999983 * 8737523485418743?", + "options": [ + "87375234854187281462100747881379", + "87375234854187281462100747881369", + "87375234854187281462100747881359", + "87375234854187281462100747881389" + ], + "answer": "87375234854187281462100747881369", + "correct_value": 87375234854187281462100747881369, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6569533359666274 * 4716853321647638", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6569533359666274 * 4716853321647638?", + "options": [ + "30987525249216831414583500360802", + "30987525249216831414583500360812", + "30987525249216831414583500360832", + "30987525249216831414583500360822" + ], + "answer": "30987525249216831414583500360812", + "correct_value": 30987525249216831414583500360812, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001267", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999999999999999 * 9029922271973114", + "ns_shortcut": "Use (10000000000000000-1)*m = 10000000000000000*m - 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 9999999999999999 * 9029922271973114?", + "options": [ + "90299222719731130970077728026906", + "90299222719731130970077728026896", + "90299222719731130970077728026886", + "90299222719731130970077728026876" + ], + "answer": "90299222719731130970077728026886", + "correct_value": 90299222719731130970077728026886, + "explanation": "Distributive: (10000000000000000-1)*9029922271973114 = 10000000000000000*9029922271973114 - 1*9029922271973114." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10000000000000047 * 3849886884253673", + "ns_shortcut": "Use (10000000000000000+47)*m, correction is 47*3849886884253673.", + "shortcut_strength": 0.999999999999906, + "pure_math": { + "question": "What is 10000000000000047 * 3849886884253673?", + "options": [ + "38498868842536910944683559922651", + "38498868842536910944683559922621", + "38498868842536910944683559922641", + "38498868842536910944683559922631" + ], + "answer": "38498868842536910944683559922631", + "correct_value": 38498868842536910944683559922631, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5317461807224890 * 6325453899000957", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5317461807224890 * 6325453899000957?", + "options": [ + "33635359521299355631295624219750", + "33635359521299355631295624219720", + "33635359521299355631295624219730", + "33635359521299355631295624219740" + ], + "answer": "33635359521299355631295624219730", + "correct_value": 33635359521299355631295624219730, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001268", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999999999999999 * 1022409849974956", + "ns_shortcut": "Use (10000000000000000-1)*m = 10000000000000000*m - 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 9999999999999999 * 1022409849974956?", + "options": [ + "10224098499749558977590150025064", + "10224098499749558977590150025044", + "10224098499749558977590150025054", + "10224098499749558977590150025034" + ], + "answer": "10224098499749558977590150025044", + "correct_value": 10224098499749558977590150025044, + "explanation": "Distributive: (10000000000000000-1)*1022409849974956 = 10000000000000000*1022409849974956 - 1*1022409849974956." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10000000000000022 * 8193255182143768", + "ns_shortcut": "Use (10000000000000000+22)*m, correction is 22*8193255182143768.", + "shortcut_strength": 0.999999999999956, + "pure_math": { + "question": "What is 10000000000000022 * 8193255182143768?", + "options": [ + "81932551821437860251614007162886", + "81932551821437860251614007162896", + "81932551821437860251614007162906", + "81932551821437860251614007162916" + ], + "answer": "81932551821437860251614007162896", + "correct_value": 81932551821437860251614007162896, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4406801110325705 * 6435918484949007", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4406801110325705 * 6435918484949007?", + "options": [ + "28361812725439013171138086324955", + "28361812725439013171138086324935", + "28361812725439013171138086324925", + "28361812725439013171138086324945" + ], + "answer": "28361812725439013171138086324935", + "correct_value": 28361812725439013171138086324935, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001269", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000002 * 4258930231094123", + "ns_shortcut": "Use (10000000000000000+2)*m = 10000000000000000*m + 2*m.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "What is 10000000000000002 * 4258930231094123?", + "options": [ + "42589302310941238517860462188246", + "42589302310941238517860462188256", + "42589302310941238517860462188236", + "42589302310941238517860462188266" + ], + "answer": "42589302310941238517860462188246", + "correct_value": 42589302310941238517860462188246, + "explanation": "Distributive: (10000000000000000+2)*4258930231094123 = 10000000000000000*4258930231094123 + 2*4258930231094123." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10000000000000046 * 4832404155096685", + "ns_shortcut": "Use (10000000000000000+46)*m, correction is 46*4832404155096685.", + "shortcut_strength": 0.999999999999908, + "pure_math": { + "question": "What is 10000000000000046 * 4832404155096685?", + "options": [ + "48324041550967072290591134447530", + "48324041550967072290591134447510", + "48324041550967072290591134447500", + "48324041550967072290591134447520" + ], + "answer": "48324041550967072290591134447510", + "correct_value": 48324041550967072290591134447510, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4601309318092579 * 4991077208697493", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4601309318092579 * 4991077208697493?", + "options": [ + "22965490067699274121043779204467", + "22965490067699274121043779204457", + "22965490067699274121043779204447", + "22965490067699274121043779204437" + ], + "answer": "22965490067699274121043779204447", + "correct_value": 22965490067699274121043779204447, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001270", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999999999999998 * 2364631286113537", + "ns_shortcut": "Use (10000000000000000-2)*m = 10000000000000000*m - 2*m.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "What is 9999999999999998 * 2364631286113537?", + "options": [ + "23646312861135365270737427772946", + "23646312861135365270737427772916", + "23646312861135365270737427772926", + "23646312861135365270737427772936" + ], + "answer": "23646312861135365270737427772926", + "correct_value": 23646312861135365270737427772926, + "explanation": "Distributive: (10000000000000000-2)*2364631286113537 = 10000000000000000*2364631286113537 - 2*2364631286113537." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999965 * 1258108965332056", + "ns_shortcut": "Use (10000000000000000-35)*m, correction is 35*1258108965332056.", + "shortcut_strength": 0.99999999999993, + "pure_math": { + "question": "What is 9999999999999965 * 1258108965332056?", + "options": [ + "12581089653320515966186213378040", + "12581089653320515966186213378030", + "12581089653320515966186213378060", + "12581089653320515966186213378050" + ], + "answer": "12581089653320515966186213378040", + "correct_value": 12581089653320515966186213378040, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6826851018508091 * 6017709655352069", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6826851018508091 * 6017709655352069?", + "options": [ + "41082007289726245517553730090269", + "41082007289726245517553730090289", + "41082007289726245517553730090299", + "41082007289726245517553730090279" + ], + "answer": "41082007289726245517553730090279", + "correct_value": 41082007289726245517553730090279, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_001271", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000002 * 5022872460994310", + "ns_shortcut": "Use (10000000000000000+2)*m = 10000000000000000*m + 2*m.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "What is 10000000000000002 * 5022872460994310?", + "options": [ + "50228724609943110045744921988620", + "50228724609943110045744921988630", + "50228724609943110045744921988610", + "50228724609943110045744921988640" + ], + "answer": "50228724609943110045744921988620", + "correct_value": 50228724609943110045744921988620, + "explanation": "Distributive: (10000000000000000+2)*5022872460994310 = 10000000000000000*5022872460994310 + 2*5022872460994310." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999959 * 8086675349306765", + "ns_shortcut": "Use (10000000000000000-41)*m, correction is 41*8086675349306765.", + "shortcut_strength": 0.999999999999918, + "pure_math": { + "question": "What is 9999999999999959 * 8086675349306765?", + "options": [ + "80866753493067318446310678422625", + "80866753493067318446310678422635", + "80866753493067318446310678422655", + "80866753493067318446310678422645" + ], + "answer": "80866753493067318446310678422635", + "correct_value": 80866753493067318446310678422635, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5413565279875586 * 6672200672390778", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5413565279875586 * 6672200672390778?", + "options": [ + "36120393900417255198533013745918", + "36120393900417255198533013745898", + "36120393900417255198533013745928", + "36120393900417255198533013745908" + ], + "answer": "36120393900417255198533013745908", + "correct_value": 36120393900417255198533013745908, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_001272", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000002 * 2892692224834830", + "ns_shortcut": "Use (10000000000000000+2)*m = 10000000000000000*m + 2*m.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "What is 10000000000000002 * 2892692224834830?", + "options": [ + "28926922248348305785384449669670", + "28926922248348305785384449669660", + "28926922248348305785384449669680", + "28926922248348305785384449669650" + ], + "answer": "28926922248348305785384449669660", + "correct_value": 28926922248348305785384449669660, + "explanation": "Distributive: (10000000000000000+2)*2892692224834830 = 10000000000000000*2892692224834830 + 2*2892692224834830." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999965 * 8131686281351558", + "ns_shortcut": "Use (10000000000000000-35)*m, correction is 35*8131686281351558.", + "shortcut_strength": 0.99999999999993, + "pure_math": { + "question": "What is 9999999999999965 * 8131686281351558?", + "options": [ + "81316862813515295390980152695480", + "81316862813515295390980152695490", + "81316862813515295390980152695470", + "81316862813515295390980152695460" + ], + "answer": "81316862813515295390980152695470", + "correct_value": 81316862813515295390980152695470, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6616732695770404 * 5664502989890281", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6616732695770404 * 5664502989890281?", + "options": [ + "37480502138496232516860727043544", + "37480502138496232516860727043534", + "37480502138496232516860727043524", + "37480502138496232516860727043514" + ], + "answer": "37480502138496232516860727043524", + "correct_value": 37480502138496232516860727043524, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001273", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999999999999999 * 3374066818926509", + "ns_shortcut": "Use (10000000000000000-1)*m = 10000000000000000*m - 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 9999999999999999 * 3374066818926509?", + "options": [ + "33740668189265086625933181073511", + "33740668189265086625933181073491", + "33740668189265086625933181073501", + "33740668189265086625933181073481" + ], + "answer": "33740668189265086625933181073491", + "correct_value": 33740668189265086625933181073491, + "explanation": "Distributive: (10000000000000000-1)*3374066818926509 = 10000000000000000*3374066818926509 - 1*3374066818926509." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999971 * 9965938273520212", + "ns_shortcut": "Use (10000000000000000-29)*m, correction is 29*9965938273520212.", + "shortcut_strength": 0.999999999999942, + "pure_math": { + "question": "What is 9999999999999971 * 9965938273520212?", + "options": [ + "99659382735201830987790067913852", + "99659382735201830987790067913872", + "99659382735201830987790067913862", + "99659382735201830987790067913842" + ], + "answer": "99659382735201830987790067913852", + "correct_value": 99659382735201830987790067913852, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4357016573459543 * 5070933563660812", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4357016573459543 * 5070933563660812?", + "options": [ + "22094141579782420457282656528926", + "22094141579782420457282656528916", + "22094141579782420457282656528936", + "22094141579782420457282656528906" + ], + "answer": "22094141579782420457282656528916", + "correct_value": 22094141579782420457282656528916, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001274", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999999999999999 * 9427623974760600", + "ns_shortcut": "Use (10000000000000000-1)*m = 10000000000000000*m - 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 9999999999999999 * 9427623974760600?", + "options": [ + "94276239747605990572376025239390", + "94276239747605990572376025239410", + "94276239747605990572376025239420", + "94276239747605990572376025239400" + ], + "answer": "94276239747605990572376025239400", + "correct_value": 94276239747605990572376025239400, + "explanation": "Distributive: (10000000000000000-1)*9427623974760600 = 10000000000000000*9427623974760600 - 1*9427623974760600." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999975 * 9923137506073334", + "ns_shortcut": "Use (10000000000000000-25)*m, correction is 25*9923137506073334.", + "shortcut_strength": 0.99999999999995, + "pure_math": { + "question": "What is 9999999999999975 * 9923137506073334?", + "options": [ + "99231375060733091921562348166660", + "99231375060733091921562348166640", + "99231375060733091921562348166670", + "99231375060733091921562348166650" + ], + "answer": "99231375060733091921562348166650", + "correct_value": 99231375060733091921562348166650, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6952845194881645 * 6300183042433254", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6952845194881645 * 6300183042433254?", + "options": [ + "43804197393456873018127342222850", + "43804197393456873018127342222820", + "43804197393456873018127342222830", + "43804197393456873018127342222840" + ], + "answer": "43804197393456873018127342222830", + "correct_value": 43804197393456873018127342222830, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001275", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999999999999998 * 4619003504800785", + "ns_shortcut": "Use (10000000000000000-2)*m = 10000000000000000*m - 2*m.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "What is 9999999999999998 * 4619003504800785?", + "options": [ + "46190035048007840761992990398440", + "46190035048007840761992990398430", + "46190035048007840761992990398450", + "46190035048007840761992990398420" + ], + "answer": "46190035048007840761992990398430", + "correct_value": 46190035048007840761992990398430, + "explanation": "Distributive: (10000000000000000-2)*4619003504800785 = 10000000000000000*4619003504800785 - 2*4619003504800785." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999978 * 5937253207598311", + "ns_shortcut": "Use (10000000000000000-22)*m, correction is 22*5937253207598311.", + "shortcut_strength": 0.999999999999956, + "pure_math": { + "question": "What is 9999999999999978 * 5937253207598311?", + "options": [ + "59372532075982979380429432837168", + "59372532075982979380429432837178", + "59372532075982979380429432837158", + "59372532075982979380429432837148" + ], + "answer": "59372532075982979380429432837158", + "correct_value": 59372532075982979380429432837158, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4660835605318799 * 6498762965001284", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4660835605318799 * 6498762965001284?", + "options": [ + "30289665817805152472396264337936", + "30289665817805152472396264337916", + "30289665817805152472396264337906", + "30289665817805152472396264337926" + ], + "answer": "30289665817805152472396264337916", + "correct_value": 30289665817805152472396264337916, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001276", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999999999999999 * 9154116851230910", + "ns_shortcut": "Use (10000000000000000-1)*m = 10000000000000000*m - 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 9999999999999999 * 9154116851230910?", + "options": [ + "91541168512309090845883148769090", + "91541168512309090845883148769100", + "91541168512309090845883148769110", + "91541168512309090845883148769080" + ], + "answer": "91541168512309090845883148769090", + "correct_value": 91541168512309090845883148769090, + "explanation": "Distributive: (10000000000000000-1)*9154116851230910 = 10000000000000000*9154116851230910 - 1*9154116851230910." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10000000000000044 * 6587606027809705", + "ns_shortcut": "Use (10000000000000000+44)*m, correction is 44*6587606027809705.", + "shortcut_strength": 0.999999999999912, + "pure_math": { + "question": "What is 10000000000000044 * 6587606027809705?", + "options": [ + "65876060278097339854665223627020", + "65876060278097339854665223627010", + "65876060278097339854665223627040", + "65876060278097339854665223627030" + ], + "answer": "65876060278097339854665223627020", + "correct_value": 65876060278097339854665223627020, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5894767962307215 * 5476745008448897", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5894767962307215 * 5476745008448897?", + "options": [ + "32284141013530515567608541891855", + "32284141013530515567608541891865", + "32284141013530515567608541891875", + "32284141013530515567608541891845" + ], + "answer": "32284141013530515567608541891855", + "correct_value": 32284141013530515567608541891855, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001277", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999999999999999 * 9776748474662196", + "ns_shortcut": "Use (10000000000000000-1)*m = 10000000000000000*m - 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 9999999999999999 * 9776748474662196?", + "options": [ + "97767484746621950223251525337804", + "97767484746621950223251525337824", + "97767484746621950223251525337794", + "97767484746621950223251525337814" + ], + "answer": "97767484746621950223251525337804", + "correct_value": 97767484746621950223251525337804, + "explanation": "Distributive: (10000000000000000-1)*9776748474662196 = 10000000000000000*9776748474662196 - 1*9776748474662196." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999965 * 6994733727541508", + "ns_shortcut": "Use (10000000000000000-35)*m, correction is 35*6994733727541508.", + "shortcut_strength": 0.99999999999993, + "pure_math": { + "question": "What is 9999999999999965 * 6994733727541508?", + "options": [ + "69947337275414835184319536047230", + "69947337275414835184319536047240", + "69947337275414835184319536047210", + "69947337275414835184319536047220" + ], + "answer": "69947337275414835184319536047220", + "correct_value": 69947337275414835184319536047220, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4457391424358193 * 4769253407246610", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4457391424358193 * 4769253407246610?", + "options": [ + "21258429238052132052774524975750", + "21258429238052132052774524975720", + "21258429238052132052774524975730", + "21258429238052132052774524975740" + ], + "answer": "21258429238052132052774524975730", + "correct_value": 21258429238052132052774524975730, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001278", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000001 * 6693680691938313", + "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 10000000000000001 * 6693680691938313?", + "options": [ + "66936806919383136693680691938303", + "66936806919383136693680691938323", + "66936806919383136693680691938333", + "66936806919383136693680691938313" + ], + "answer": "66936806919383136693680691938313", + "correct_value": 66936806919383136693680691938313, + "explanation": "Distributive: (10000000000000000+1)*6693680691938313 = 10000000000000000*6693680691938313 + 1*6693680691938313." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999950 * 5072867841864335", + "ns_shortcut": "Use (10000000000000000-50)*m, correction is 50*5072867841864335.", + "shortcut_strength": 0.9999999999999, + "pure_math": { + "question": "What is 9999999999999950 * 5072867841864335?", + "options": [ + "50728678418643096356607906783270", + "50728678418643096356607906783260", + "50728678418643096356607906783240", + "50728678418643096356607906783250" + ], + "answer": "50728678418643096356607906783250", + "correct_value": 50728678418643096356607906783250, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4421601049445143 * 6154705742013148", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4421601049445143 * 6154705742013148?", + "options": [ + "27213653367911382746709210740164", + "27213653367911382746709210740154", + "27213653367911382746709210740184", + "27213653367911382746709210740174" + ], + "answer": "27213653367911382746709210740164", + "correct_value": 27213653367911382746709210740164, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001279", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000001 * 1402731624738029", + "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 10000000000000001 * 1402731624738029?", + "options": [ + "14027316247380291402731624738019", + "14027316247380291402731624738029", + "14027316247380291402731624738039", + "14027316247380291402731624738049" + ], + "answer": "14027316247380291402731624738029", + "correct_value": 14027316247380291402731624738029, + "explanation": "Distributive: (10000000000000000+1)*1402731624738029 = 10000000000000000*1402731624738029 + 1*1402731624738029." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10000000000000026 * 6975611998330910", + "ns_shortcut": "Use (10000000000000000+26)*m, correction is 26*6975611998330910.", + "shortcut_strength": 0.999999999999948, + "pure_math": { + "question": "What is 10000000000000026 * 6975611998330910?", + "options": [ + "69756119983309281365911956603660", + "69756119983309281365911956603680", + "69756119983309281365911956603670", + "69756119983309281365911956603650" + ], + "answer": "69756119983309281365911956603660", + "correct_value": 69756119983309281365911956603660, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6177028245978908 * 4231515706557818", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6177028245978908 * 4231515706557818?", + "options": [ + "26138192042711038088844510502754", + "26138192042711038088844510502764", + "26138192042711038088844510502734", + "26138192042711038088844510502744" + ], + "answer": "26138192042711038088844510502744", + "correct_value": 26138192042711038088844510502744, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_001280", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000001 * 1443330178587762", + "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 10000000000000001 * 1443330178587762?", + "options": [ + "14433301785877621443330178587762", + "14433301785877621443330178587782", + "14433301785877621443330178587752", + "14433301785877621443330178587772" + ], + "answer": "14433301785877621443330178587762", + "correct_value": 14433301785877621443330178587762, + "explanation": "Distributive: (10000000000000000+1)*1443330178587762 = 10000000000000000*1443330178587762 + 1*1443330178587762." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10000000000000047 * 2536309613708211", + "ns_shortcut": "Use (10000000000000000+47)*m, correction is 47*2536309613708211.", + "shortcut_strength": 0.999999999999906, + "pure_math": { + "question": "What is 10000000000000047 * 2536309613708211?", + "options": [ + "25363096137082229206551844285917", + "25363096137082229206551844285937", + "25363096137082229206551844285927", + "25363096137082229206551844285907" + ], + "answer": "25363096137082229206551844285917", + "correct_value": 25363096137082229206551844285917, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4168769638901135 * 6877213979510415", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4168769638901135 * 6877213979510415?", + "options": [ + "28669520838009470376205887821025", + "28669520838009470376205887821045", + "28669520838009470376205887821035", + "28669520838009470376205887821015" + ], + "answer": "28669520838009470376205887821025", + "correct_value": 28669520838009470376205887821025, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001281", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999999999999998 * 8948678252428046", + "ns_shortcut": "Use (10000000000000000-2)*m = 10000000000000000*m - 2*m.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "What is 9999999999999998 * 8948678252428046?", + "options": [ + "89486782524280442102643495143928", + "89486782524280442102643495143898", + "89486782524280442102643495143908", + "89486782524280442102643495143918" + ], + "answer": "89486782524280442102643495143908", + "correct_value": 89486782524280442102643495143908, + "explanation": "Distributive: (10000000000000000-2)*8948678252428046 = 10000000000000000*8948678252428046 - 2*8948678252428046." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999954 * 5150263370449804", + "ns_shortcut": "Use (10000000000000000-46)*m, correction is 46*5150263370449804.", + "shortcut_strength": 0.999999999999908, + "pure_math": { + "question": "What is 9999999999999954 * 5150263370449804?", + "options": [ + "51502633704497803087884959309006", + "51502633704497803087884959309036", + "51502633704497803087884959309026", + "51502633704497803087884959309016" + ], + "answer": "51502633704497803087884959309016", + "correct_value": 51502633704497803087884959309016, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5855542967303095 * 6130333741911067", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5855542967303095 * 6130333741911067?", + "options": [ + "35896432629668215016820323852375", + "35896432629668215016820323852355", + "35896432629668215016820323852365", + "35896432629668215016820323852385" + ], + "answer": "35896432629668215016820323852365", + "correct_value": 35896432629668215016820323852365, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001282", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000002 * 8712188127172187", + "ns_shortcut": "Use (10000000000000000+2)*m = 10000000000000000*m + 2*m.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "What is 10000000000000002 * 8712188127172187?", + "options": [ + "87121881271721887424376254344394", + "87121881271721887424376254344374", + "87121881271721887424376254344384", + "87121881271721887424376254344364" + ], + "answer": "87121881271721887424376254344374", + "correct_value": 87121881271721887424376254344374, + "explanation": "Distributive: (10000000000000000+2)*8712188127172187 = 10000000000000000*8712188127172187 + 2*8712188127172187." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10000000000000031 * 2834947370025633", + "ns_shortcut": "Use (10000000000000000+31)*m, correction is 31*2834947370025633.", + "shortcut_strength": 0.999999999999938, + "pure_math": { + "question": "What is 10000000000000031 * 2834947370025633?", + "options": [ + "28349473700256417883368470794633", + "28349473700256417883368470794643", + "28349473700256417883368470794613", + "28349473700256417883368470794623" + ], + "answer": "28349473700256417883368470794623", + "correct_value": 28349473700256417883368470794623, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4984619018219265 * 6245536713628459", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4984619018219265 * 6245536713628459?", + "options": [ + "31131621081739064124947006062655", + "31131621081739064124947006062625", + "31131621081739064124947006062635", + "31131621081739064124947006062645" + ], + "answer": "31131621081739064124947006062635", + "correct_value": 31131621081739064124947006062635, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001283", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000002 * 5080314992659676", + "ns_shortcut": "Use (10000000000000000+2)*m = 10000000000000000*m + 2*m.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "What is 10000000000000002 * 5080314992659676?", + "options": [ + "50803149926596770160629985319342", + "50803149926596770160629985319352", + "50803149926596770160629985319372", + "50803149926596770160629985319362" + ], + "answer": "50803149926596770160629985319352", + "correct_value": 50803149926596770160629985319352, + "explanation": "Distributive: (10000000000000000+2)*5080314992659676 = 10000000000000000*5080314992659676 + 2*5080314992659676." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10000000000000016 * 4912166597522484", + "ns_shortcut": "Use (10000000000000000+16)*m, correction is 16*4912166597522484.", + "shortcut_strength": 0.999999999999968, + "pure_math": { + "question": "What is 10000000000000016 * 4912166597522484?", + "options": [ + "49121665975224918594665560359764", + "49121665975224918594665560359734", + "49121665975224918594665560359754", + "49121665975224918594665560359744" + ], + "answer": "49121665975224918594665560359744", + "correct_value": 49121665975224918594665560359744, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6520037536829451 * 4615662249162375", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6520037536829451 * 4615662249162375?", + "options": [ + "30094291121865335227137981106115", + "30094291121865335227137981106145", + "30094291121865335227137981106135", + "30094291121865335227137981106125" + ], + "answer": "30094291121865335227137981106125", + "correct_value": 30094291121865335227137981106125, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_001284", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000001 * 2474276053813833", + "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 10000000000000001 * 2474276053813833?", + "options": [ + "24742760538138332474276053813823", + "24742760538138332474276053813853", + "24742760538138332474276053813833", + "24742760538138332474276053813843" + ], + "answer": "24742760538138332474276053813833", + "correct_value": 24742760538138332474276053813833, + "explanation": "Distributive: (10000000000000000+1)*2474276053813833 = 10000000000000000*2474276053813833 + 1*2474276053813833." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10000000000000040 * 7511057170397543", + "ns_shortcut": "Use (10000000000000000+40)*m, correction is 40*7511057170397543.", + "shortcut_strength": 0.99999999999992, + "pure_math": { + "question": "What is 10000000000000040 * 7511057170397543?", + "options": [ + "75110571703975730442286815901740", + "75110571703975730442286815901730", + "75110571703975730442286815901710", + "75110571703975730442286815901720" + ], + "answer": "75110571703975730442286815901720", + "correct_value": 75110571703975730442286815901720, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6420880883391922 * 6625948362483753", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6420880883391922 * 6625948362483753?", + "options": [ + "42544425175013938969915256443286", + "42544425175013938969915256443266", + "42544425175013938969915256443276", + "42544425175013938969915256443256" + ], + "answer": "42544425175013938969915256443266", + "correct_value": 42544425175013938969915256443266, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001285", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000002 * 2867228057095123", + "ns_shortcut": "Use (10000000000000000+2)*m = 10000000000000000*m + 2*m.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "What is 10000000000000002 * 2867228057095123?", + "options": [ + "28672280570951235734456114190246", + "28672280570951235734456114190236", + "28672280570951235734456114190256", + "28672280570951235734456114190266" + ], + "answer": "28672280570951235734456114190246", + "correct_value": 28672280570951235734456114190246, + "explanation": "Distributive: (10000000000000000+2)*2867228057095123 = 10000000000000000*2867228057095123 + 2*2867228057095123." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999954 * 4832819212624722", + "ns_shortcut": "Use (10000000000000000-46)*m, correction is 46*4832819212624722.", + "shortcut_strength": 0.999999999999908, + "pure_math": { + "question": "What is 9999999999999954 * 4832819212624722?", + "options": [ + "48328192126246997690316219262778", + "48328192126246997690316219262798", + "48328192126246997690316219262808", + "48328192126246997690316219262788" + ], + "answer": "48328192126246997690316219262788", + "correct_value": 48328192126246997690316219262788, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6577421179497447 * 6453424290600615", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6577421179497447 * 6453424290600615?", + "options": [ + "42446889609279772284511489129915", + "42446889609279772284511489129905", + "42446889609279772284511489129895", + "42446889609279772284511489129925" + ], + "answer": "42446889609279772284511489129905", + "correct_value": 42446889609279772284511489129905, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001286", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000002 * 7671846297191454", + "ns_shortcut": "Use (10000000000000000+2)*m = 10000000000000000*m + 2*m.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "What is 10000000000000002 * 7671846297191454?", + "options": [ + "76718462971914555343692594382928", + "76718462971914555343692594382898", + "76718462971914555343692594382918", + "76718462971914555343692594382908" + ], + "answer": "76718462971914555343692594382908", + "correct_value": 76718462971914555343692594382908, + "explanation": "Distributive: (10000000000000000+2)*7671846297191454 = 10000000000000000*7671846297191454 + 2*7671846297191454." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999964 * 8478435942327698", + "ns_shortcut": "Use (10000000000000000-36)*m, correction is 36*8478435942327698.", + "shortcut_strength": 0.999999999999928, + "pure_math": { + "question": "What is 9999999999999964 * 8478435942327698?", + "options": [ + "84784359423276674776306076202882", + "84784359423276674776306076202862", + "84784359423276674776306076202892", + "84784359423276674776306076202872" + ], + "answer": "84784359423276674776306076202872", + "correct_value": 84784359423276674776306076202872, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6313136572019354 * 4423533873620911", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6313136572019354 * 4423533873620911?", + "options": [ + "27926373475122612372647151111504", + "27926373475122612372647151111494", + "27926373475122612372647151111514", + "27926373475122612372647151111484" + ], + "answer": "27926373475122612372647151111494", + "correct_value": 27926373475122612372647151111494, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001287", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000001 * 9993094115021722", + "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 10000000000000001 * 9993094115021722?", + "options": [ + "99930941150217229993094115021742", + "99930941150217229993094115021732", + "99930941150217229993094115021712", + "99930941150217229993094115021722" + ], + "answer": "99930941150217229993094115021722", + "correct_value": 99930941150217229993094115021722, + "explanation": "Distributive: (10000000000000000+1)*9993094115021722 = 10000000000000000*9993094115021722 + 1*9993094115021722." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10000000000000023 * 5323077888782658", + "ns_shortcut": "Use (10000000000000000+23)*m, correction is 23*5323077888782658.", + "shortcut_strength": 0.999999999999954, + "pure_math": { + "question": "What is 10000000000000023 * 5323077888782658?", + "options": [ + "53230778887826702430791442001134", + "53230778887826702430791442001154", + "53230778887826702430791442001144", + "53230778887826702430791442001124" + ], + "answer": "53230778887826702430791442001134", + "correct_value": 53230778887826702430791442001134, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6660722383924499 * 4916011565770359", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6660722383924499 * 4916011565770359?", + "options": [ + "32744188275758354605788628125141", + "32744188275758354605788628125161", + "32744188275758354605788628125151", + "32744188275758354605788628125131" + ], + "answer": "32744188275758354605788628125141", + "correct_value": 32744188275758354605788628125141, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001288", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999999999999998 * 9973186274612113", + "ns_shortcut": "Use (10000000000000000-2)*m = 10000000000000000*m - 2*m.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "What is 9999999999999998 * 9973186274612113?", + "options": [ + "99731862746121110053627450775764", + "99731862746121110053627450775774", + "99731862746121110053627450775794", + "99731862746121110053627450775784" + ], + "answer": "99731862746121110053627450775774", + "correct_value": 99731862746121110053627450775774, + "explanation": "Distributive: (10000000000000000-2)*9973186274612113 = 10000000000000000*9973186274612113 - 2*9973186274612113." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999980 * 5168202248385586", + "ns_shortcut": "Use (10000000000000000-20)*m, correction is 20*5168202248385586.", + "shortcut_strength": 0.99999999999996, + "pure_math": { + "question": "What is 9999999999999980 * 5168202248385586?", + "options": [ + "51682022483855756635955032288280", + "51682022483855756635955032288300", + "51682022483855756635955032288290", + "51682022483855756635955032288270" + ], + "answer": "51682022483855756635955032288280", + "correct_value": 51682022483855756635955032288280, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4682576869666766 * 6857153346748540", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4682576869666766 * 6857153346748540?", + "options": [ + "32109147653242766471919397021650", + "32109147653242766471919397021640", + "32109147653242766471919397021630", + "32109147653242766471919397021660" + ], + "answer": "32109147653242766471919397021640", + "correct_value": 32109147653242766471919397021640, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001289", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000002 * 8378430433899730", + "ns_shortcut": "Use (10000000000000000+2)*m = 10000000000000000*m + 2*m.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "What is 10000000000000002 * 8378430433899730?", + "options": [ + "83784304338997316756860867799450", + "83784304338997316756860867799480", + "83784304338997316756860867799460", + "83784304338997316756860867799470" + ], + "answer": "83784304338997316756860867799460", + "correct_value": 83784304338997316756860867799460, + "explanation": "Distributive: (10000000000000000+2)*8378430433899730 = 10000000000000000*8378430433899730 + 2*8378430433899730." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999957 * 5846527593448002", + "ns_shortcut": "Use (10000000000000000-43)*m, correction is 43*5846527593448002.", + "shortcut_strength": 0.999999999999914, + "pure_math": { + "question": "What is 9999999999999957 * 5846527593448002?", + "options": [ + "58465275934479768599313481735924", + "58465275934479768599313481735934", + "58465275934479768599313481735914", + "58465275934479768599313481735904" + ], + "answer": "58465275934479768599313481735914", + "correct_value": 58465275934479768599313481735914, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5871192653641243 * 5961350132853739", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5871192653641243 * 5961350132853739?", + "options": [ + "35000235105794120383621097157587", + "35000235105794120383621097157597", + "35000235105794120383621097157567", + "35000235105794120383621097157577" + ], + "answer": "35000235105794120383621097157577", + "correct_value": 35000235105794120383621097157577, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_001290", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000001 * 4550244648356241", + "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 10000000000000001 * 4550244648356241?", + "options": [ + "45502446483562414550244648356261", + "45502446483562414550244648356241", + "45502446483562414550244648356231", + "45502446483562414550244648356251" + ], + "answer": "45502446483562414550244648356241", + "correct_value": 45502446483562414550244648356241, + "explanation": "Distributive: (10000000000000000+1)*4550244648356241 = 10000000000000000*4550244648356241 + 1*4550244648356241." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10000000000000026 * 4237294704401376", + "ns_shortcut": "Use (10000000000000000+26)*m, correction is 26*4237294704401376.", + "shortcut_strength": 0.999999999999948, + "pure_math": { + "question": "What is 10000000000000026 * 4237294704401376?", + "options": [ + "42372947044013870169662314435776", + "42372947044013870169662314435766", + "42372947044013870169662314435796", + "42372947044013870169662314435786" + ], + "answer": "42372947044013870169662314435776", + "correct_value": 42372947044013870169662314435776, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6763941457901719 * 6450938499450458", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6763941457901719 * 6450938499450458?", + "options": [ + "43633770358807258396623273537312", + "43633770358807258396623273537322", + "43633770358807258396623273537292", + "43633770358807258396623273537302" + ], + "answer": "43633770358807258396623273537302", + "correct_value": 43633770358807258396623273537302, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_001291", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000002 * 4813441212099304", + "ns_shortcut": "Use (10000000000000000+2)*m = 10000000000000000*m + 2*m.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "What is 10000000000000002 * 4813441212099304?", + "options": [ + "48134412120993049626882424198608", + "48134412120993049626882424198628", + "48134412120993049626882424198598", + "48134412120993049626882424198618" + ], + "answer": "48134412120993049626882424198608", + "correct_value": 48134412120993049626882424198608, + "explanation": "Distributive: (10000000000000000+2)*4813441212099304 = 10000000000000000*4813441212099304 + 2*4813441212099304." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10000000000000030 * 5024193653620045", + "ns_shortcut": "Use (10000000000000000+30)*m, correction is 30*5024193653620045.", + "shortcut_strength": 0.99999999999994, + "pure_math": { + "question": "What is 10000000000000030 * 5024193653620045?", + "options": [ + "50241936536200600725809608601360", + "50241936536200600725809608601350", + "50241936536200600725809608601370", + "50241936536200600725809608601340" + ], + "answer": "50241936536200600725809608601350", + "correct_value": 50241936536200600725809608601350, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6037158240001338 * 4325011403484697", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6037158240001338 * 4325011403484697?", + "options": [ + "26110778232647390072711142524576", + "26110778232647390072711142524596", + "26110778232647390072711142524586", + "26110778232647390072711142524606" + ], + "answer": "26110778232647390072711142524586", + "correct_value": 26110778232647390072711142524586, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001292", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000001 * 4635354010835348", + "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 10000000000000001 * 4635354010835348?", + "options": [ + "46353540108353484635354010835338", + "46353540108353484635354010835368", + "46353540108353484635354010835358", + "46353540108353484635354010835348" + ], + "answer": "46353540108353484635354010835348", + "correct_value": 46353540108353484635354010835348, + "explanation": "Distributive: (10000000000000000+1)*4635354010835348 = 10000000000000000*4635354010835348 + 1*4635354010835348." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10000000000000019 * 3443640307357999", + "ns_shortcut": "Use (10000000000000000+19)*m, correction is 19*3443640307357999.", + "shortcut_strength": 0.999999999999962, + "pure_math": { + "question": "What is 10000000000000019 * 3443640307357999?", + "options": [ + "34436403073580055429165839802001", + "34436403073580055429165839801971", + "34436403073580055429165839801981", + "34436403073580055429165839801991" + ], + "answer": "34436403073580055429165839801981", + "correct_value": 34436403073580055429165839801981, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6062293240837032 * 5955826575703993", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6062293240837032 * 5955826575703993?", + "options": [ + "36105967193487882435221984668766", + "36105967193487882435221984668786", + "36105967193487882435221984668776", + "36105967193487882435221984668796" + ], + "answer": "36105967193487882435221984668776", + "correct_value": 36105967193487882435221984668776, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001293", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999999999999998 * 5950225159180903", + "ns_shortcut": "Use (10000000000000000-2)*m = 10000000000000000*m - 2*m.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "What is 9999999999999998 * 5950225159180903?", + "options": [ + "59502251591809018099549681638214", + "59502251591809018099549681638204", + "59502251591809018099549681638194", + "59502251591809018099549681638184" + ], + "answer": "59502251591809018099549681638194", + "correct_value": 59502251591809018099549681638194, + "explanation": "Distributive: (10000000000000000-2)*5950225159180903 = 10000000000000000*5950225159180903 - 2*5950225159180903." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999955 * 2346224016906136", + "ns_shortcut": "Use (10000000000000000-45)*m, correction is 45*2346224016906136.", + "shortcut_strength": 0.99999999999991, + "pure_math": { + "question": "What is 9999999999999955 * 2346224016906136?", + "options": [ + "23462240169061254419919239223890", + "23462240169061254419919239223880", + "23462240169061254419919239223870", + "23462240169061254419919239223900" + ], + "answer": "23462240169061254419919239223880", + "correct_value": 23462240169061254419919239223880, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5717637492083665 * 6757737914354763", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5717637492083665 * 6757737914354763?", + "options": [ + "38638295660790064060178887246385", + "38638295660790064060178887246395", + "38638295660790064060178887246405", + "38638295660790064060178887246415" + ], + "answer": "38638295660790064060178887246395", + "correct_value": 38638295660790064060178887246395, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001294", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999999999999999 * 7172716200124277", + "ns_shortcut": "Use (10000000000000000-1)*m = 10000000000000000*m - 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 9999999999999999 * 7172716200124277?", + "options": [ + "71727162001242762827283799875723", + "71727162001242762827283799875743", + "71727162001242762827283799875733", + "71727162001242762827283799875713" + ], + "answer": "71727162001242762827283799875723", + "correct_value": 71727162001242762827283799875723, + "explanation": "Distributive: (10000000000000000-1)*7172716200124277 = 10000000000000000*7172716200124277 - 1*7172716200124277." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999951 * 8516256153203202", + "ns_shortcut": "Use (10000000000000000-49)*m, correction is 49*8516256153203202.", + "shortcut_strength": 0.999999999999902, + "pure_math": { + "question": "What is 9999999999999951 * 8516256153203202?", + "options": [ + "85162561532031602703448493043102", + "85162561532031602703448493043112", + "85162561532031602703448493043122", + "85162561532031602703448493043092" + ], + "answer": "85162561532031602703448493043102", + "correct_value": 85162561532031602703448493043102, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5456357042405382 * 5506557711119340", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5456357042405382 * 5506557711119340?", + "options": [ + "30045744946477671889441260287890", + "30045744946477671889441260287870", + "30045744946477671889441260287880", + "30045744946477671889441260287900" + ], + "answer": "30045744946477671889441260287880", + "correct_value": 30045744946477671889441260287880, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001295", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000001 * 3599012612734735", + "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 10000000000000001 * 3599012612734735?", + "options": [ + "35990126127347353599012612734735", + "35990126127347353599012612734745", + "35990126127347353599012612734725", + "35990126127347353599012612734755" + ], + "answer": "35990126127347353599012612734735", + "correct_value": 35990126127347353599012612734735, + "explanation": "Distributive: (10000000000000000+1)*3599012612734735 = 10000000000000000*3599012612734735 + 1*3599012612734735." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10000000000000023 * 2347019034476577", + "ns_shortcut": "Use (10000000000000000+23)*m, correction is 23*2347019034476577.", + "shortcut_strength": 0.999999999999954, + "pure_math": { + "question": "What is 10000000000000023 * 2347019034476577?", + "options": [ + "23470190344765823981437792961291", + "23470190344765823981437792961261", + "23470190344765823981437792961271", + "23470190344765823981437792961281" + ], + "answer": "23470190344765823981437792961271", + "correct_value": 23470190344765823981437792961271, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6111385739331078 * 5086269191456967", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6111385739331078 * 5086269191456967?", + "options": [ + "31084153003069120587362802720446", + "31084153003069120587362802720416", + "31084153003069120587362802720426", + "31084153003069120587362802720436" + ], + "answer": "31084153003069120587362802720426", + "correct_value": 31084153003069120587362802720426, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001296", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000001 * 4404454257196990", + "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 10000000000000001 * 4404454257196990?", + "options": [ + "44044542571969904404454257196980", + "44044542571969904404454257197010", + "44044542571969904404454257197000", + "44044542571969904404454257196990" + ], + "answer": "44044542571969904404454257196990", + "correct_value": 44044542571969904404454257196990, + "explanation": "Distributive: (10000000000000000+1)*4404454257196990 = 10000000000000000*4404454257196990 + 1*4404454257196990." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999959 * 9565690484335143", + "ns_shortcut": "Use (10000000000000000-41)*m, correction is 41*9565690484335143.", + "shortcut_strength": 0.999999999999918, + "pure_math": { + "question": "What is 9999999999999959 * 9565690484335143?", + "options": [ + "95656904843351037806690142259127", + "95656904843351037806690142259137", + "95656904843351037806690142259147", + "95656904843351037806690142259157" + ], + "answer": "95656904843351037806690142259137", + "correct_value": 95656904843351037806690142259137, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4515465193548196 * 6649740480831356", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4515465193548196 * 6649740480831356?", + "options": [ + "30026671687322432853621534033776", + "30026671687322432853621534033786", + "30026671687322432853621534033796", + "30026671687322432853621534033766" + ], + "answer": "30026671687322432853621534033776", + "correct_value": 30026671687322432853621534033776, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001297", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000001 * 1572206525590862", + "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 10000000000000001 * 1572206525590862?", + "options": [ + "15722065255908621572206525590852", + "15722065255908621572206525590862", + "15722065255908621572206525590882", + "15722065255908621572206525590872" + ], + "answer": "15722065255908621572206525590862", + "correct_value": 15722065255908621572206525590862, + "explanation": "Distributive: (10000000000000000+1)*1572206525590862 = 10000000000000000*1572206525590862 + 1*1572206525590862." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10000000000000017 * 5128191308201640", + "ns_shortcut": "Use (10000000000000000+17)*m, correction is 17*5128191308201640.", + "shortcut_strength": 0.999999999999966, + "pure_math": { + "question": "What is 10000000000000017 * 5128191308201640?", + "options": [ + "51281913082016487179252239427880", + "51281913082016487179252239427890", + "51281913082016487179252239427870", + "51281913082016487179252239427900" + ], + "answer": "51281913082016487179252239427880", + "correct_value": 51281913082016487179252239427880, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6231744209687154 * 5961076271859974", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6231744209687154 * 5961076271859974?", + "options": [ + "37147902540666880037904234574006", + "37147902540666880037904234574016", + "37147902540666880037904234573996", + "37147902540666880037904234573986" + ], + "answer": "37147902540666880037904234573996", + "correct_value": 37147902540666880037904234573996, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001298", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999999999999998 * 5758831058390411", + "ns_shortcut": "Use (10000000000000000-2)*m = 10000000000000000*m - 2*m.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "What is 9999999999999998 * 5758831058390411?", + "options": [ + "57588310583904098482337883219188", + "57588310583904098482337883219178", + "57588310583904098482337883219168", + "57588310583904098482337883219198" + ], + "answer": "57588310583904098482337883219178", + "correct_value": 57588310583904098482337883219178, + "explanation": "Distributive: (10000000000000000-2)*5758831058390411 = 10000000000000000*5758831058390411 - 2*5758831058390411." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10000000000000026 * 1000305577513003", + "ns_shortcut": "Use (10000000000000000+26)*m, correction is 26*1000305577513003.", + "shortcut_strength": 0.999999999999948, + "pure_math": { + "question": "What is 10000000000000026 * 1000305577513003?", + "options": [ + "10003055775130056007945015338098", + "10003055775130056007945015338078", + "10003055775130056007945015338088", + "10003055775130056007945015338068" + ], + "answer": "10003055775130056007945015338078", + "correct_value": 10003055775130056007945015338078, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4928328456374384 * 4379690494815326", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4928328456374384 * 4379690494815326?", + "options": [ + "21584553295710777636927597009194", + "21584553295710777636927597009174", + "21584553295710777636927597009184", + "21584553295710777636927597009204" + ], + "answer": "21584553295710777636927597009184", + "correct_value": 21584553295710777636927597009184, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001299", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000001 * 4901325936247151", + "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 10000000000000001 * 4901325936247151?", + "options": [ + "49013259362471514901325936247151", + "49013259362471514901325936247161", + "49013259362471514901325936247171", + "49013259362471514901325936247141" + ], + "answer": "49013259362471514901325936247151", + "correct_value": 49013259362471514901325936247151, + "explanation": "Distributive: (10000000000000000+1)*4901325936247151 = 10000000000000000*4901325936247151 + 1*4901325936247151." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10000000000000022 * 8169869467556913", + "ns_shortcut": "Use (10000000000000000+22)*m, correction is 22*8169869467556913.", + "shortcut_strength": 0.999999999999956, + "pure_math": { + "question": "What is 10000000000000022 * 8169869467556913?", + "options": [ + "81698694675569309737128286252076", + "81698694675569309737128286252106", + "81698694675569309737128286252086", + "81698694675569309737128286252096" + ], + "answer": "81698694675569309737128286252086", + "correct_value": 81698694675569309737128286252086, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6054126300933369 * 4204178868261372", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6054126300933369 * 4204178868261372?", + "options": [ + "25452629860169457725373848522268", + "25452629860169457725373848522278", + "25452629860169457725373848522288", + "25452629860169457725373848522258" + ], + "answer": "25452629860169457725373848522268", + "correct_value": 25452629860169457725373848522268, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001300", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000002 * 2160793514766419", + "ns_shortcut": "Use (10000000000000000+2)*m = 10000000000000000*m + 2*m.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "What is 10000000000000002 * 2160793514766419?", + "options": [ + "21607935147664194321587029532828", + "21607935147664194321587029532838", + "21607935147664194321587029532848", + "21607935147664194321587029532858" + ], + "answer": "21607935147664194321587029532838", + "correct_value": 21607935147664194321587029532838, + "explanation": "Distributive: (10000000000000000+2)*2160793514766419 = 10000000000000000*2160793514766419 + 2*2160793514766419." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9999999999999970 * 2843490753480827", + "ns_shortcut": "Use (10000000000000000-30)*m, correction is 30*2843490753480827.", + "shortcut_strength": 0.99999999999994, + "pure_math": { + "question": "What is 9999999999999970 * 2843490753480827?", + "options": [ + "28434907534808184695277395575200", + "28434907534808184695277395575210", + "28434907534808184695277395575180", + "28434907534808184695277395575190" + ], + "answer": "28434907534808184695277395575190", + "correct_value": 28434907534808184695277395575190, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6603921942806788 * 4891363152584986", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6603921942806788 * 4891363152584986?", + "options": [ + "32302180453592576160310547684958", + "32302180453592576160310547684968", + "32302180453592576160310547684988", + "32302180453592576160310547684978" + ], + "answer": "32302180453592576160310547684968", + "correct_value": 32302180453592576160310547684968, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_001301", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10000000000000001 * 3371397224183750", + "ns_shortcut": "Use (10000000000000000+1)*m = 10000000000000000*m + 1*m.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "What is 10000000000000001 * 3371397224183750?", + "options": [ + "33713972241837503371397224183740", + "33713972241837503371397224183750", + "33713972241837503371397224183770", + "33713972241837503371397224183760" + ], + "answer": "33713972241837503371397224183750", + "correct_value": 33713972241837503371397224183750, + "explanation": "Distributive: (10000000000000000+1)*3371397224183750 = 10000000000000000*3371397224183750 + 1*3371397224183750." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10000000000000033 * 4901458774180398", + "ns_shortcut": "Use (10000000000000000+33)*m, correction is 33*4901458774180398.", + "shortcut_strength": 0.999999999999934, + "pure_math": { + "question": "What is 10000000000000033 * 4901458774180398?", + "options": [ + "49014587741804141748139547953144", + "49014587741804141748139547953154", + "49014587741804141748139547953134", + "49014587741804141748139547953124" + ], + "answer": "49014587741804141748139547953134", + "correct_value": 49014587741804141748139547953134, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4833249658143153 * 6261721011777479", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4833249658143153 * 6261721011777479?", + "options": [ + "30264460939561298496851163451287", + "30264460939561298496851163451307", + "30264460939561298496851163451297", + "30264460939561298496851163451277" + ], + "answer": "30264460939561298496851163451287", + "correct_value": 30264460939561298496851163451287, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "relative_distance_001302", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 3156830228486078/6566646687092828 or 4548715704741623/8842405860316785?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.033683566889417926, + "pure_math": { + "question": "Which is larger: 3156830228486078/6566646687092828 or 4548715704741623/8842405860316785?", + "options": [ + "826917412147113/1931820504787809", + "2032708981357408/3456041580915706", + "3156830228486078/6566646687092828", + "4548715704741623/8842405860316785" + ], + "answer": "4548715704741623/8842405860316785", + "correct_value": 0.5144205973575005, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 3372175200288107/6861628584742188 or 2434697431897746/4835965254004392?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.012002297163120124, + "pure_math": { + "question": "Which is larger: 3372175200288107/6861628584742188 or 2434697431897746/4835965254004392?", + "options": [ + "4349766789706744/9811428863908402", + "2434697431897746/4835965254004392", + "1101892974897070/2293131082977623", + "3372175200288107/6861628584742188" + ], + "answer": "2434697431897746/4835965254004392", + "correct_value": 0.5034563533891625, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 3794324659519859/6898772108217924 or 5297536326472213/9133683321503815?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 3794324659519859/6898772108217924 or 5297536326472213/9133683321503815?", + "options": [ + "3843524577931784/7577230160805086", + "3969910162369072/7203693675667891", + "5297536326472213/9133683321503815", + "3794324659519859/6898772108217924" + ], + "answer": "5297536326472213/9133683321503815", + "correct_value": 0.5800000000000001, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001303", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 1628148390326898/3343200627235451 or 2259459937403775/4373538981659057?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.029617612703456753, + "pure_math": { + "question": "Which is larger: 1628148390326898/3343200627235451 or 2259459937403775/4373538981659057?", + "options": [ + "2259459937403775/4373538981659057", + "1706426004228856/3802526850057808", + "1324042127974059/2584332021447318", + "1628148390326898/3343200627235451" + ], + "answer": "2259459937403775/4373538981659057", + "correct_value": 0.516620509587106, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 4257923265569541/8760931988387643 or 2140105390002006/4172364584159390?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.026911277928638455, + "pure_math": { + "question": "Which is larger: 4257923265569541/8760931988387643 or 2140105390002006/4172364584159390?", + "options": [ + "4257923265569541/8760931988387643", + "2140105390002006/4172364584159390", + "4942317274958244/8522405866564943", + "3637092241011150/8257310160871239" + ], + "answer": "2140105390002006/4172364584159390", + "correct_value": 0.5129238701064219, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 5281050516198242/9601910029451349 or 3648829720203973/6291085724489609?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 5281050516198242/9601910029451349 or 3648829720203973/6291085724489609?", + "options": [ + "5281050516198242/9601910029451349", + "3278877006788082/5067270096273020", + "3588548795549949/6389746851508991", + "3648829720203973/6291085724489609" + ], + "answer": "3648829720203973/6291085724489609", + "correct_value": 0.58, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001304", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 3220662534796948/6647440010120226 or 1928230730472980/3769291125634051?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.027066558791819018, + "pure_math": { + "question": "Which is larger: 3220662534796948/6647440010120226 or 1928230730472980/3769291125634051?", + "options": [ + "1928230730472980/3769291125634051", + "3220662534796948/6647440010120226", + "1237112765199482/2391592955849234", + "3088640644904944/5470023248328239" + ], + "answer": "1928230730472980/3769291125634051", + "correct_value": 0.511563226666028, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 1963505072338930/4018644540248816 or 1538172129233847/3019625868580428?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.020792782492592865, + "pure_math": { + "question": "Which is larger: 1963505072338930/4018644540248816 or 1538172129233847/3019625868580428?", + "options": [ + "1963505072338930/4018644540248816", + "2324972554392452/5289483760989002", + "1538172129233847/3019625868580428", + "2117296715849548/4892148725709180" + ], + "answer": "1538172129233847/3019625868580428", + "correct_value": 0.5093916253793934, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 2575054336946853/4681916976267005 or 694250715235600/1196983991785517?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 2575054336946853/4681916976267005 or 694250715235600/1196983991785517?", + "options": [ + "4389391886449678/9107898491873354", + "6172700772410794/9809068654205697", + "694250715235600/1196983991785517", + "2575054336946853/4681916976267005" + ], + "answer": "694250715235600/1196983991785517", + "correct_value": 0.5800000000000001, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001305", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 1454204997919891/3040755933152225 or 3445807942413052/6646985639840719?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.040163605219574294, + "pure_math": { + "question": "Which is larger: 1454204997919891/3040755933152225 or 3445807942413052/6646985639840719?", + "options": [ + "3129640954618884/5207434698905855", + "1454204997919891/3040755933152225", + "4094780089717400/9194811767424842", + "3445807942413052/6646985639840719" + ], + "answer": "3445807942413052/6646985639840719", + "correct_value": 0.5184015927098683, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 3930189047528952/7916166648814665 or 3560589629168386/6935027885098984?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.016944807837905584, + "pure_math": { + "question": "Which is larger: 3930189047528952/7916166648814665 or 3560589629168386/6935027885098984?", + "options": [ + "3930189047528952/7916166648814665", + "3237663950679270/7648144013436969", + "3560589629168386/6935027885098984", + "2841855633889217/5609524112941494" + ], + "answer": "3560589629168386/6935027885098984", + "correct_value": 0.5134210976741539, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 1155139038892897/2100252797987085 or 3874143305783707/6679557423765012?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 1155139038892897/2100252797987085 or 3874143305783707/6679557423765012?", + "options": [ + "3874143305783707/6679557423765012", + "1155139038892897/2100252797987085", + "2069188180744777/3390764182335194", + "3230498624662145/6424690693927703" + ], + "answer": "3874143305783707/6679557423765012", + "correct_value": 0.58, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001306", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 3333052801968624/6860639450354117 or 5132723570042286/9855194266382395?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03499157319173252, + "pure_math": { + "question": "Which is larger: 3333052801968624/6860639450354117 or 5132723570042286/9855194266382395?", + "options": [ + "3333052801968624/6860639450354117", + "5132723570042286/9855194266382395", + "2978177391015356/6730991151732789", + "3904620046064212/6999986778416726" + ], + "answer": "5132723570042286/9855194266382395", + "correct_value": 0.5208140429611628, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 1842256977819895/3783352512245763 or 3071786111278982/6017767289138701?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.02351508958239512, + "pure_math": { + "question": "Which is larger: 1842256977819895/3783352512245763 or 3071786111278982/6017767289138701?", + "options": [ + "3071786111278982/6017767289138701", + "638144502256096/1538954996921627", + "1842256977819895/3783352512245763", + "4263662399372258/7562434907645223" + ], + "answer": "3071786111278982/6017767289138701", + "correct_value": 0.5104527914901532, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 811009880479418/1474563419053486 or 1400907195323288/2415357233316013?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 811009880479418/1474563419053486 or 1400907195323288/2415357233316013?", + "options": [ + "811009880479418/1474563419053486", + "2979281035625189/4671084778567335", + "1400907195323288/2415357233316013", + "936965484664366/1743303226046354" + ], + "answer": "1400907195323288/2415357233316013", + "correct_value": 0.5800000000000002, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001307", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 1229537758691493/2519410450378818 or 3007403250535453/5758234843786562?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.0342526907308775, + "pure_math": { + "question": "Which is larger: 1229537758691493/2519410450378818 or 3007403250535453/5758234843786562?", + "options": [ + "1229537758691493/2519410450378818", + "2911071022476600/6470847518888681", + "3007403250535453/5758234843786562", + "2474542771774016/4846757333950487" + ], + "answer": "3007403250535453/5758234843786562", + "correct_value": 0.5222786725658818, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 3051744962993482/6177937792904941 or 3261333844822568/6335216347540372?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.020819692315944716, + "pure_math": { + "question": "Which is larger: 3051744962993482/6177937792904941 or 3261333844822568/6335216347540372?", + "options": [ + "2496008933528442/5541114818369478", + "3051744962993482/6177937792904941", + "3261333844822568/6335216347540372", + "2071934556275850/4951643436497568" + ], + "answer": "3261333844822568/6335216347540372", + "correct_value": 0.514794391526782, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 4041795351853063/7348718821551023 or 1524160051863781/2627862158385829?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 4041795351853063/7348718821551023 or 1524160051863781/2627862158385829?", + "options": [ + "3056244361502256/5049600719720822", + "4041795351853063/7348718821551023", + "4601343183321240/9564830446286305", + "1524160051863781/2627862158385829" + ], + "answer": "1524160051863781/2627862158385829", + "correct_value": 0.5800000000000001, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001308", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 621972244946824/1321343939535526 or 4986469870644520/9682642702775566?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.04427874323819353, + "pure_math": { + "question": "Which is larger: 621972244946824/1321343939535526 or 4986469870644520/9682642702775566?", + "options": [ + "621972244946824/1321343939535526", + "2075686528182336/4770939494887476", + "4368710044953868/7799564731614446", + "4986469870644520/9682642702775566" + ], + "answer": "4986469870644520/9682642702775566", + "correct_value": 0.5149905892133281, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 3290698361029717/6683556800234541 or 1667560642537979/3254087518089096?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.020093716767469794, + "pure_math": { + "question": "Which is larger: 3290698361029717/6683556800234541 or 1667560642537979/3254087518089096?", + "options": [ + "3870771005817290/7303656306789064", + "3290698361029717/6683556800234541", + "1667560642537979/3254087518089096", + "850669737178215/1474455459034722" + ], + "answer": "1667560642537979/3254087518089096", + "correct_value": 0.5124510736936859, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 3710668912477579/6746670749959234 or 1855793764419956/3199644421413716?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 3710668912477579/6746670749959234 or 1855793764419956/3199644421413716?", + "options": [ + "3710668912477579/6746670749959234", + "1855793764419956/3199644421413716", + "1880510494246346/3725201813978980", + "1909594229114240/2840493379938222" + ], + "answer": "1855793764419956/3199644421413716", + "correct_value": 0.5800000000000002, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001309", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 3624172306526417/7483531551441204 or 3014998201968400/5888582815017812?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.02772107135150481, + "pure_math": { + "question": "Which is larger: 3624172306526417/7483531551441204 or 3014998201968400/5888582815017812?", + "options": [ + "3624172306526417/7483531551441204", + "3180084534395234/5294796712005516", + "3014998201968400/5888582815017812", + "1679752661713731/2959045653674508" + ], + "answer": "3014998201968400/5888582815017812", + "correct_value": 0.5120074382377995, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 2473348213231356/5084378278631433 or 2941843566423737/5744432217471448?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.025660553828265376, + "pure_math": { + "question": "Which is larger: 2473348213231356/5084378278631433 or 2941843566423737/5744432217471448?", + "options": [ + "5150453885676395/9854837634193719", + "2473348213231356/5084378278631433", + "2941843566423737/5744432217471448", + "1022830830242634/2391075853590735" + ], + "answer": "2941843566423737/5744432217471448", + "correct_value": 0.5121208598260145, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 3837831986636319/6977876339338760 or 4615591470243281/7957916328005657?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 3837831986636319/6977876339338760 or 4615591470243281/7957916328005657?", + "options": [ + "4655253455983604/8549222969853635", + "1803559305690937/3277752941741463", + "4615591470243281/7957916328005657", + "3837831986636319/6977876339338760" + ], + "answer": "4615591470243281/7957916328005657", + "correct_value": 0.58, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001310", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 4093817194031311/8548352818694374 or 4350635498424729/8368514596708697?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.04098015925861709, + "pure_math": { + "question": "Which is larger: 4093817194031311/8548352818694374 or 4350635498424729/8368514596708697?", + "options": [ + "4591492187946326/9003276877623646", + "4350635498424729/8368514596708697", + "1595663420655602/2745857036705163", + "4093817194031311/8548352818694374" + ], + "answer": "4350635498424729/8368514596708697", + "correct_value": 0.5198814494672467, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 4706781351734568/9567613416409113 or 3414157819514092/6784340129631510?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.011291591281408886, + "pure_math": { + "question": "Which is larger: 4706781351734568/9567613416409113 or 3414157819514092/6784340129631510?", + "options": [ + "3300191058131380/5805226944840016", + "3414157819514092/6784340129631510", + "2874179384824560/5973215343387199", + "4706781351734568/9567613416409113" + ], + "answer": "3414157819514092/6784340129631510", + "correct_value": 0.5032409570095554, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 1989139181926951/3616616694412638 or 1231469667569955/2123223564775784?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 1989139181926951/3616616694412638 or 1231469667569955/2123223564775784?", + "options": [ + "1231469667569955/2123223564775784", + "1989139181926951/3616616694412638", + "5063973183873033/9755263462960607", + "5770094070614239/8588805202473470" + ], + "answer": "1231469667569955/2123223564775784", + "correct_value": 0.5800000000000002, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001311", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 3400950152504542/6946881746689933 or 5099402649947071/9987378353437411?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.021019708378544943, + "pure_math": { + "question": "Which is larger: 3400950152504542/6946881746689933 or 5099402649947071/9987378353437411?", + "options": [ + "3400950152504542/6946881746689933", + "3608246193151117/8778455359131540", + "5099402649947071/9987378353437411", + "4509407413009303/9974769704465236" + ], + "answer": "5099402649947071/9987378353437411", + "correct_value": 0.5105847069658658, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 3800087307234718/7777935815831479 or 1564605571542898/3105797740800920?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.015196579313861758, + "pure_math": { + "question": "Which is larger: 3800087307234718/7777935815831479 or 1564605571542898/3105797740800920?", + "options": [ + "3372565788121730/5945949483890539", + "1564605571542898/3105797740800920", + "3800087307234718/7777935815831479", + "1332634088041347/3136037756952679" + ], + "answer": "1564605571542898/3105797740800920", + "correct_value": 0.5037693057048265, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 714715855461390/1299483373566162 or 3196356830879319/5510960053240205?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 714715855461390/1299483373566162 or 3196356830879319/5510960053240205?", + "options": [ + "4386475992409966/8443955736178859", + "3196356830879319/5510960053240205", + "714715855461390/1299483373566162", + "1362388578538712/2030383546674213" + ], + "answer": "3196356830879319/5510960053240205", + "correct_value": 0.5800000000000001, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001312", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 1809049684448755/3793677693721527 or 2097520633542843/4100768123955861?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03463548349140205, + "pure_math": { + "question": "Which is larger: 1809049684448755/3793677693721527 or 2097520633542843/4100768123955861?", + "options": [ + "1809049684448755/3793677693721527", + "2359104552781194/5267865246457108", + "2097520633542843/4100768123955861", + "2938563784889089/6589269830009136" + ], + "answer": "2097520633542843/4100768123955861", + "correct_value": 0.5114945712949606, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 733723295160259/1484406961123012 or 1922388678223931/3816535393576681?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.00941279172194115, + "pure_math": { + "question": "Which is larger: 733723295160259/1484406961123012 or 1922388678223931/3816535393576681?", + "options": [ + "1922388678223931/3816535393576681", + "733723295160259/1484406961123012", + "4112475697612940/8052288475087787", + "2689646493658724/4725255150779860" + ], + "answer": "1922388678223931/3816535393576681", + "correct_value": 0.5036999477220508, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 4850925502184806/8819864549426919 or 3225083018516997/5560487962960339?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 4850925502184806/8819864549426919 or 3225083018516997/5560487962960339?", + "options": [ + "2088582831520776/3870837424383381", + "5295912612133356/8794836921300009", + "3225083018516997/5560487962960339", + "4850925502184806/8819864549426919" + ], + "answer": "3225083018516997/5560487962960339", + "correct_value": 0.5800000000000001, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001313", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 3832135623781000/7990113075730424 or 1217161281479986/2355849665581661?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03704521738239802, + "pure_math": { + "question": "Which is larger: 3832135623781000/7990113075730424 or 1217161281479986/2355849665581661?", + "options": [ + "4746046912840130/8457970966921286", + "3316142736988988/5528718857816152", + "1217161281479986/2355849665581661", + "3832135623781000/7990113075730424" + ], + "answer": "1217161281479986/2355849665581661", + "correct_value": 0.5166549034356435, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 3646445341004400/7328529707232140 or 3264228631497663/6418934773087698?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.010962730743062421, + "pure_math": { + "question": "Which is larger: 3646445341004400/7328529707232140 or 3264228631497663/6418934773087698?", + "options": [ + "1242313365642037/2374622536615810", + "3264228631497663/6418934773087698", + "3646445341004400/7328529707232140", + "1619090798040308/2797047622732739" + ], + "answer": "3264228631497663/6418934773087698", + "correct_value": 0.5085312044583173, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 3681699039027596/6693998252777446 or 4813112772027922/8298470296599866?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 3681699039027596/6693998252777446 or 4813112772027922/8298470296599866?", + "options": [ + "562339164345563/1024278819423814", + "4106561054545598/6405105331827948", + "4813112772027922/8298470296599866", + "3681699039027596/6693998252777446" + ], + "answer": "4813112772027922/8298470296599866", + "correct_value": 0.58, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001314", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 2328099629426930/4802289850285624 or 1085692948671045/2052630831071951?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.044138040415096835, + "pure_math": { + "question": "Which is larger: 2328099629426930/4802289850285624 or 1085692948671045/2052630831071951?", + "options": [ + "2328099629426930/4802289850285624", + "3112887959791793/6474412118515400", + "4736166046620089/9637331951400662", + "1085692948671045/2052630831071951" + ], + "answer": "1085692948671045/2052630831071951", + "correct_value": 0.5289275266824579, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 1651586676297210/3374259162052785 or 909990441952020/1786374564022702?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.0199398347584554, + "pure_math": { + "question": "Which is larger: 1651586676297210/3374259162052785 or 909990441952020/1786374564022702?", + "options": [ + "2438843441109301/5066384740074769", + "909990441952020/1786374564022702", + "1651586676297210/3374259162052785", + "1362638275094165/2419192855137853" + ], + "answer": "909990441952020/1786374564022702", + "correct_value": 0.509406291535427, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 3737414519217084/6795299125849243 or 5265740317699619/9078862616723482?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 3737414519217084/6795299125849243 or 5265740317699619/9078862616723482?", + "options": [ + "3983293953208783/6283020137650891", + "2482314485567700/4343196485347070", + "3737414519217084/6795299125849243", + "5265740317699619/9078862616723482" + ], + "answer": "5265740317699619/9078862616723482", + "correct_value": 0.58, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001315", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 2571455766889687/5314128421507481 or 1474968005701703/2852085805696115?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03326374088668799, + "pure_math": { + "question": "Which is larger: 2571455766889687/5314128421507481 or 1474968005701703/2852085805696115?", + "options": [ + "5003619631207823/8166942172815245", + "1958557955976782/3253129134936374", + "1474968005701703/2852085805696115", + "2571455766889687/5314128421507481" + ], + "answer": "1474968005701703/2852085805696115", + "correct_value": 0.5171541482924299, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 2151123623073786/4403882701826968 or 1763829668391729/3434867733523728?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.0250466046441154, + "pure_math": { + "question": "Which is larger: 2151123623073786/4403882701826968 or 1763829668391729/3434867733523728?", + "options": [ + "1763829668391729/3434867733523728", + "3927104112885930/7596147399798533", + "2466573936728772/4895933282394038", + "2151123623073786/4403882701826968" + ], + "answer": "1763829668391729/3434867733523728", + "correct_value": 0.5135073036952922, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 4961589883296470/9021072515084490 or 4064394552799012/7007576815170711?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 4961589883296470/9021072515084490 or 4064394552799012/7007576815170711?", + "options": [ + "5653773258249729/8688835813771749", + "4961589883296470/9021072515084490", + "3920541137415851/7537515983576144", + "4064394552799012/7007576815170711" + ], + "answer": "4064394552799012/7007576815170711", + "correct_value": 0.58, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001316", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 4400235113990462/9133296478578571 or 4292907636683648/8228332662690457?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.0399431526564924, + "pure_math": { + "question": "Which is larger: 4400235113990462/9133296478578571 or 4292907636683648/8228332662690457?", + "options": [ + "4292907636683648/8228332662690457", + "2801801662865984/4676285543733454", + "4400235113990462/9133296478578571", + "4350315373265544/8401711986879224" + ], + "answer": "4292907636683648/8228332662690457", + "correct_value": 0.5217226639546165, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 713362764052804/1453633670149995 or 1198132274424776/2385567644476762?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.011497479632861074, + "pure_math": { + "question": "Which is larger: 713362764052804/1453633670149995 or 1198132274424776/2385567644476762?", + "options": [ + "713362764052804/1453633670149995", + "1198132274424776/2385567644476762", + "3089087600357570/7153904165941948", + "2561250561506201/4374257384172332" + ], + "answer": "1198132274424776/2385567644476762", + "correct_value": 0.5022420039937991, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 2194415693531519/3989846715511852 or 4132098666261770/7124308045278914?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 2194415693531519/3989846715511852 or 4132098666261770/7124308045278914?", + "options": [ + "4132098666261770/7124308045278914", + "4396017839217536/7522750990228412", + "5298331555144480/9107148567063869", + "2194415693531519/3989846715511852" + ], + "answer": "4132098666261770/7124308045278914", + "correct_value": 0.58, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001317", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 899353677509414/1876610312189529 or 1953773664469476/3752265578237291?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.04144802814689097, + "pure_math": { + "question": "Which is larger: 899353677509414/1876610312189529 or 1953773664469476/3752265578237291?", + "options": [ + "1953773664469476/3752265578237291", + "2378704202311726/5560404585212965", + "3005285997303963/6282496631690872", + "899353677509414/1876610312189529" + ], + "answer": "1953773664469476/3752265578237291", + "correct_value": 0.5206917324299054, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 3475729804195854/7055115658396344 or 4282882197468053/8345242411101316?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.020558599623763885, + "pure_math": { + "question": "Which is larger: 3475729804195854/7055115658396344 or 4282882197468053/8345242411101316?", + "options": [ + "3475729804195854/7055115658396344", + "1823694826992158/3285178990916632", + "3598167828063282/8347552758408573", + "4282882197468053/8345242411101316" + ], + "answer": "4282882197468053/8345242411101316", + "correct_value": 0.5132124372769231, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 4161586802510395/7566521459109808 or 3913488263172457/6747393557193891?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 4161586802510395/7566521459109808 or 3913488263172457/6747393557193891?", + "options": [ + "3913488263172457/6747393557193891", + "2994999338814920/5429687432046532", + "3062128785755546/5961157524320735", + "4161586802510395/7566521459109808" + ], + "answer": "3913488263172457/6747393557193891", + "correct_value": 0.5800000000000001, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001318", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 2544934428522855/5197074050005155 or 2941839518116505/5625160278864343?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03329278547857595, + "pure_math": { + "question": "Which is larger: 2544934428522855/5197074050005155 or 2941839518116505/5625160278864343?", + "options": [ + "2941839518116505/5625160278864343", + "2681454867017476/6106115904144565", + "2544934428522855/5197074050005155", + "4864991466173950/8938345843954084" + ], + "answer": "2941839518116505/5625160278864343", + "correct_value": 0.522978790341318, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 2297377069965648/4664001722315931 or 3552671990262190/7071392335349837?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.009824256835011758, + "pure_math": { + "question": "Which is larger: 2297377069965648/4664001722315931 or 3552671990262190/7071392335349837?", + "options": [ + "3878071165744446/9227115706937936", + "2297377069965648/4664001722315931", + "3341703628809730/7190330958350584", + "3552671990262190/7071392335349837" + ], + "answer": "3552671990262190/7071392335349837", + "correct_value": 0.5024006336775304, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 5183878942571820/9425234441039672 or 3676087368763872/6338081670282538?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 5183878942571820/9425234441039672 or 3676087368763872/6338081670282538?", + "options": [ + "3273457772714356/5244026943640966", + "5183878942571820/9425234441039672", + "5359704138248868/9807528949030234", + "3676087368763872/6338081670282538" + ], + "answer": "3676087368763872/6338081670282538", + "correct_value": 0.58, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001319", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 910646369219713/1871334268902897 or 3089540991586700/6044075948185964?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.024539007810614244, + "pure_math": { + "question": "Which is larger: 910646369219713/1871334268902897 or 3089540991586700/6044075948185964?", + "options": [ + "4859218225158536/9258901846692782", + "910646369219713/1871334268902897", + "2224000349275634/5259140082640715", + "3089540991586700/6044075948185964" + ], + "answer": "3089540991586700/6044075948185964", + "correct_value": 0.5111684595085173, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 3355513585936439/6915585738415527 or 572184954916716/1111106539763659?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.02975825568422308, + "pure_math": { + "question": "Which is larger: 3355513585936439/6915585738415527 or 572184954916716/1111106539763659?", + "options": [ + "2793163766124851/5586676880153876", + "572184954916716/1111106539763659", + "3355513585936439/6915585738415527", + "4194418494223416/8725643450669567" + ], + "answer": "572184954916716/1111106539763659", + "correct_value": 0.5149685781153122, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 601789337951075/1094162432638318 or 883439868815099/1523172187612239?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 601789337951075/1094162432638318 or 883439868815099/1523172187612239?", + "options": [ + "883439868815099/1523172187612239", + "716686942101947/1241282289313064", + "601789337951075/1094162432638318", + "3782438002925814/6085027624888039" + ], + "answer": "883439868815099/1523172187612239", + "correct_value": 0.5800000000000003, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001320", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 3950685376183907/8071990796507909 or 4004028028668073/7741107408862846?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.02781092486936687, + "pure_math": { + "question": "Which is larger: 3950685376183907/8071990796507909 or 4004028028668073/7741107408862846?", + "options": [ + "4004028028668073/7741107408862846", + "644310969319432/1218456905875296", + "5060906674020351/8237340062704028", + "3950685376183907/8071990796507909" + ], + "answer": "4004028028668073/7741107408862846", + "correct_value": 0.5172422777758947, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 1953786165390229/3984967560087258 or 1642719039332376/3267801120498895?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.012409495526573944, + "pure_math": { + "question": "Which is larger: 1953786165390229/3984967560087258 or 1642719039332376/3267801120498895?", + "options": [ + "3250293045313914/7241735290492555", + "1953786165390229/3984967560087258", + "1642719039332376/3267801120498895", + "5865852235968630/9948416552852361" + ], + "answer": "1642719039332376/3267801120498895", + "correct_value": 0.5026985972394741, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 2103244890203102/3824081618551094 or 5424461217143410/9352519339902431?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 2103244890203102/3824081618551094 or 5424461217143410/9352519339902431?", + "options": [ + "2103244890203102/3824081618551094", + "2578300913909242/4669849232015448", + "1027935006989502/1607955914991646", + "5424461217143410/9352519339902431" + ], + "answer": "5424461217143410/9352519339902431", + "correct_value": 0.58, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001321", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 3818190347847252/7859509924251032 or 4331070142258588/8443016131360446?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.027171507678161178, + "pure_math": { + "question": "Which is larger: 3818190347847252/7859509924251032 or 4331070142258588/8443016131360446?", + "options": [ + "3818190347847252/7859509924251032", + "4331070142258588/8443016131360446", + "5465112892269918/9951476506088829", + "5199941974409009/9946252436873207" + ], + "answer": "4331070142258588/8443016131360446", + "correct_value": 0.51297665133807, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 2452928009910006/5041099278184445 or 2745053558374209/5444258490046201?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.017624796149114608, + "pure_math": { + "question": "Which is larger: 2452928009910006/5041099278184445 or 2745053558374209/5444258490046201?", + "options": [ + "4236284112346408/8532523817990687", + "2745053558374209/5444258490046201", + "838150938621901/1554976177127418", + "2452928009910006/5041099278184445" + ], + "answer": "2745053558374209/5444258490046201", + "correct_value": 0.5042107319836157, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 1475079906462510/2681963466295472 or 5761457835990353/9933547993086815?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 1475079906462510/2681963466295472 or 5761457835990353/9933547993086815?", + "options": [ + "1475079906462510/2681963466295472", + "2086013543328417/3831223799933664", + "4632794301576571/8917894668890351", + "5761457835990353/9933547993086815" + ], + "answer": "5761457835990353/9933547993086815", + "correct_value": 0.5800000000000001, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001322", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 2607013388966683/5321421034883884 or 2811022505423814/5470967110370684?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.023897988804157777, + "pure_math": { + "question": "Which is larger: 2607013388966683/5321421034883884 or 2811022505423814/5470967110370684?", + "options": [ + "2607013388966683/5321421034883884", + "2227438836053826/5106690397020671", + "2811022505423814/5470967110370684", + "3345048941385950/5864483503007188" + ], + "answer": "2811022505423814/5470967110370684", + "correct_value": 0.5138072389606001, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 4550154378899129/9280124164396468 or 3656219672547532/7117577908780174?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.02337694675591645, + "pure_math": { + "question": "Which is larger: 4550154378899129/9280124164396468 or 3656219672547532/7117577908780174?", + "options": [ + "1089162586293145/2039023606313356", + "4550154378899129/9280124164396468", + "1395077101973392/2507816574500963", + "3656219672547532/7117577908780174" + ], + "answer": "3656219672547532/7117577908780174", + "correct_value": 0.5136887462850608, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 4624885607478622/8408882922688403 or 5640818746280330/9725549562552294?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 4624885607478622/8408882922688403 or 5640818746280330/9725549562552294?", + "options": [ + "5640818746280330/9725549562552294", + "5444070713551233/9769947318420437", + "4624885607478622/8408882922688403", + "3333529742952483/5330390373562849" + ], + "answer": "5640818746280330/9725549562552294", + "correct_value": 0.58, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001323", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 4412182227556401/9018022577686047 or 2640276677828445/5043640536194472?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03422357416440208, + "pure_math": { + "question": "Which is larger: 4412182227556401/9018022577686047 or 2640276677828445/5043640536194472?", + "options": [ + "2640276677828445/5043640536194472", + "3427153018122302/7353643867046593", + "2326940021900240/4139572061383055", + "4412182227556401/9018022577686047" + ], + "answer": "2640276677828445/5043640536194472", + "correct_value": 0.5234862910790599, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 2019538362979077/4079083105366376 or 1003585043493176/1968628573859549?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.014692769154363461, + "pure_math": { + "question": "Which is larger: 2019538362979077/4079083105366376 or 1003585043493176/1968628573859549?", + "options": [ + "590355372323102/1038420137814867", + "2864613691644326/6889008812509295", + "1003585043493176/1968628573859549", + "2019538362979077/4079083105366376" + ], + "answer": "1003585043493176/1968628573859549", + "correct_value": 0.5097889245433539, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 2686119213140158/4883853114800287 or 5047443207912410/8702488289504155?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 2686119213140158/4883853114800287 or 5047443207912410/8702488289504155?", + "options": [ + "5047443207912410/8702488289504155", + "1742620330191822/3274268029688133", + "2686119213140158/4883853114800287", + "5662467448121169/9078087709099722" + ], + "answer": "5047443207912410/8702488289504155", + "correct_value": 0.58, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001324", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 3762873463966898/7736193396426065 or 606334248785514/1138338364057463?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.04624994584783848, + "pure_math": { + "question": "Which is larger: 3762873463966898/7736193396426065 or 606334248785514/1138338364057463?", + "options": [ + "2965846903386792/5168387757701654", + "3192815450819356/6803868512465385", + "606334248785514/1138338364057463", + "3762873463966898/7736193396426065" + ], + "answer": "606334248785514/1138338364057463", + "correct_value": 0.5326485234357844, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 4821209642193859/9717811369442923 or 893899496711254/1745751479412805?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.01592187919644822, + "pure_math": { + "question": "Which is larger: 4821209642193859/9717811369442923 or 893899496711254/1745751479412805?", + "options": [ + "4821209642193859/9717811369442923", + "3465142081056995/7818059129193969", + "893899496711254/1745751479412805", + "645029535122236/1134123340523095" + ], + "answer": "893899496711254/1745751479412805", + "correct_value": 0.5120428120799434, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 3458260023419260/6287745497125927 or 3147890474551938/5427397369917135?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 3458260023419260/6287745497125927 or 3147890474551938/5427397369917135?", + "options": [ + "1522455454985539/3170811620377464", + "3147890474551938/5427397369917135", + "3458260023419260/6287745497125927", + "1922696657445385/3010293319102051" + ], + "answer": "3147890474551938/5427397369917135", + "correct_value": 0.58, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001325", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 2334093559660546/4820774190763534 or 4650408078602261/8956359906833583?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03505569364100736, + "pure_math": { + "question": "Which is larger: 2334093559660546/4820774190763534 or 4650408078602261/8956359906833583?", + "options": [ + "2456153839037698/5372274207447582", + "4650408078602261/8956359906833583", + "2334093559660546/4820774190763534", + "2252191763471898/4683139030731783" + ], + "answer": "4650408078602261/8956359906833583", + "correct_value": 0.5192297012376715, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 4919930319518848/9937768064140201 or 3886063331178023/7649442921273683?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.012945156001739289, + "pure_math": { + "question": "Which is larger: 4919930319518848/9937768064140201 or 3886063331178023/7649442921273683?", + "options": [ + "4257079773957318/9505916388177287", + "3886063331178023/7649442921273683", + "4919930319518848/9937768064140201", + "3084607375181382/7281707838544679" + ], + "answer": "3886063331178023/7649442921273683", + "correct_value": 0.5080191291277676, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 4894498530814632/8899088237844785 or 1750770861012757/3018570450021995?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 4894498530814632/8899088237844785 or 1750770861012757/3018570450021995?", + "options": [ + "2607026480660291/4745470332838100", + "1750770861012757/3018570450021995", + "4894498530814632/8899088237844785", + "4168885193157192/6357465287205160" + ], + "answer": "1750770861012757/3018570450021995", + "correct_value": 0.58, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001326", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 534837061794351/1130789586333723 or 2875399962050632/5553475060032459?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.04478924850978466, + "pure_math": { + "question": "Which is larger: 534837061794351/1130789586333723 or 2875399962050632/5553475060032459?", + "options": [ + "534837061794351/1130789586333723", + "1121067616966639/2058479517514305", + "2875399962050632/5553475060032459", + "5748208199720642/9585993587552760" + ], + "answer": "2875399962050632/5553475060032459", + "correct_value": 0.5177658909003592, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 4366255093163884/8883576839284705 or 2212322675925314/4392211548616846?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.01219477544067954, + "pure_math": { + "question": "Which is larger: 4366255093163884/8883576839284705 or 2212322675925314/4392211548616846?", + "options": [ + "2212322675925314/4392211548616846", + "1000963882704460/1870504935150829", + "4366255093163884/8883576839284705", + "3523346741834662/5873339912067862" + ], + "answer": "2212322675925314/4392211548616846", + "correct_value": 0.5036921950223454, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 2378677727339270/4324868595162308 or 2243102837060376/3867418684586855?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 2378677727339270/4324868595162308 or 2243102837060376/3867418684586855?", + "options": [ + "1790604976805330/3038728371130342", + "6117100713363486/9924275641047440", + "2378677727339270/4324868595162308", + "2243102837060376/3867418684586855" + ], + "answer": "2243102837060376/3867418684586855", + "correct_value": 0.5800000000000001, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001327", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 2735390981649663/5691084238440991 or 4295109095246166/8351862018618441?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.033624678104418104, + "pure_math": { + "question": "Which is larger: 2735390981649663/5691084238440991 or 4295109095246166/8351862018618441?", + "options": [ + "617236428868181/1223862747558723", + "2031456334205041/3580982707859069", + "2735390981649663/5691084238440991", + "4295109095246166/8351862018618441" + ], + "answer": "4295109095246166/8351862018618441", + "correct_value": 0.5142696425864397, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 2290601367143787/4703775571670791 or 5028021769094710/9903621976464959?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.02072444034688009, + "pure_math": { + "question": "Which is larger: 2290601367143787/4703775571670791 or 5028021769094710/9903621976464959?", + "options": [ + "2290601367143787/4703775571670791", + "5028021769094710/9903621976464959", + "5887091354936243/9985533119019792", + "948755934326244/2120434006407571" + ], + "answer": "5028021769094710/9903621976464959", + "correct_value": 0.5076952433204073, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 4760153649387807/8654824817068740 or 1006626787530420/1735563426776585?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 4760153649387807/8654824817068740 or 1006626787530420/1735563426776585?", + "options": [ + "4760153649387807/8654824817068740", + "1006626787530420/1735563426776585", + "6273355377898368/9687264767622686", + "3595322834925330/6092785559050952" + ], + "answer": "1006626787530420/1735563426776585", + "correct_value": 0.5800000000000004, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001328", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 3398067461346230/7100579156150744 or 2867940035574290/5608231587068869?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03281844367140946, + "pure_math": { + "question": "Which is larger: 3398067461346230/7100579156150744 or 2867940035574290/5608231587068869?", + "options": [ + "2867940035574290/5608231587068869", + "3398067461346230/7100579156150744", + "4038835917224014/8130230740243585", + "3526847817698832/7330477268960436" + ], + "answer": "2867940035574290/5608231587068869", + "correct_value": 0.5113804576449763, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 4663702514573769/9502206491512501 or 879787191995726/1736207779474908?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.01592714804397516, + "pure_math": { + "question": "Which is larger: 4663702514573769/9502206491512501 or 879787191995726/1736207779474908?", + "options": [ + "4663702514573769/9502206491512501", + "3701906347095936/6961560148816440", + "1569702331953697/2883135099564296", + "879787191995726/1736207779474908" + ], + "answer": "879787191995726/1736207779474908", + "correct_value": 0.5067292074119178, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 2185370478211734/3973400869475879 or 4217972244057987/7272365938031013?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 2185370478211734/3973400869475879 or 4217972244057987/7272365938031013?", + "options": [ + "2185370478211734/3973400869475879", + "4217972244057987/7272365938031013", + "1366538836238242/2048130517685016", + "1027053370563694/1880209476152048" + ], + "answer": "4217972244057987/7272365938031013", + "correct_value": 0.58, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001329", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 4118043687492473/8543280461730780 or 2492270044633080/4833345504534993?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03361941732807627, + "pure_math": { + "question": "Which is larger: 4118043687492473/8543280461730780 or 2492270044633080/4833345504534993?", + "options": [ + "3408642894745590/7075401161197349", + "4118043687492473/8543280461730780", + "2492270044633080/4833345504534993", + "1570119869193244/3050770287516046" + ], + "answer": "2492270044633080/4833345504534993", + "correct_value": 0.5156407797238275, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 2190970081489660/4447369192746225 or 669651770074654/1319716273614575?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.014776939393268351, + "pure_math": { + "question": "Which is larger: 2190970081489660/4447369192746225 or 669651770074654/1319716273614575?", + "options": [ + "3888459134820571/6997808022412214", + "2190970081489660/4447369192746225", + "962367813007527/1755782766088950", + "669651770074654/1319716273614575" + ], + "answer": "669651770074654/1319716273614575", + "correct_value": 0.5074210142461475, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 4609971706783611/8381766739606565 or 2005897578690569/3458444101190635?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 4609971706783611/8381766739606565 or 2005897578690569/3458444101190635?", + "options": [ + "4572201298847648/8871086570862480", + "4609971706783611/8381766739606565", + "2005897578690569/3458444101190635", + "4903830923497090/9207406918926752" + ], + "answer": "2005897578690569/3458444101190635", + "correct_value": 0.5800000000000002, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001330", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 523121356736527/1122113587913851 or 3338924647595120/6440772830084977?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.05221147681703825, + "pure_math": { + "question": "Which is larger: 523121356736527/1122113587913851 or 3338924647595120/6440772830084977?", + "options": [ + "2516100279844157/4764622874148550", + "3338924647595120/6440772830084977", + "523121356736527/1122113587913851", + "4231780825652356/7576969195842783" + ], + "answer": "3338924647595120/6440772830084977", + "correct_value": 0.518404349241901, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 1357678798992610/2745623415921136 or 4695003609586796/9146054663174490?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.018848138944960513, + "pure_math": { + "question": "Which is larger: 1357678798992610/2745623415921136 or 4695003609586796/9146054663174490?", + "options": [ + "4695003609586796/9146054663174490", + "454714514028190/1016516926150135", + "923295564680861/1588040509772651", + "1357678798992610/2745623415921136" + ], + "answer": "4695003609586796/9146054663174490", + "correct_value": 0.5133364912513233, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 4572946460884796/8314448110699629 or 4864606572552940/8387252711298173?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 4572946460884796/8314448110699629 or 4864606572552940/8387252711298173?", + "options": [ + "4572946460884796/8314448110699629", + "4864606572552940/8387252711298173", + "3883979111385546/5724015006102090", + "4069740355347359/7648103724929683" + ], + "answer": "4864606572552940/8387252711298173", + "correct_value": 0.58, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001331", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 2337137697930081/4820992535670831 or 4069499609858078/7841469570042440?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03418803601306808, + "pure_math": { + "question": "Which is larger: 2337137697930081/4820992535670831 or 4069499609858078/7841469570042440?", + "options": [ + "2767752600646682/6310717749500977", + "4069499609858078/7841469570042440", + "4555053149955965/7929667348677205", + "2337137697930081/4820992535670831" + ], + "answer": "4069499609858078/7841469570042440", + "correct_value": 0.5189715490837584, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 4151980129238030/8434121218697235 or 3175755140307945/6324186912728718?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.009876559808911778, + "pure_math": { + "question": "Which is larger: 4151980129238030/8434121218697235 or 3175755140307945/6324186912728718?", + "options": [ + "1328590436815466/2339383448759758", + "3175755140307945/6324186912728718", + "4151980129238030/8434121218697235", + "3120226302436280/6722619281185166" + ], + "answer": "3175755140307945/6324186912728718", + "correct_value": 0.5021602277307916, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 5082015476290782/9240028138710513 or 3892008974015782/6710360300027210?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 5082015476290782/9240028138710513 or 3892008974015782/6710360300027210?", + "options": [ + "3892008974015782/6710360300027210", + "1571366238504614/2798510633030528", + "3697932573277365/7609545816765941", + "5082015476290782/9240028138710513" + ], + "answer": "3892008974015782/6710360300027210", + "correct_value": 0.5800000000000001, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001332", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 4686843661616532/9773225767096842 or 3130737125680045/6057647470054147?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.037264381486155684, + "pure_math": { + "question": "Which is larger: 4686843661616532/9773225767096842 or 3130737125680045/6057647470054147?", + "options": [ + "3130737125680045/6057647470054147", + "4686843661616532/9773225767096842", + "3935216554476600/6604596303808471", + "3891441734539914/9037461188341525" + ], + "answer": "3130737125680045/6057647470054147", + "correct_value": 0.516823922348862, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 2067200038734377/4173786374762110 or 1402267345056736/2766726222716309?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.011551001856902499, + "pure_math": { + "question": "Which is larger: 2067200038734377/4173786374762110 or 1402267345056736/2766726222716309?", + "options": [ + "4288792382854759/9184574065742254", + "2067200038734377/4173786374762110", + "1402267345056736/2766726222716309", + "2411653069979286/4249720249540360" + ], + "answer": "1402267345056736/2766726222716309", + "correct_value": 0.5068327084708879, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 4718238855131213/8578616100238569 or 1693957204831094/2920615870398438?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 4718238855131213/8578616100238569 or 1693957204831094/2920615870398438?", + "options": [ + "1829589709164500/2727899493299769", + "1693957204831094/2920615870398438", + "4718238855131213/8578616100238569", + "1493079285869078/2694572429954983" + ], + "answer": "1693957204831094/2920615870398438", + "correct_value": 0.58, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001333", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 1625216608778890/3324879232130241 or 3407637128664700/6552641746983606?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03123549665135572, + "pure_math": { + "question": "Which is larger: 1625216608778890/3324879232130241 or 3407637128664700/6552641746983606?", + "options": [ + "4549412076835150/9033955399039167", + "3407637128664700/6552641746983606", + "1625216608778890/3324879232130241", + "5274042311726651/8871265304802566" + ], + "answer": "3407637128664700/6552641746983606", + "correct_value": 0.5200402006157816, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 3735810553504051/7606840451308897 or 566130854510678/1113219859552959?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.017440610262984435, + "pure_math": { + "question": "Which is larger: 3735810553504051/7606840451308897 or 566130854510678/1113219859552959?", + "options": [ + "3217103839739998/6320995120518642", + "3735810553504051/7606840451308897", + "2789267296907942/6190547494016540", + "566130854510678/1113219859552959" + ], + "answer": "566130854510678/1113219859552959", + "correct_value": 0.5085526005060867, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 1303975793000345/2370865078182445 or 600568012159500/1035462089930171?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 1303975793000345/2370865078182445 or 600568012159500/1035462089930171?", + "options": [ + "616500614384905/1209863167333742", + "1303975793000345/2370865078182445", + "3502853659716746/6331636881884912", + "600568012159500/1035462089930171" + ], + "answer": "600568012159500/1035462089930171", + "correct_value": 0.5800000000000007, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001334", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 2019145687252705/4146076980235333 or 4228380328927378/8128192755779531?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03321012607834173, + "pure_math": { + "question": "Which is larger: 2019145687252705/4146076980235333 or 4228380328927378/8128192755779531?", + "options": [ + "2019145687252705/4146076980235333", + "4228380328927378/8128192755779531", + "4127586459268053/7566662875029198", + "1672708367847358/3478675003084052" + ], + "answer": "4228380328927378/8128192755779531", + "correct_value": 0.5202116209560604, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 2455303244156940/4961613497846706 or 1169803089163102/2286598516469136?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.016731105597211915, + "pure_math": { + "question": "Which is larger: 2455303244156940/4961613497846706 or 1169803089163102/2286598516469136?", + "options": [ + "2981824855060226/5445646880425277", + "4663823983708041/9947797342963520", + "2455303244156940/4961613497846706", + "1169803089163102/2286598516469136" + ], + "answer": "1169803089163102/2286598516469136", + "correct_value": 0.5115909420642238, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 1601841727843736/2912439505170428 or 3222896302702020/5556717763279344?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 1601841727843736/2912439505170428 or 3222896302702020/5556717763279344?", + "options": [ + "3222896302702020/5556717763279344", + "1601841727843736/2912439505170428", + "5156941608335202/9403899579418829", + "1871193465246943/3878904190860438" + ], + "answer": "3222896302702020/5556717763279344", + "correct_value": 0.5800000000000001, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001335", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 1084059988850201/2273968299967842 or 2286429987581696/4464879691951442?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03536610497019438, + "pure_math": { + "question": "Which is larger: 1084059988850201/2273968299967842 or 2286429987581696/4464879691951442?", + "options": [ + "949012624262388/1623787856898585", + "2286429987581696/4464879691951442", + "2940171283250512/6162002287297619", + "1084059988850201/2273968299967842" + ], + "answer": "2286429987581696/4464879691951442", + "correct_value": 0.5120921828427538, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 954540566866734/1967430496322492 or 2205827926865648/4339317640379422?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.023164030428434457, + "pure_math": { + "question": "Which is larger: 954540566866734/1967430496322492 or 2205827926865648/4339317640379422?", + "options": [ + "774203620250975/1568367134274264", + "1005171808766176/2042160626843561", + "954540566866734/1967430496322492", + "2205827926865648/4339317640379422" + ], + "answer": "2205827926865648/4339317640379422", + "correct_value": 0.5083352060562164, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 2090560450309943/3801019000563532 or 5226476081713045/9011165658125939?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 2090560450309943/3801019000563532 or 5226476081713045/9011165658125939?", + "options": [ + "1822688025030948/3097854428266356", + "2090560450309943/3801019000563532", + "5226476081713045/9011165658125939", + "2753562731215419/5613939836078244" + ], + "answer": "5226476081713045/9011165658125939", + "correct_value": 0.5800000000000001, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001336", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 3184319397227633/6537363710836453 or 4735064900814904/9084226096404750?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03414497850706527, + "pure_math": { + "question": "Which is larger: 3184319397227633/6537363710836453 or 4735064900814904/9084226096404750?", + "options": [ + "1361303571343678/2583866449998175", + "2147287797774732/4919877706598877", + "4735064900814904/9084226096404750", + "3184319397227633/6537363710836453" + ], + "answer": "4735064900814904/9084226096404750", + "correct_value": 0.521240318169634, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 4280842766102084/8790908275021653 or 1649883149166605/3208234898409106?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.027302560588856217, + "pure_math": { + "question": "Which is larger: 4280842766102084/8790908275021653 or 1649883149166605/3208234898409106?", + "options": [ + "907394138304993/1739573698928892", + "4280842766102084/8790908275021653", + "1474013443098467/3418804474348912", + "1649883149166605/3208234898409106" + ], + "answer": "1649883149166605/3208234898409106", + "correct_value": 0.514265071421281, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 858876027344172/1561592776989402 or 976992479307539/1684469791909549?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 858876027344172/1561592776989402 or 976992479307539/1684469791909549?", + "options": [ + "2656181287220544/4424188846685126", + "976992479307539/1684469791909549", + "3275163012097004/4832456372172441", + "858876027344172/1561592776989402" + ], + "answer": "976992479307539/1684469791909549", + "correct_value": 0.5800000000000003, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001337", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 3357867569362663/6896901737119165 or 3508689182343676/6854721606247319?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.024998489820744763, + "pure_math": { + "question": "Which is larger: 3357867569362663/6896901737119165 or 3508689182343676/6854721606247319?", + "options": [ + "3508689182343676/6854721606247319", + "3357867569362663/6896901737119165", + "3113051997313280/5985360217372915", + "3937898031415452/7008676083650705" + ], + "answer": "3508689182343676/6854721606247319", + "correct_value": 0.5118645780079376, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 3918171302961720/8043202777600170 or 1537694615852214/3031369449113607?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.020120024910669054, + "pure_math": { + "question": "Which is larger: 3918171302961720/8043202777600170 or 1537694615852214/3031369449113607?", + "options": [ + "1537694615852214/3031369449113607", + "4069095967496376/7697547514485231", + "4406720150529692/9765137645209172", + "3918171302961720/8043202777600170" + ], + "answer": "1537694615852214/3031369449113607", + "correct_value": 0.5072607089518061, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 3965166756125506/7209394102046374 or 3985850659649933/6872156309741264?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 3965166756125506/7209394102046374 or 3985850659649933/6872156309741264?", + "options": [ + "3985850659649933/6872156309741264", + "3965166756125506/7209394102046374", + "1098279609546152/2274623490065534", + "1236082494589244/2008505140825774" + ], + "answer": "3985850659649933/6872156309741264", + "correct_value": 0.58, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001338", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 868845075584592/1833935462398168 or 2830049248898475/5519126244147128?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03901135052470528, + "pure_math": { + "question": "Which is larger: 868845075584592/1833935462398168 or 2830049248898475/5519126244147128?", + "options": [ + "2457027297834989/5222233173903642", + "2163939436660623/3879160627252800", + "2830049248898475/5519126244147128", + "868845075584592/1833935462398168" + ], + "answer": "2830049248898475/5519126244147128", + "correct_value": 0.5127712474233869, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 3025658439119534/6159427615997753 or 4576296843995681/9047154908565248?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.014603216533017382, + "pure_math": { + "question": "Which is larger: 3025658439119534/6159427615997753 or 4576296843995681/9047154908565248?", + "options": [ + "2316060690529954/4035889938519996", + "3259067151437114/6550281156545806", + "3025658439119534/6159427615997753", + "4576296843995681/9047154908565248" + ], + "answer": "4576296843995681/9047154908565248", + "correct_value": 0.505827178847479, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 1304687838849693/2372159706999441 or 3684375518214571/6352371583128570?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 1304687838849693/2372159706999441 or 3684375518214571/6352371583128570?", + "options": [ + "5217493705867827/9439566958230191", + "3684375518214571/6352371583128570", + "1161653333803923/1854000496766655", + "1304687838849693/2372159706999441" + ], + "answer": "3684375518214571/6352371583128570", + "correct_value": 0.5800000000000001, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001339", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 1764942955006914/3609737891832292 or 1501284997370891/2853901654670184?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03710714156055317, + "pure_math": { + "question": "Which is larger: 1764942955006914/3609737891832292 or 1501284997370891/2853901654670184?", + "options": [ + "2512882737604272/5493221901481943", + "5869632320999520/9404599401982372", + "1764942955006914/3609737891832292", + "1501284997370891/2853901654670184" + ], + "answer": "1501284997370891/2853901654670184", + "correct_value": 0.5260465072137847, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 3610604033953450/7294690953850435 or 3482223572809720/6824231230581686?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.015310122834740314, + "pure_math": { + "question": "Which is larger: 3610604033953450/7294690953850435 or 3482223572809720/6824231230581686?", + "options": [ + "3482223572809720/6824231230581686", + "3515815408424384/5940989860873307", + "1889832051803909/3765293520207879", + "3610604033953450/7294690953850435" + ], + "answer": "3482223572809720/6824231230581686", + "correct_value": 0.5102733854041609, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 1649492839039788/2999077889163250 or 5738162613041919/9893383815589517?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 1649492839039788/2999077889163250 or 5738162613041919/9893383815589517?", + "options": [ + "5738162613041919/9893383815589517", + "3448778944830642/5244954294335088", + "4401721037423114/8187151841345460", + "1649492839039788/2999077889163250" + ], + "answer": "5738162613041919/9893383815589517", + "correct_value": 0.58, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001340", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 4346702105191448/9061446599882290 or 3223360345624141/6244385429075352?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03650952274721364, + "pure_math": { + "question": "Which is larger: 4346702105191448/9061446599882290 or 3223360345624141/6244385429075352?", + "options": [ + "3223360345624141/6244385429075352", + "3502200693007688/5815011153392818", + "3597839034805179/6201464759292738", + "4346702105191448/9061446599882290" + ], + "answer": "3223360345624141/6244385429075352", + "correct_value": 0.5162013751770357, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 3477685179767593/7025794002303377 or 2069756507867359/4050048134799183?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.01605670821560251, + "pure_math": { + "question": "Which is larger: 3477685179767593/7025794002303377 or 2069756507867359/4050048134799183?", + "options": [ + "1944551476003737/3197434332903640", + "3948955896284518/7752868886091889", + "2069756507867359/4050048134799183", + "3477685179767593/7025794002303377" + ], + "answer": "2069756507867359/4050048134799183", + "correct_value": 0.5110449157587569, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 4857509660182905/8831835745787100 or 4980380569899551/8586863051550951?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 4857509660182905/8831835745787100 or 4980380569899551/8586863051550951?", + "options": [ + "4857509660182905/8831835745787100", + "4980380569899551/8586863051550951", + "3461399778938974/6486995465200153", + "3716951044735863/6536425135240954" + ], + "answer": "4980380569899551/8586863051550951", + "correct_value": 0.58, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001341", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 1020076506990537/2122399223857063 or 932765900007092/1757835664258900?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.05000892709898208, + "pure_math": { + "question": "Which is larger: 1020076506990537/2122399223857063 or 932765900007092/1757835664258900?", + "options": [ + "5189866683564466/8476172031212809", + "2514334797134233/4143770615781618", + "932765900007092/1757835664258900", + "1020076506990537/2122399223857063" + ], + "answer": "932765900007092/1757835664258900", + "correct_value": 0.5306331638232771, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 4882122712277066/9949157996743252 or 2793273194906070/5468860323376772?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.0200525273221292, + "pure_math": { + "question": "Which is larger: 4882122712277066/9949157996743252 or 2793273194906070/5468860323376772?", + "options": [ + "4573935773120604/9851389365326994", + "2793273194906070/5468860323376772", + "4087262072370951/8283477726814000", + "4882122712277066/9949157996743252" + ], + "answer": "2793273194906070/5468860323376772", + "correct_value": 0.5107596518722847, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 3735413737163654/6791661340297551 or 2444359227159502/4214412460619831?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 3735413737163654/6791661340297551 or 2444359227159502/4214412460619831?", + "options": [ + "3178052751053222/5990155524625409", + "2444359227159502/4214412460619831", + "3451430251610944/5853556535967252", + "3735413737163654/6791661340297551" + ], + "answer": "2444359227159502/4214412460619831", + "correct_value": 0.58, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001342", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 3842046505581364/8014540785646586 or 2079021598737597/4074142207125622?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.030912282707871108, + "pure_math": { + "question": "Which is larger: 3842046505581364/8014540785646586 or 2079021598737597/4074142207125622?", + "options": [ + "2546441418693976/4211496326731502", + "2079021598737597/4074142207125622", + "2571173230323056/5078702610485445", + "3842046505581364/8014540785646586" + ], + "answer": "2079021598737597/4074142207125622", + "correct_value": 0.5102967675260366, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 4566998352606383/9384341596401488 or 1837927661125912/3652609981355893?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.01652045454787282, + "pure_math": { + "question": "Which is larger: 4566998352606383/9384341596401488 or 1837927661125912/3652609981355893?", + "options": [ + "1837927661125912/3652609981355893", + "4628647679015465/9023928657999394", + "1430096137084976/3048819136382965", + "4566998352606383/9384341596401488" + ], + "answer": "1837927661125912/3652609981355893", + "correct_value": 0.5031820179289033, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 3433027144391820/6241867535257853 or 5279657858942153/9102858377486473?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 3433027144391820/6241867535257853 or 5279657858942153/9102858377486473?", + "options": [ + "5279657858942153/9102858377486473", + "3433027144391820/6241867535257853", + "5410745186934916/9203139415666231", + "5820333535559982/9885994348382550" + ], + "answer": "5279657858942153/9102858377486473", + "correct_value": 0.5799999999999998, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001343", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 2699181860479841/5570746350276855 or 2260496818825161/4309799787303831?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.039973715375536956, + "pure_math": { + "question": "Which is larger: 2699181860479841/5570746350276855 or 2260496818825161/4309799787303831?", + "options": [ + "2195712951693809/4851865973747759", + "2260496818825161/4309799787303831", + "2699181860479841/5570746350276855", + "2526950718672560/5804439968004486" + ], + "answer": "2260496818825161/4309799787303831", + "correct_value": 0.5245015848495612, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 2411359745650468/4906083988565643 or 892614852989565/1762269871574573?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.015010311186909653, + "pure_math": { + "question": "Which is larger: 2411359745650468/4906083988565643 or 892614852989565/1762269871574573?", + "options": [ + "801296250844713/1637655130544094", + "892614852989565/1762269871574573", + "3548756253224950/6813772661910614", + "2411359745650468/4906083988565643" + ], + "answer": "892614852989565/1762269871574573", + "correct_value": 0.506514278764819, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 848228667648923/1542233941179859 or 1471728980427774/2537463759358231?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 848228667648923/1542233941179859 or 1471728980427774/2537463759358231?", + "options": [ + "1471728980427774/2537463759358231", + "912475414456964/1450986385541645", + "2873846704156870/4674876294853463", + "848228667648923/1542233941179859" + ], + "answer": "1471728980427774/2537463759358231", + "correct_value": 0.58, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001344", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 3621708825890507/7452671663685288 or 1492794305435217/2890616693142384?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.030466481071862173, + "pure_math": { + "question": "Which is larger: 3621708825890507/7452671663685288 or 1492794305435217/2890616693142384?", + "options": [ + "3621708825890507/7452671663685288", + "3262680298056836/5922730934314292", + "1492794305435217/2890616693142384", + "2813884828699646/5983305954626777" + ], + "answer": "1492794305435217/2890616693142384", + "correct_value": 0.5164276221668128, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 2952429997147698/6049155652411894 or 2857179685663812/5625730634265959?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.019804012344002753, + "pure_math": { + "question": "Which is larger: 2952429997147698/6049155652411894 or 2857179685663812/5625730634265959?", + "options": [ + "2952429997147698/6049155652411894", + "4468144572402968/9328792895390672", + "2278259587298962/5314973427394324", + "2857179685663812/5625730634265959" + ], + "answer": "2857179685663812/5625730634265959", + "correct_value": 0.5078770868020087, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 4159767551635229/7563213730245870 or 1660393523062978/2862747453556858?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 4159767551635229/7563213730245870 or 1660393523062978/2862747453556858?", + "options": [ + "1660393523062978/2862747453556858", + "3072491440442170/4733018207296292", + "5231864442496922/8031614635696683", + "4159767551635229/7563213730245870" + ], + "answer": "1660393523062978/2862747453556858", + "correct_value": 0.5800000000000001, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001345", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 4017905752275166/8280230891777500 or 3551818234881982/6954246519204120?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.025500128915934617, + "pure_math": { + "question": "Which is larger: 4017905752275166/8280230891777500 or 3551818234881982/6954246519204120?", + "options": [ + "3384677958801852/7237252955759563", + "4259549853349834/8597784492212904", + "3551818234881982/6954246519204120", + "4017905752275166/8280230891777500" + ], + "answer": "3551818234881982/6954246519204120", + "correct_value": 0.510740915651128, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 1702967706089885/3452247204125610 or 4513505898424737/8832293174558281?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.01773059171617042, + "pure_math": { + "question": "Which is larger: 1702967706089885/3452247204125610 or 4513505898424737/8832293174558281?", + "options": [ + "4513505898424737/8832293174558281", + "4593299210122059/8454688449273046", + "2661428196781989/5967407350846953", + "1702967706089885/3452247204125610" + ], + "answer": "4513505898424737/8832293174558281", + "correct_value": 0.5110231068219115, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 1006131235611399/1829329519293452 or 5723939706188107/9868861562393289?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 1006131235611399/1829329519293452 or 5723939706188107/9868861562393289?", + "options": [ + "5723939706188107/9868861562393289", + "2188054346681618/3340912773128747", + "1006131235611399/1829329519293452", + "3544117965402570/7266390895576074" + ], + "answer": "5723939706188107/9868861562393289", + "correct_value": 0.58, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001346", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 1826661504330556/3750083738054417 or 1889569696386313/3662348912254606?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.0288458931676861, + "pure_math": { + "question": "Which is larger: 1826661504330556/3750083738054417 or 1889569696386313/3662348912254606?", + "options": [ + "1826661504330556/3750083738054417", + "1889569696386313/3662348912254606", + "5499674714660168/9598858562068483", + "4992537151528782/9357006574357913" + ], + "answer": "1889569696386313/3662348912254606", + "correct_value": 0.5159447506663315, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 3091795815947612/6319241858847855 or 3286129086726898/6490623644348618?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.017021759928805613, + "pure_math": { + "question": "Which is larger: 3091795815947612/6319241858847855 or 3286129086726898/6490623644348618?", + "options": [ + "3286129086726898/6490623644348618", + "3564361663965602/7638636160073444", + "3181952539084172/6815574434214378", + "3091795815947612/6319241858847855" + ], + "answer": "3286129086726898/6490623644348618", + "correct_value": 0.506288650642705, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 682109100591654/1240198364712097 or 2588761817061519/4463382443209516?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 682109100591654/1240198364712097 or 2588761817061519/4463382443209516?", + "options": [ + "931714019916183/1519752294519979", + "4072118219743160/6398357034550668", + "682109100591654/1240198364712097", + "2588761817061519/4463382443209516" + ], + "answer": "2588761817061519/4463382443209516", + "correct_value": 0.58, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001347", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 1096250179524652/2296558316495842 or 1784808749420023/3400676061909729?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.0474945858257278, + "pure_math": { + "question": "Which is larger: 1096250179524652/2296558316495842 or 1784808749420023/3400676061909729?", + "options": [ + "3036772285432694/5436869902447668", + "1096250179524652/2296558316495842", + "802898003638481/1851625148795914", + "1784808749420023/3400676061909729" + ], + "answer": "1784808749420023/3400676061909729", + "correct_value": 0.5248393898528877, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 699403959181126/1406969445565431 or 4744875408322916/9407134962829650?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.007291519755757092, + "pure_math": { + "question": "Which is larger: 699403959181126/1406969445565431 or 4744875408322916/9407134962829650?", + "options": [ + "699403959181126/1406969445565431", + "1425867556058326/2406777180737023", + "2553390980863545/5211276509125470", + "4744875408322916/9407134962829650" + ], + "answer": "4744875408322916/9407134962829650", + "correct_value": 0.5043911272742775, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 3866653637213065/7030279340387390 or 4671024546168615/8053490596842440?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 3866653637213065/7030279340387390 or 4671024546168615/8053490596842440?", + "options": [ + "1833486152750768/3450457265323523", + "4671024546168615/8053490596842440", + "3866653637213065/7030279340387390", + "3331074583081438/6078826257788332" + ], + "answer": "4671024546168615/8053490596842440", + "correct_value": 0.58, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001348", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 2298717635817654/4805662446333514 or 5001350293010689/9725754866200039?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.035902522616023225, + "pure_math": { + "question": "Which is larger: 2298717635817654/4805662446333514 or 5001350293010689/9725754866200039?", + "options": [ + "5001350293010689/9725754866200039", + "5799553835734272/9540859078332400", + "3818012474684758/8057882924126315", + "2298717635817654/4805662446333514" + ], + "answer": "5001350293010689/9725754866200039", + "correct_value": 0.5142377493382961, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 3347449571050627/6806356823192244 or 2715721504533959/5332752313458164?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.01744102436029371, + "pure_math": { + "question": "Which is larger: 3347449571050627/6806356823192244 or 2715721504533959/5332752313458164?", + "options": [ + "2715721504533959/5332752313458164", + "3859215684066802/6656396771908862", + "1517759020020946/3193750157580736", + "3347449571050627/6806356823192244" + ], + "answer": "2715721504533959/5332752313458164", + "correct_value": 0.5092532607749932, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 4883093768562807/8878352306477831 or 3905760067025311/6734069081078122?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 4883093768562807/8878352306477831 or 3905760067025311/6734069081078122?", + "options": [ + "4883093768562807/8878352306477831", + "3709512976487030/5843888615522402", + "5237938060975568/9998715313347575", + "3905760067025311/6734069081078122" + ], + "answer": "3905760067025311/6734069081078122", + "correct_value": 0.5800000000000001, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001349", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 1027571917473488/2152901084379104 or 3734325433337088/7289257027522810?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.035008967445614936, + "pure_math": { + "question": "Which is larger: 1027571917473488/2152901084379104 or 3734325433337088/7289257027522810?", + "options": [ + "1027571917473488/2152901084379104", + "4529271578469340/9331157566977177", + "1392319536516942/2719151490458338", + "3734325433337088/7289257027522810" + ], + "answer": "3734325433337088/7289257027522810", + "correct_value": 0.5123053583152584, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 557411526323285/1130709848730534 or 3546689171745906/6899542062418101?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.02107218229689406, + "pure_math": { + "question": "Which is larger: 557411526323285/1130709848730534 or 3546689171745906/6899542062418101?", + "options": [ + "557411526323285/1130709848730534", + "3731970706143090/6113008614288064", + "1621094735634453/2672551080042966", + "3546689171745906/6899542062418101" + ], + "answer": "3546689171745906/6899542062418101", + "correct_value": 0.5140470395948117, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 5342268111289720/9713214747799490 or 2618287552669848/4514288883913531?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 5342268111289720/9713214747799490 or 2618287552669848/4514288883913531?", + "options": [ + "1925161218557222/3125537690952804", + "2618287552669848/4514288883913531", + "5342268111289720/9713214747799490", + "4722548331758751/9353302870040205" + ], + "answer": "2618287552669848/4514288883913531", + "correct_value": 0.58, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001350", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 3448731705750732/7182974452955876 or 3291552320887094/6360664293114701?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03735977369852833, + "pure_math": { + "question": "Which is larger: 3448731705750732/7182974452955876 or 3291552320887094/6360664293114701?", + "options": [ + "3448731705750732/7182974452955876", + "986241500587478/2176322891567777", + "3291552320887094/6360664293114701", + "5302167276349741/9154611338433432" + ], + "answer": "3291552320887094/6360664293114701", + "correct_value": 0.5174856224451488, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 4424126540464797/9053339842894726 or 4674456197221998/9140524968174877?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.022725675403263257, + "pure_math": { + "question": "Which is larger: 4424126540464797/9053339842894726 or 4674456197221998/9140524968174877?", + "options": [ + "3341409808678908/6179155380632538", + "4674456197221998/9140524968174877", + "4614244916286202/9993913919507941", + "4424126540464797/9053339842894726" + ], + "answer": "4674456197221998/9140524968174877", + "correct_value": 0.5113990950735693, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 860250341689244/1564091530344079 or 5469455236399968/9430095235172361?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 860250341689244/1564091530344079 or 5469455236399968/9430095235172361?", + "options": [ + "860250341689244/1564091530344079", + "5806163335720849/9450590292868555", + "5469455236399968/9430095235172361", + "6170480191789436/9488027472527746" + ], + "answer": "5469455236399968/9430095235172361", + "correct_value": 0.5799999999999998, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_001351", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 4080460046716636/8330135989229760 or 1615192380814155/3090076401254617?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.032859907584392556, + "pure_math": { + "question": "Which is larger: 4080460046716636/8330135989229760 or 1615192380814155/3090076401254617?", + "options": [ + "2716179641619403/4597548158142664", + "4080460046716636/8330135989229760", + "3086518073099383/6853099662633411", + "1615192380814155/3090076401254617" + ], + "answer": "1615192380814155/3090076401254617", + "correct_value": 0.5227030568429838, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 1696212746935578/3487303918873317 or 2140037932990052/4206847130692648?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.022306923920217547, + "pure_math": { + "question": "Which is larger: 1696212746935578/3487303918873317 or 2140037932990052/4206847130692648?", + "options": [ + "3224031964394206/5515766130666183", + "1696212746935578/3487303918873317", + "2140037932990052/4206847130692648", + "4048408288742299/7238870431254089" + ], + "answer": "2140037932990052/4206847130692648", + "correct_value": 0.5087035175052105, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 3053434746819050/5551699539670999 or 3586256826101075/6183201424312199?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 3053434746819050/5551699539670999 or 3586256826101075/6183201424312199?", + "options": [ + "4741776921689394/7992634183251650", + "2164937090226504/3626730178380152", + "3586256826101075/6183201424312199", + "3053434746819050/5551699539670999" + ], + "answer": "3586256826101075/6183201424312199", + "correct_value": 0.58, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "cancellation_identity_001352", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9353983493802460 + 5630721528224774 - 5630721528224775", + "ns_shortcut": "Near-cancellation: 5630721528224774 - 5630721528224775 = -1, so answer ≈ 9353983493802460 + -1.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "Evaluate: 9353983493802460 + 5630721528224774 - 5630721528224775", + "options": [ + "9353983493802469", + "9353983493802459", + "9353983493802449", + "9353983493802479" + ], + "answer": "9353983493802459", + "correct_value": 9353983493802459, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6932464695698539 + 8723964766500443 - 8380847847347378", + "ns_shortcut": "Near-cancellation: 8723964766500443 - 8380847847347378 = 343116919153065, offset is manageable.", + "shortcut_strength": 0.9606696120013443, + "pure_math": { + "question": "Evaluate: 6932464695698539 + 8723964766500443 - 8380847847347378", + "options": [ + "7275581614851604", + "7275581614851624", + "7275581614851614", + "7275581614851594" + ], + "answer": "7275581614851604", + "correct_value": 7275581614851604, + "explanation": "B-C = 343116919153065; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9353983493802460 + 5630721528224774 - 8551746009954387", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 9353983493802460 + 5630721528224774 - 8551746009954387", + "options": [ + "6432959012072847", + "6432959012072867", + "6432959012072837", + "6432959012072857" + ], + "answer": "6432959012072847", + "correct_value": 6432959012072847, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001353", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4910714678897393 + 4527555332700942 - 4527555332700941", + "ns_shortcut": "Near-cancellation: 4527555332700942 - 4527555332700941 = 1, so answer ≈ 4910714678897393 + 1.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "Evaluate: 4910714678897393 + 4527555332700942 - 4527555332700941", + "options": [ + "4910714678897404", + "4910714678897384", + "4910714678897394", + "4910714678897414" + ], + "answer": "4910714678897394", + "correct_value": 4910714678897394, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6117261882326184 + 6172180626665470 - 6449815536146337", + "ns_shortcut": "Near-cancellation: 6172180626665470 - 6449815536146337 = -277634909480867, offset is manageable.", + "shortcut_strength": 0.9550183433904365, + "pure_math": { + "question": "Evaluate: 6117261882326184 + 6172180626665470 - 6449815536146337", + "options": [ + "5839626972845317", + "5839626972845337", + "5839626972845327", + "5839626972845307" + ], + "answer": "5839626972845317", + "correct_value": 5839626972845317, + "explanation": "B-C = -277634909480867; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4910714678897393 + 4527555332700942 - 2108691550966365", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 4910714678897393 + 4527555332700942 - 2108691550966365", + "options": [ + "7329578460631980", + "7329578460631960", + "7329578460631970", + "7329578460631990" + ], + "answer": "7329578460631970", + "correct_value": 7329578460631970, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001354", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8960759563483501 + 4679561405662685 - 4679561405662686", + "ns_shortcut": "Near-cancellation: 4679561405662685 - 4679561405662686 = -1, so answer ≈ 8960759563483501 + -1.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "Evaluate: 8960759563483501 + 4679561405662685 - 4679561405662686", + "options": [ + "8960759563483490", + "8960759563483500", + "8960759563483510", + "8960759563483520" + ], + "answer": "8960759563483500", + "correct_value": 8960759563483500, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1131468034302009 + 6652232431651857 - 6927018998142107", + "ns_shortcut": "Near-cancellation: 6652232431651857 - 6927018998142107 = -274786566490250, offset is manageable.", + "shortcut_strength": 0.9586925788728017, + "pure_math": { + "question": "Evaluate: 1131468034302009 + 6652232431651857 - 6927018998142107", + "options": [ + "856681467811779", + "856681467811769", + "856681467811749", + "856681467811759" + ], + "answer": "856681467811759", + "correct_value": 856681467811759, + "explanation": "B-C = -274786566490250; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8960759563483501 + 4679561405662685 - 6136976713782523", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 8960759563483501 + 4679561405662685 - 6136976713782523", + "options": [ + "7503344255363653", + "7503344255363673", + "7503344255363663", + "7503344255363683" + ], + "answer": "7503344255363663", + "correct_value": 7503344255363663, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001355", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7491770029949762 + 6367214608309250 - 6367214608309251", + "ns_shortcut": "Near-cancellation: 6367214608309250 - 6367214608309251 = -1, so answer ≈ 7491770029949762 + -1.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "Evaluate: 7491770029949762 + 6367214608309250 - 6367214608309251", + "options": [ + "7491770029949761", + "7491770029949751", + "7491770029949771", + "7491770029949781" + ], + "answer": "7491770029949761", + "correct_value": 7491770029949761, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2435945251662927 + 4700585321318475 - 4876028297576304", + "ns_shortcut": "Near-cancellation: 4700585321318475 - 4876028297576304 = -175442976257829, offset is manageable.", + "shortcut_strength": 0.9626763553334208, + "pure_math": { + "question": "Evaluate: 2435945251662927 + 4700585321318475 - 4876028297576304", + "options": [ + "2260502275405088", + "2260502275405108", + "2260502275405118", + "2260502275405098" + ], + "answer": "2260502275405098", + "correct_value": 2260502275405098, + "explanation": "B-C = -175442976257829; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7491770029949762 + 6367214608309250 - 4069256424506097", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 7491770029949762 + 6367214608309250 - 4069256424506097", + "options": [ + "9789728213752905", + "9789728213752925", + "9789728213752935", + "9789728213752915" + ], + "answer": "9789728213752915", + "correct_value": 9789728213752915, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_001356", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3047125066342880 + 7650052116765667 - 7650052116765665", + "ns_shortcut": "Near-cancellation: 7650052116765667 - 7650052116765665 = 2, so answer ≈ 3047125066342880 + 2.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "Evaluate: 3047125066342880 + 7650052116765667 - 7650052116765665", + "options": [ + "3047125066342882", + "3047125066342872", + "3047125066342902", + "3047125066342892" + ], + "answer": "3047125066342882", + "correct_value": 3047125066342882, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4097188869600102 + 3822104829843905 - 3715016059969892", + "ns_shortcut": "Near-cancellation: 3822104829843905 - 3715016059969892 = 107088769874013, offset is manageable.", + "shortcut_strength": 0.971981728748558, + "pure_math": { + "question": "Evaluate: 4097188869600102 + 3822104829843905 - 3715016059969892", + "options": [ + "4204277639474105", + "4204277639474135", + "4204277639474115", + "4204277639474125" + ], + "answer": "4204277639474115", + "correct_value": 4204277639474115, + "explanation": "B-C = 107088769874013; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3047125066342880 + 7650052116765667 - 2822989013939863", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 3047125066342880 + 7650052116765667 - 2822989013939863", + "options": [ + "7874188169168704", + "7874188169168674", + "7874188169168694", + "7874188169168684" + ], + "answer": "7874188169168684", + "correct_value": 7874188169168684, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_001357", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6035876363532134 + 4163007159408050 - 4163007159408052", + "ns_shortcut": "Near-cancellation: 4163007159408050 - 4163007159408052 = -2, so answer ≈ 6035876363532134 + -2.", + "shortcut_strength": 0.9999999999999996, + "pure_math": { + "question": "Evaluate: 6035876363532134 + 4163007159408050 - 4163007159408052", + "options": [ + "6035876363532142", + "6035876363532122", + "6035876363532132", + "6035876363532152" + ], + "answer": "6035876363532132", + "correct_value": 6035876363532132, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9062968696989280 + 4468478095973296 - 4570133501018174", + "ns_shortcut": "Near-cancellation: 4468478095973296 - 4570133501018174 = -101655405044878, offset is manageable.", + "shortcut_strength": 0.9772505531275887, + "pure_math": { + "question": "Evaluate: 9062968696989280 + 4468478095973296 - 4570133501018174", + "options": [ + "8961313291944402", + "8961313291944392", + "8961313291944412", + "8961313291944422" + ], + "answer": "8961313291944402", + "correct_value": 8961313291944402, + "explanation": "B-C = -101655405044878; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6035876363532134 + 4163007159408050 - 1647854337727897", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 6035876363532134 + 4163007159408050 - 1647854337727897", + "options": [ + "8551029185212277", + "8551029185212297", + "8551029185212287", + "8551029185212307" + ], + "answer": "8551029185212287", + "correct_value": 8551029185212287, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001358", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8968831756244665 + 2254422786222179 - 2254422786222177", + "ns_shortcut": "Near-cancellation: 2254422786222179 - 2254422786222177 = 2, so answer ≈ 8968831756244665 + 2.", + "shortcut_strength": 0.9999999999999991, + "pure_math": { + "question": "Evaluate: 8968831756244665 + 2254422786222179 - 2254422786222177", + "options": [ + "8968831756244677", + "8968831756244667", + "8968831756244657", + "8968831756244687" + ], + "answer": "8968831756244667", + "correct_value": 8968831756244667, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7118494637116223 + 7566864667992923 - 7917281360912575", + "ns_shortcut": "Near-cancellation: 7566864667992923 - 7917281360912575 = -350416692919652, offset is manageable.", + "shortcut_strength": 0.9536906356470363, + "pure_math": { + "question": "Evaluate: 7118494637116223 + 7566864667992923 - 7917281360912575", + "options": [ + "6768077944196561", + "6768077944196591", + "6768077944196571", + "6768077944196581" + ], + "answer": "6768077944196571", + "correct_value": 6768077944196571, + "explanation": "B-C = -350416692919652; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8968831756244665 + 2254422786222179 - 1452360350818663", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 8968831756244665 + 2254422786222179 - 1452360350818663", + "options": [ + "9770894191648181", + "9770894191648171", + "9770894191648201", + "9770894191648191" + ], + "answer": "9770894191648181", + "correct_value": 9770894191648181, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001359", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1262944667178215 + 9950983607937084 - 9950983607937085", + "ns_shortcut": "Near-cancellation: 9950983607937084 - 9950983607937085 = -1, so answer ≈ 1262944667178215 + -1.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "Evaluate: 1262944667178215 + 9950983607937084 - 9950983607937085", + "options": [ + "1262944667178234", + "1262944667178214", + "1262944667178224", + "1262944667178204" + ], + "answer": "1262944667178214", + "correct_value": 1262944667178214, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1825069733510673 + 3337740315817758 - 3462818393281205", + "ns_shortcut": "Near-cancellation: 3337740315817758 - 3462818393281205 = -125078077463447, offset is manageable.", + "shortcut_strength": 0.96252612077977, + "pure_math": { + "question": "Evaluate: 1825069733510673 + 3337740315817758 - 3462818393281205", + "options": [ + "1699991656047226", + "1699991656047246", + "1699991656047216", + "1699991656047236" + ], + "answer": "1699991656047226", + "correct_value": 1699991656047226, + "explanation": "B-C = -125078077463447; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1262944667178215 + 9950983607937084 - 15979112010600517", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 1262944667178215 + 9950983607937084 - 15979112010600517", + "options": [ + "-4765183735485218", + "-4765183735485232", + "-4765183735485202", + "-4765183735485212" + ], + "answer": "-4765183735485218", + "correct_value": -4765183735485218, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001360", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7232083922624080 + 5715921555175029 - 5715921555175028", + "ns_shortcut": "Near-cancellation: 5715921555175029 - 5715921555175028 = 1, so answer ≈ 7232083922624080 + 1.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "Evaluate: 7232083922624080 + 5715921555175029 - 5715921555175028", + "options": [ + "7232083922624091", + "7232083922624101", + "7232083922624081", + "7232083922624071" + ], + "answer": "7232083922624081", + "correct_value": 7232083922624081, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1049152041646239 + 5267674693859319 - 5426348742776347", + "ns_shortcut": "Near-cancellation: 5267674693859319 - 5426348742776347 = -158674048917028, offset is manageable.", + "shortcut_strength": 0.969877781347812, + "pure_math": { + "question": "Evaluate: 1049152041646239 + 5267674693859319 - 5426348742776347", + "options": [ + "890477992729211", + "890477992729231", + "890477992729221", + "890477992729201" + ], + "answer": "890477992729211", + "correct_value": 890477992729211, + "explanation": "B-C = -158674048917028; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7232083922624080 + 5715921555175029 - 8853656161363979", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 7232083922624080 + 5715921555175029 - 8853656161363979", + "options": [ + "4094349316435140", + "4094349316435120", + "4094349316435150", + "4094349316435130" + ], + "answer": "4094349316435130", + "correct_value": 4094349316435130, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_001361", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3091024518766016 + 1844576646325284 - 1844576646325286", + "ns_shortcut": "Near-cancellation: 1844576646325284 - 1844576646325286 = -2, so answer ≈ 3091024518766016 + -2.", + "shortcut_strength": 0.9999999999999989, + "pure_math": { + "question": "Evaluate: 3091024518766016 + 1844576646325284 - 1844576646325286", + "options": [ + "3091024518766014", + "3091024518766024", + "3091024518766034", + "3091024518766004" + ], + "answer": "3091024518766014", + "correct_value": 3091024518766014, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7180261019784450 + 2148455851236618 - 2206230806703453", + "ns_shortcut": "Near-cancellation: 2148455851236618 - 2206230806703453 = -57774955466835, offset is manageable.", + "shortcut_strength": 0.9731086140617781, + "pure_math": { + "question": "Evaluate: 7180261019784450 + 2148455851236618 - 2206230806703453", + "options": [ + "7122486064317605", + "7122486064317625", + "7122486064317615", + "7122486064317635" + ], + "answer": "7122486064317615", + "correct_value": 7122486064317615, + "explanation": "B-C = -57774955466835; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3091024518766016 + 1844576646325284 - 2908834662791459", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 3091024518766016 + 1844576646325284 - 2908834662791459", + "options": [ + "2026766502299861", + "2026766502299851", + "2026766502299841", + "2026766502299831" + ], + "answer": "2026766502299841", + "correct_value": 2026766502299841, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001362", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5898260678145156 + 1969656921692371 - 1969656921692369", + "ns_shortcut": "Near-cancellation: 1969656921692371 - 1969656921692369 = 2, so answer ≈ 5898260678145156 + 2.", + "shortcut_strength": 0.999999999999999, + "pure_math": { + "question": "Evaluate: 5898260678145156 + 1969656921692371 - 1969656921692369", + "options": [ + "5898260678145168", + "5898260678145178", + "5898260678145158", + "5898260678145148" + ], + "answer": "5898260678145158", + "correct_value": 5898260678145158, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1258754971128922 + 8791475070802598 - 8606538127542485", + "ns_shortcut": "Near-cancellation: 8791475070802598 - 8606538127542485 = 184936943260113, offset is manageable.", + "shortcut_strength": 0.9789640598681434, + "pure_math": { + "question": "Evaluate: 1258754971128922 + 8791475070802598 - 8606538127542485", + "options": [ + "1443691914389035", + "1443691914389045", + "1443691914389055", + "1443691914389025" + ], + "answer": "1443691914389035", + "correct_value": 1443691914389035, + "explanation": "B-C = 184936943260113; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5898260678145156 + 1969656921692371 - 2658362626388779", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 5898260678145156 + 1969656921692371 - 2658362626388779", + "options": [ + "5209554973448748", + "5209554973448768", + "5209554973448758", + "5209554973448738" + ], + "answer": "5209554973448748", + "correct_value": 5209554973448748, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001363", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7502420366077232 + 8831652800854053 - 8831652800854054", + "ns_shortcut": "Near-cancellation: 8831652800854053 - 8831652800854054 = -1, so answer ≈ 7502420366077232 + -1.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "Evaluate: 7502420366077232 + 8831652800854053 - 8831652800854054", + "options": [ + "7502420366077251", + "7502420366077231", + "7502420366077241", + "7502420366077221" + ], + "answer": "7502420366077231", + "correct_value": 7502420366077231, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4406749603386068 + 3433358178924696 - 3303039207991090", + "ns_shortcut": "Near-cancellation: 3433358178924696 - 3303039207991090 = 130318970933606, offset is manageable.", + "shortcut_strength": 0.962043292851426, + "pure_math": { + "question": "Evaluate: 4406749603386068 + 3433358178924696 - 3303039207991090", + "options": [ + "4537068574319664", + "4537068574319684", + "4537068574319694", + "4537068574319674" + ], + "answer": "4537068574319674", + "correct_value": 4537068574319674, + "explanation": "B-C = 130318970933606; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7502420366077232 + 8831652800854053 - 12014950909439814", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 7502420366077232 + 8831652800854053 - 12014950909439814", + "options": [ + "4319122257491481", + "4319122257491491", + "4319122257491471", + "4319122257491461" + ], + "answer": "4319122257491471", + "correct_value": 4319122257491471, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001364", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6597401479046184 + 1533322980861231 - 1533322980861229", + "ns_shortcut": "Near-cancellation: 1533322980861231 - 1533322980861229 = 2, so answer ≈ 6597401479046184 + 2.", + "shortcut_strength": 0.9999999999999987, + "pure_math": { + "question": "Evaluate: 6597401479046184 + 1533322980861231 - 1533322980861229", + "options": [ + "6597401479046206", + "6597401479046176", + "6597401479046186", + "6597401479046196" + ], + "answer": "6597401479046186", + "correct_value": 6597401479046186, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8350627460268120 + 1272733644786462 - 1241234753080512", + "ns_shortcut": "Near-cancellation: 1272733644786462 - 1241234753080512 = 31498891705950, offset is manageable.", + "shortcut_strength": 0.9752509947112816, + "pure_math": { + "question": "Evaluate: 8350627460268120 + 1272733644786462 - 1241234753080512", + "options": [ + "8382126351974060", + "8382126351974090", + "8382126351974070", + "8382126351974080" + ], + "answer": "8382126351974070", + "correct_value": 8382126351974070, + "explanation": "B-C = 31498891705950; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6597401479046184 + 1533322980861231 - 2034843403417515", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 6597401479046184 + 1533322980861231 - 2034843403417515", + "options": [ + "6095881056489920", + "6095881056489910", + "6095881056489900", + "6095881056489890" + ], + "answer": "6095881056489900", + "correct_value": 6095881056489900, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001365", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1292608973276385 + 3971746765883373 - 3971746765883374", + "ns_shortcut": "Near-cancellation: 3971746765883373 - 3971746765883374 = -1, so answer ≈ 1292608973276385 + -1.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "Evaluate: 1292608973276385 + 3971746765883373 - 3971746765883374", + "options": [ + "1292608973276384", + "1292608973276374", + "1292608973276404", + "1292608973276394" + ], + "answer": "1292608973276384", + "correct_value": 1292608973276384, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9250212932742928 + 1094878449760473 - 1071046220038691", + "ns_shortcut": "Near-cancellation: 1094878449760473 - 1071046220038691 = 23832229721782, offset is manageable.", + "shortcut_strength": 0.9782329904045551, + "pure_math": { + "question": "Evaluate: 9250212932742928 + 1094878449760473 - 1071046220038691", + "options": [ + "9274045162464720", + "9274045162464730", + "9274045162464710", + "9274045162464700" + ], + "answer": "9274045162464710", + "correct_value": 9274045162464710, + "explanation": "B-C = 23832229721782; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1292608973276385 + 3971746765883373 - 1584479035761489", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 1292608973276385 + 3971746765883373 - 1584479035761489", + "options": [ + "3679876703398269", + "3679876703398279", + "3679876703398259", + "3679876703398289" + ], + "answer": "3679876703398269", + "correct_value": 3679876703398269, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001366", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9428576322309430 + 6922867663852684 - 6922867663852683", + "ns_shortcut": "Near-cancellation: 6922867663852684 - 6922867663852683 = 1, so answer ≈ 9428576322309430 + 1.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "Evaluate: 9428576322309430 + 6922867663852684 - 6922867663852683", + "options": [ + "9428576322309421", + "9428576322309451", + "9428576322309431", + "9428576322309441" + ], + "answer": "9428576322309431", + "correct_value": 9428576322309431, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1144964561939826 + 7485357783684748 - 7701007639517961", + "ns_shortcut": "Near-cancellation: 7485357783684748 - 7701007639517961 = -215649855833213, offset is manageable.", + "shortcut_strength": 0.971190441116489, + "pure_math": { + "question": "Evaluate: 1144964561939826 + 7485357783684748 - 7701007639517961", + "options": [ + "929314706106633", + "929314706106623", + "929314706106613", + "929314706106603" + ], + "answer": "929314706106613", + "correct_value": 929314706106613, + "explanation": "B-C = -215649855833213; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9428576322309430 + 6922867663852684 - 4637948985835889", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 9428576322309430 + 6922867663852684 - 4637948985835889", + "options": [ + "11713495000326215", + "11713495000326225", + "11713495000326245", + "11713495000326235" + ], + "answer": "11713495000326225", + "correct_value": 11713495000326225, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001367", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3218621651172795 + 2193585393187522 - 2193585393187520", + "ns_shortcut": "Near-cancellation: 2193585393187522 - 2193585393187520 = 2, so answer ≈ 3218621651172795 + 2.", + "shortcut_strength": 0.9999999999999991, + "pure_math": { + "question": "Evaluate: 3218621651172795 + 2193585393187522 - 2193585393187520", + "options": [ + "3218621651172797", + "3218621651172787", + "3218621651172807", + "3218621651172817" + ], + "answer": "3218621651172797", + "correct_value": 3218621651172797, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1803572945425034 + 7256356429152883 - 7088111589818837", + "ns_shortcut": "Near-cancellation: 7256356429152883 - 7088111589818837 = 168244839334046, offset is manageable.", + "shortcut_strength": 0.9768141434373163, + "pure_math": { + "question": "Evaluate: 1803572945425034 + 7256356429152883 - 7088111589818837", + "options": [ + "1971817784759100", + "1971817784759070", + "1971817784759090", + "1971817784759080" + ], + "answer": "1971817784759080", + "correct_value": 1971817784759080, + "explanation": "B-C = 168244839334046; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3218621651172795 + 2193585393187522 - 1205843842814193", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 3218621651172795 + 2193585393187522 - 1205843842814193", + "options": [ + "4206363201546124", + "4206363201546134", + "4206363201546114", + "4206363201546144" + ], + "answer": "4206363201546124", + "correct_value": 4206363201546124, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001368", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9064145529201681 + 5695786894384243 - 5695786894384244", + "ns_shortcut": "Near-cancellation: 5695786894384243 - 5695786894384244 = -1, so answer ≈ 9064145529201681 + -1.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "Evaluate: 9064145529201681 + 5695786894384243 - 5695786894384244", + "options": [ + "9064145529201670", + "9064145529201690", + "9064145529201680", + "9064145529201700" + ], + "answer": "9064145529201680", + "correct_value": 9064145529201680, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1841587699957485 + 5394487508839473 - 5175655578119812", + "ns_shortcut": "Near-cancellation: 5394487508839473 - 5175655578119812 = 218831930719661, offset is manageable.", + "shortcut_strength": 0.9594341574874202, + "pure_math": { + "question": "Evaluate: 1841587699957485 + 5394487508839473 - 5175655578119812", + "options": [ + "2060419630677136", + "2060419630677156", + "2060419630677146", + "2060419630677166" + ], + "answer": "2060419630677146", + "correct_value": 2060419630677146, + "explanation": "B-C = 218831930719661; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9064145529201681 + 5695786894384243 - 8493965405820755", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 9064145529201681 + 5695786894384243 - 8493965405820755", + "options": [ + "6265967017765179", + "6265967017765159", + "6265967017765189", + "6265967017765169" + ], + "answer": "6265967017765169", + "correct_value": 6265967017765169, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_001369", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1142310823752700 + 2324754350027802 - 2324754350027804", + "ns_shortcut": "Near-cancellation: 2324754350027802 - 2324754350027804 = -2, so answer ≈ 1142310823752700 + -2.", + "shortcut_strength": 0.9999999999999991, + "pure_math": { + "question": "Evaluate: 1142310823752700 + 2324754350027802 - 2324754350027804", + "options": [ + "1142310823752708", + "1142310823752698", + "1142310823752718", + "1142310823752688" + ], + "answer": "1142310823752698", + "correct_value": 1142310823752698, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9803016209186119 + 8949311547580064 - 9145349842732507", + "ns_shortcut": "Near-cancellation: 8949311547580064 - 9145349842732507 = -196038295152443, offset is manageable.", + "shortcut_strength": 0.9780945948623888, + "pure_math": { + "question": "Evaluate: 9803016209186119 + 8949311547580064 - 9145349842732507", + "options": [ + "9606977914033666", + "9606977914033696", + "9606977914033676", + "9606977914033686" + ], + "answer": "9606977914033676", + "correct_value": 9606977914033676, + "explanation": "B-C = -196038295152443; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1142310823752700 + 2324754350027802 - 1445243519684742", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 1142310823752700 + 2324754350027802 - 1445243519684742", + "options": [ + "2021821654095770", + "2021821654095780", + "2021821654095760", + "2021821654095750" + ], + "answer": "2021821654095760", + "correct_value": 2021821654095760, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001370", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1482564244448419 + 5534401978133828 - 5534401978133826", + "ns_shortcut": "Near-cancellation: 5534401978133828 - 5534401978133826 = 2, so answer ≈ 1482564244448419 + 2.", + "shortcut_strength": 0.9999999999999997, + "pure_math": { + "question": "Evaluate: 1482564244448419 + 5534401978133828 - 5534401978133826", + "options": [ + "1482564244448411", + "1482564244448421", + "1482564244448441", + "1482564244448431" + ], + "answer": "1482564244448421", + "correct_value": 1482564244448421, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5211999502552818 + 8497077976829595 - 8680881856976673", + "ns_shortcut": "Near-cancellation: 8497077976829595 - 8680881856976673 = -183803880147078, offset is manageable.", + "shortcut_strength": 0.9783685779219295, + "pure_math": { + "question": "Evaluate: 5211999502552818 + 8497077976829595 - 8680881856976673", + "options": [ + "5028195622405730", + "5028195622405750", + "5028195622405760", + "5028195622405740" + ], + "answer": "5028195622405740", + "correct_value": 5028195622405740, + "explanation": "B-C = -183803880147078; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1482564244448419 + 5534401978133828 - 7920395035450401", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 1482564244448419 + 5534401978133828 - 7920395035450401", + "options": [ + "-903428812868154", + "-903428812868166", + "-903428812868136", + "-903428812868146" + ], + "answer": "-903428812868154", + "correct_value": -903428812868154, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001371", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8007726343681213 + 2025996751533766 - 2025996751533768", + "ns_shortcut": "Near-cancellation: 2025996751533766 - 2025996751533768 = -2, so answer ≈ 8007726343681213 + -2.", + "shortcut_strength": 0.999999999999999, + "pure_math": { + "question": "Evaluate: 8007726343681213 + 2025996751533766 - 2025996751533768", + "options": [ + "8007726343681201", + "8007726343681221", + "8007726343681211", + "8007726343681231" + ], + "answer": "8007726343681211", + "correct_value": 8007726343681211, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7557425138598869 + 2527862048526318 - 2629100737794445", + "ns_shortcut": "Near-cancellation: 2527862048526318 - 2629100737794445 = -101238689268127, offset is manageable.", + "shortcut_strength": 0.9599508646735107, + "pure_math": { + "question": "Evaluate: 7557425138598869 + 2527862048526318 - 2629100737794445", + "options": [ + "7456186449330732", + "7456186449330762", + "7456186449330752", + "7456186449330742" + ], + "answer": "7456186449330742", + "correct_value": 7456186449330742, + "explanation": "B-C = -101238689268127; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8007726343681213 + 2025996751533766 - 2724968178798653", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 8007726343681213 + 2025996751533766 - 2724968178798653", + "options": [ + "7308754916416346", + "7308754916416316", + "7308754916416326", + "7308754916416336" + ], + "answer": "7308754916416326", + "correct_value": 7308754916416326, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001372", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3862450851421550 + 1093243959548208 - 1093243959548209", + "ns_shortcut": "Near-cancellation: 1093243959548208 - 1093243959548209 = -1, so answer ≈ 3862450851421550 + -1.", + "shortcut_strength": 0.9999999999999991, + "pure_math": { + "question": "Evaluate: 3862450851421550 + 1093243959548208 - 1093243959548209", + "options": [ + "3862450851421569", + "3862450851421559", + "3862450851421539", + "3862450851421549" + ], + "answer": "3862450851421549", + "correct_value": 3862450851421549, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8965705948669061 + 3141782249628854 - 3025476866879805", + "ns_shortcut": "Near-cancellation: 3141782249628854 - 3025476866879805 = 116305382749049, offset is manageable.", + "shortcut_strength": 0.9629810809572216, + "pure_math": { + "question": "Evaluate: 8965705948669061 + 3141782249628854 - 3025476866879805", + "options": [ + "9082011331418120", + "9082011331418110", + "9082011331418100", + "9082011331418130" + ], + "answer": "9082011331418110", + "correct_value": 9082011331418110, + "explanation": "B-C = 116305382749049; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3862450851421550 + 1093243959548208 - 1541153624055630", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 3862450851421550 + 1093243959548208 - 1541153624055630", + "options": [ + "3414541186914138", + "3414541186914118", + "3414541186914128", + "3414541186914148" + ], + "answer": "3414541186914128", + "correct_value": 3414541186914128, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001373", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3848154280890826 + 4600765302376846 - 4600765302376844", + "ns_shortcut": "Near-cancellation: 4600765302376846 - 4600765302376844 = 2, so answer ≈ 3848154280890826 + 2.", + "shortcut_strength": 0.9999999999999996, + "pure_math": { + "question": "Evaluate: 3848154280890826 + 4600765302376846 - 4600765302376844", + "options": [ + "3848154280890838", + "3848154280890848", + "3848154280890828", + "3848154280890818" + ], + "answer": "3848154280890828", + "correct_value": 3848154280890828, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4280455298074716 + 6410053414153669 - 6635532181248744", + "ns_shortcut": "Near-cancellation: 6410053414153669 - 6635532181248744 = -225478767095075, offset is manageable.", + "shortcut_strength": 0.9648241984072694, + "pure_math": { + "question": "Evaluate: 4280455298074716 + 6410053414153669 - 6635532181248744", + "options": [ + "4054976530979641", + "4054976530979631", + "4054976530979661", + "4054976530979651" + ], + "answer": "4054976530979641", + "correct_value": 4054976530979641, + "explanation": "B-C = -225478767095075; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3848154280890826 + 4600765302376846 - 6899925794043708", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 3848154280890826 + 4600765302376846 - 6899925794043708", + "options": [ + "1548993789223954", + "1548993789223964", + "1548993789223974", + "1548993789223984" + ], + "answer": "1548993789223964", + "correct_value": 1548993789223964, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001374", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7403970341146625 + 2316814310994107 - 2316814310994108", + "ns_shortcut": "Near-cancellation: 2316814310994107 - 2316814310994108 = -1, so answer ≈ 7403970341146625 + -1.", + "shortcut_strength": 0.9999999999999996, + "pure_math": { + "question": "Evaluate: 7403970341146625 + 2316814310994107 - 2316814310994108", + "options": [ + "7403970341146634", + "7403970341146624", + "7403970341146644", + "7403970341146614" + ], + "answer": "7403970341146624", + "correct_value": 7403970341146624, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1782830431497243 + 8519139115344043 - 8137311997997429", + "ns_shortcut": "Near-cancellation: 8519139115344043 - 8137311997997429 = 381827117346614, offset is manageable.", + "shortcut_strength": 0.9551800819100494, + "pure_math": { + "question": "Evaluate: 1782830431497243 + 8519139115344043 - 8137311997997429", + "options": [ + "2164657548843867", + "2164657548843857", + "2164657548843847", + "2164657548843877" + ], + "answer": "2164657548843857", + "correct_value": 2164657548843857, + "explanation": "B-C = 381827117346614; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7403970341146625 + 2316814310994107 - 1427528204429881", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 7403970341146625 + 2316814310994107 - 1427528204429881", + "options": [ + "8293256447710871", + "8293256447710841", + "8293256447710861", + "8293256447710851" + ], + "answer": "8293256447710851", + "correct_value": 8293256447710851, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_001375", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3202864002931054 + 4380372396287022 - 4380372396287024", + "ns_shortcut": "Near-cancellation: 4380372396287022 - 4380372396287024 = -2, so answer ≈ 3202864002931054 + -2.", + "shortcut_strength": 0.9999999999999996, + "pure_math": { + "question": "Evaluate: 3202864002931054 + 4380372396287022 - 4380372396287024", + "options": [ + "3202864002931042", + "3202864002931062", + "3202864002931072", + "3202864002931052" + ], + "answer": "3202864002931052", + "correct_value": 3202864002931052, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2473971916805469 + 4584160127129365 - 4778623548761421", + "ns_shortcut": "Near-cancellation: 4584160127129365 - 4778623548761421 = -194463421632056, offset is manageable.", + "shortcut_strength": 0.9575792694323201, + "pure_math": { + "question": "Evaluate: 2473971916805469 + 4584160127129365 - 4778623548761421", + "options": [ + "2279508495173433", + "2279508495173423", + "2279508495173403", + "2279508495173413" + ], + "answer": "2279508495173413", + "correct_value": 2279508495173413, + "explanation": "B-C = -194463421632056; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3202864002931054 + 4380372396287022 - 6075170890392789", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 3202864002931054 + 4380372396287022 - 6075170890392789", + "options": [ + "1508065508825277", + "1508065508825307", + "1508065508825287", + "1508065508825297" + ], + "answer": "1508065508825287", + "correct_value": 1508065508825287, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001376", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5955519397453081 + 2774301413890383 - 2774301413890385", + "ns_shortcut": "Near-cancellation: 2774301413890383 - 2774301413890385 = -2, so answer ≈ 5955519397453081 + -2.", + "shortcut_strength": 0.9999999999999993, + "pure_math": { + "question": "Evaluate: 5955519397453081 + 2774301413890383 - 2774301413890385", + "options": [ + "5955519397453089", + "5955519397453079", + "5955519397453099", + "5955519397453069" + ], + "answer": "5955519397453079", + "correct_value": 5955519397453079, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7691829430599997 + 2506943551435156 - 2408011304108513", + "ns_shortcut": "Near-cancellation: 2506943551435156 - 2408011304108513 = 98932247326643, offset is manageable.", + "shortcut_strength": 0.9605367072306008, + "pure_math": { + "question": "Evaluate: 7691829430599997 + 2506943551435156 - 2408011304108513", + "options": [ + "7790761677926640", + "7790761677926630", + "7790761677926650", + "7790761677926660" + ], + "answer": "7790761677926640", + "correct_value": 7790761677926640, + "explanation": "B-C = 98932247326643; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5955519397453081 + 2774301413890383 - 1305494528035585", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 5955519397453081 + 2774301413890383 - 1305494528035585", + "options": [ + "7424326283307899", + "7424326283307889", + "7424326283307879", + "7424326283307869" + ], + "answer": "7424326283307879", + "correct_value": 7424326283307879, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001377", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1113763376748804 + 7747167989588306 - 7747167989588305", + "ns_shortcut": "Near-cancellation: 7747167989588306 - 7747167989588305 = 1, so answer ≈ 1113763376748804 + 1.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "Evaluate: 1113763376748804 + 7747167989588306 - 7747167989588305", + "options": [ + "1113763376748805", + "1113763376748825", + "1113763376748815", + "1113763376748795" + ], + "answer": "1113763376748805", + "correct_value": 1113763376748805, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5054839549442162 + 2991950401165783 - 2867772101155726", + "ns_shortcut": "Near-cancellation: 2991950401165783 - 2867772101155726 = 124178300010057, offset is manageable.", + "shortcut_strength": 0.9584958694630525, + "pure_math": { + "question": "Evaluate: 5054839549442162 + 2991950401165783 - 2867772101155726", + "options": [ + "5179017849452219", + "5179017849452239", + "5179017849452229", + "5179017849452209" + ], + "answer": "5179017849452219", + "correct_value": 5179017849452219, + "explanation": "B-C = 124178300010057; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1113763376748804 + 7747167989588306 - 10435175686655470", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 1113763376748804 + 7747167989588306 - 10435175686655470", + "options": [ + "-1574244320318340", + "-1574244320318350", + "-1574244320318370", + "-1574244320318360" + ], + "answer": "-1574244320318360", + "correct_value": -1574244320318360, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_001378", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1079581076760174 + 2028079693443884 - 2028079693443882", + "ns_shortcut": "Near-cancellation: 2028079693443884 - 2028079693443882 = 2, so answer ≈ 1079581076760174 + 2.", + "shortcut_strength": 0.999999999999999, + "pure_math": { + "question": "Evaluate: 1079581076760174 + 2028079693443884 - 2028079693443882", + "options": [ + "1079581076760176", + "1079581076760196", + "1079581076760166", + "1079581076760186" + ], + "answer": "1079581076760176", + "correct_value": 1079581076760176, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1780464232782456 + 4141059571836620 - 4000708423812371", + "ns_shortcut": "Near-cancellation: 4141059571836620 - 4000708423812371 = 140351148024249, offset is manageable.", + "shortcut_strength": 0.9661074308182431, + "pure_math": { + "question": "Evaluate: 1780464232782456 + 4141059571836620 - 4000708423812371", + "options": [ + "1920815380806725", + "1920815380806715", + "1920815380806705", + "1920815380806695" + ], + "answer": "1920815380806705", + "correct_value": 1920815380806705, + "explanation": "B-C = 140351148024249; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1079581076760174 + 2028079693443884 - 3153038774013756", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 1079581076760174 + 2028079693443884 - 3153038774013756", + "options": [ + "-45378003809692", + "-45378003809712", + "-45378003809698", + "-45378003809682" + ], + "answer": "-45378003809698", + "correct_value": -45378003809698, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001379", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8035349198224981 + 7786699376607413 - 7786699376607415", + "ns_shortcut": "Near-cancellation: 7786699376607413 - 7786699376607415 = -2, so answer ≈ 8035349198224981 + -2.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "Evaluate: 8035349198224981 + 7786699376607413 - 7786699376607415", + "options": [ + "8035349198224979", + "8035349198224989", + "8035349198224969", + "8035349198224999" + ], + "answer": "8035349198224979", + "correct_value": 8035349198224979, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7278997814478573 + 6520945870093188 - 6223631879683651", + "ns_shortcut": "Near-cancellation: 6520945870093188 - 6223631879683651 = 297313990409537, offset is manageable.", + "shortcut_strength": 0.9544063090949583, + "pure_math": { + "question": "Evaluate: 7278997814478573 + 6520945870093188 - 6223631879683651", + "options": [ + "7576311804888120", + "7576311804888100", + "7576311804888130", + "7576311804888110" + ], + "answer": "7576311804888110", + "correct_value": 7576311804888110, + "explanation": "B-C = 297313990409537; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8035349198224981 + 7786699376607413 - 11672472527289751", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 8035349198224981 + 7786699376607413 - 11672472527289751", + "options": [ + "4149576047542653", + "4149576047542663", + "4149576047542643", + "4149576047542633" + ], + "answer": "4149576047542643", + "correct_value": 4149576047542643, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001380", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4853154548046597 + 1704789094515621 - 1704789094515622", + "ns_shortcut": "Near-cancellation: 1704789094515621 - 1704789094515622 = -1, so answer ≈ 4853154548046597 + -1.", + "shortcut_strength": 0.9999999999999994, + "pure_math": { + "question": "Evaluate: 4853154548046597 + 1704789094515621 - 1704789094515622", + "options": [ + "4853154548046616", + "4853154548046596", + "4853154548046606", + "4853154548046586" + ], + "answer": "4853154548046596", + "correct_value": 4853154548046596, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6055747953863791 + 2969118782577616 - 3034035219801003", + "ns_shortcut": "Near-cancellation: 2969118782577616 - 3034035219801003 = -64916437223387, offset is manageable.", + "shortcut_strength": 0.9781361265826387, + "pure_math": { + "question": "Evaluate: 6055747953863791 + 2969118782577616 - 3034035219801003", + "options": [ + "5990831516640424", + "5990831516640404", + "5990831516640394", + "5990831516640414" + ], + "answer": "5990831516640404", + "correct_value": 5990831516640404, + "explanation": "B-C = -64916437223387; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4853154548046597 + 1704789094515621 - 1073387913227803", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 4853154548046597 + 1704789094515621 - 1073387913227803", + "options": [ + "5484555729334415", + "5484555729334435", + "5484555729334425", + "5484555729334405" + ], + "answer": "5484555729334415", + "correct_value": 5484555729334415, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001381", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2970404482430650 + 1227517044682432 - 1227517044682431", + "ns_shortcut": "Near-cancellation: 1227517044682432 - 1227517044682431 = 1, so answer ≈ 2970404482430650 + 1.", + "shortcut_strength": 0.9999999999999992, + "pure_math": { + "question": "Evaluate: 2970404482430650 + 1227517044682432 - 1227517044682431", + "options": [ + "2970404482430641", + "2970404482430651", + "2970404482430661", + "2970404482430671" + ], + "answer": "2970404482430651", + "correct_value": 2970404482430651, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2140840669508292 + 2039676315182189 - 2082861053129637", + "ns_shortcut": "Near-cancellation: 2039676315182189 - 2082861053129637 = -43184737947448, offset is manageable.", + "shortcut_strength": 0.9788276514140968, + "pure_math": { + "question": "Evaluate: 2140840669508292 + 2039676315182189 - 2082861053129637", + "options": [ + "2097655931560834", + "2097655931560844", + "2097655931560864", + "2097655931560854" + ], + "answer": "2097655931560844", + "correct_value": 2097655931560844, + "explanation": "B-C = -43184737947448; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2970404482430650 + 1227517044682432 - 1757055685248408", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 2970404482430650 + 1227517044682432 - 1757055685248408", + "options": [ + "2440865841864664", + "2440865841864674", + "2440865841864694", + "2440865841864684" + ], + "answer": "2440865841864674", + "correct_value": 2440865841864674, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001382", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7191462578948404 + 9355341997717434 - 9355341997717433", + "ns_shortcut": "Near-cancellation: 9355341997717434 - 9355341997717433 = 1, so answer ≈ 7191462578948404 + 1.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "Evaluate: 7191462578948404 + 9355341997717434 - 9355341997717433", + "options": [ + "7191462578948425", + "7191462578948415", + "7191462578948405", + "7191462578948395" + ], + "answer": "7191462578948405", + "correct_value": 7191462578948405, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9560508097372620 + 2409537388782123 - 2357324478019387", + "ns_shortcut": "Near-cancellation: 2409537388782123 - 2357324478019387 = 52212910762736, offset is manageable.", + "shortcut_strength": 0.9783307322783954, + "pure_math": { + "question": "Evaluate: 9560508097372620 + 2409537388782123 - 2357324478019387", + "options": [ + "9612721008135356", + "9612721008135346", + "9612721008135366", + "9612721008135376" + ], + "answer": "9612721008135356", + "correct_value": 9612721008135356, + "explanation": "B-C = 52212910762736; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7191462578948404 + 9355341997717434 - 5626432004096479", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 7191462578948404 + 9355341997717434 - 5626432004096479", + "options": [ + "10920372572569379", + "10920372572569349", + "10920372572569369", + "10920372572569359" + ], + "answer": "10920372572569359", + "correct_value": 10920372572569359, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_001383", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8255621704717552 + 4747859145104852 - 4747859145104854", + "ns_shortcut": "Near-cancellation: 4747859145104852 - 4747859145104854 = -2, so answer ≈ 8255621704717552 + -2.", + "shortcut_strength": 0.9999999999999996, + "pure_math": { + "question": "Evaluate: 8255621704717552 + 4747859145104852 - 4747859145104854", + "options": [ + "8255621704717560", + "8255621704717540", + "8255621704717550", + "8255621704717570" + ], + "answer": "8255621704717550", + "correct_value": 8255621704717550, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1141528761137667 + 4712169457298319 - 4595582492692089", + "ns_shortcut": "Near-cancellation: 4712169457298319 - 4595582492692089 = 116586964606230, offset is manageable.", + "shortcut_strength": 0.9752583251381893, + "pure_math": { + "question": "Evaluate: 1141528761137667 + 4712169457298319 - 4595582492692089", + "options": [ + "1258115725743917", + "1258115725743897", + "1258115725743907", + "1258115725743887" + ], + "answer": "1258115725743897", + "correct_value": 1258115725743897, + "explanation": "B-C = 116586964606230; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8255621704717552 + 4747859145104852 - 7525467078345214", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 8255621704717552 + 4747859145104852 - 7525467078345214", + "options": [ + "5478013771477210", + "5478013771477200", + "5478013771477190", + "5478013771477180" + ], + "answer": "5478013771477190", + "correct_value": 5478013771477190, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001384", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5316228509104056 + 8581527927386372 - 8581527927386371", + "ns_shortcut": "Near-cancellation: 8581527927386372 - 8581527927386371 = 1, so answer ≈ 5316228509104056 + 1.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "Evaluate: 5316228509104056 + 8581527927386372 - 8581527927386371", + "options": [ + "5316228509104057", + "5316228509104077", + "5316228509104067", + "5316228509104047" + ], + "answer": "5316228509104057", + "correct_value": 5316228509104057, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9893192993666332 + 5441588940057179 - 5212679305221599", + "ns_shortcut": "Near-cancellation: 5441588940057179 - 5212679305221599 = 228909634835580, offset is manageable.", + "shortcut_strength": 0.957933310039186, + "pure_math": { + "question": "Evaluate: 9893192993666332 + 5441588940057179 - 5212679305221599", + "options": [ + "10122102628501902", + "10122102628501922", + "10122102628501912", + "10122102628501932" + ], + "answer": "10122102628501912", + "correct_value": 10122102628501912, + "explanation": "B-C = 228909634835580; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5316228509104056 + 8581527927386372 - 3524658435521293", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 5316228509104056 + 8581527927386372 - 3524658435521293", + "options": [ + "10373098000969155", + "10373098000969145", + "10373098000969125", + "10373098000969135" + ], + "answer": "10373098000969135", + "correct_value": 10373098000969135, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_001385", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2980587838640542 + 7151121106508808 - 7151121106508809", + "ns_shortcut": "Near-cancellation: 7151121106508808 - 7151121106508809 = -1, so answer ≈ 2980587838640542 + -1.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "Evaluate: 2980587838640542 + 7151121106508808 - 7151121106508809", + "options": [ + "2980587838640551", + "2980587838640541", + "2980587838640561", + "2980587838640531" + ], + "answer": "2980587838640541", + "correct_value": 2980587838640541, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4453139839135248 + 1669509554860676 - 1747305902488522", + "ns_shortcut": "Near-cancellation: 1669509554860676 - 1747305902488522 = -77796347627846, offset is manageable.", + "shortcut_strength": 0.953401675718868, + "pure_math": { + "question": "Evaluate: 4453139839135248 + 1669509554860676 - 1747305902488522", + "options": [ + "4375343491507392", + "4375343491507422", + "4375343491507402", + "4375343491507412" + ], + "answer": "4375343491507402", + "correct_value": 4375343491507402, + "explanation": "B-C = -77796347627846; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2980587838640542 + 7151121106508808 - 10051562699049249", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 2980587838640542 + 7151121106508808 - 10051562699049249", + "options": [ + "80146246100091", + "80146246100121", + "80146246100111", + "80146246100101" + ], + "answer": "80146246100101", + "correct_value": 80146246100101, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_001386", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3009093781510667 + 1580186860456278 - 1580186860456280", + "ns_shortcut": "Near-cancellation: 1580186860456278 - 1580186860456280 = -2, so answer ≈ 3009093781510667 + -2.", + "shortcut_strength": 0.9999999999999988, + "pure_math": { + "question": "Evaluate: 3009093781510667 + 1580186860456278 - 1580186860456280", + "options": [ + "3009093781510655", + "3009093781510675", + "3009093781510665", + "3009093781510685" + ], + "answer": "3009093781510665", + "correct_value": 3009093781510665, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3449114411267930 + 9631175605839490 - 9999999999999999", + "ns_shortcut": "Near-cancellation: 9631175605839490 - 9999999999999999 = -368824394160509, offset is manageable.", + "shortcut_strength": 0.960515520160038, + "pure_math": { + "question": "Evaluate: 3449114411267930 + 9631175605839490 - 9999999999999999", + "options": [ + "3080290017107441", + "3080290017107431", + "3080290017107421", + "3080290017107411" + ], + "answer": "3080290017107421", + "correct_value": 3080290017107421, + "explanation": "B-C = -368824394160509; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3009093781510667 + 1580186860456278 - 1033914490604551", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 3009093781510667 + 1580186860456278 - 1033914490604551", + "options": [ + "3555366151362384", + "3555366151362394", + "3555366151362404", + "3555366151362414" + ], + "answer": "3555366151362394", + "correct_value": 3555366151362394, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001387", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2734602079244710 + 1788173831169177 - 1788173831169179", + "ns_shortcut": "Near-cancellation: 1788173831169177 - 1788173831169179 = -2, so answer ≈ 2734602079244710 + -2.", + "shortcut_strength": 0.9999999999999989, + "pure_math": { + "question": "Evaluate: 2734602079244710 + 1788173831169177 - 1788173831169179", + "options": [ + "2734602079244708", + "2734602079244718", + "2734602079244728", + "2734602079244698" + ], + "answer": "2734602079244708", + "correct_value": 2734602079244708, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5410018196170384 + 8340354497092603 - 8694468992565301", + "ns_shortcut": "Near-cancellation: 8340354497092603 - 8694468992565301 = -354114495472698, offset is manageable.", + "shortcut_strength": 0.9575420330639255, + "pure_math": { + "question": "Evaluate: 5410018196170384 + 8340354497092603 - 8694468992565301", + "options": [ + "5055903700697706", + "5055903700697686", + "5055903700697696", + "5055903700697676" + ], + "answer": "5055903700697686", + "correct_value": 5055903700697686, + "explanation": "B-C = -354114495472698; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2734602079244710 + 1788173831169177 - 2861752771858599", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 2734602079244710 + 1788173831169177 - 2861752771858599", + "options": [ + "1661023138555298", + "1661023138555288", + "1661023138555308", + "1661023138555278" + ], + "answer": "1661023138555288", + "correct_value": 1661023138555288, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001388", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2452228583048936 + 8820490899666081 - 8820490899666082", + "ns_shortcut": "Near-cancellation: 8820490899666081 - 8820490899666082 = -1, so answer ≈ 2452228583048936 + -1.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "Evaluate: 2452228583048936 + 8820490899666081 - 8820490899666082", + "options": [ + "2452228583048955", + "2452228583048925", + "2452228583048945", + "2452228583048935" + ], + "answer": "2452228583048935", + "correct_value": 2452228583048935, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7391138933240773 + 6988743124604126 - 6651533549824121", + "ns_shortcut": "Near-cancellation: 6988743124604126 - 6651533549824121 = 337209574780005, offset is manageable.", + "shortcut_strength": 0.9517496109432257, + "pure_math": { + "question": "Evaluate: 7391138933240773 + 6988743124604126 - 6651533549824121", + "options": [ + "7728348508020788", + "7728348508020798", + "7728348508020768", + "7728348508020778" + ], + "answer": "7728348508020778", + "correct_value": 7728348508020778, + "explanation": "B-C = 337209574780005; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2452228583048936 + 8820490899666081 - 3846955157861624", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 2452228583048936 + 8820490899666081 - 3846955157861624", + "options": [ + "7425764324853413", + "7425764324853403", + "7425764324853383", + "7425764324853393" + ], + "answer": "7425764324853393", + "correct_value": 7425764324853393, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_001389", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5436186151273914 + 6691128927295273 - 6691128927295272", + "ns_shortcut": "Near-cancellation: 6691128927295273 - 6691128927295272 = 1, so answer ≈ 5436186151273914 + 1.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "Evaluate: 5436186151273914 + 6691128927295273 - 6691128927295272", + "options": [ + "5436186151273925", + "5436186151273935", + "5436186151273915", + "5436186151273905" + ], + "answer": "5436186151273915", + "correct_value": 5436186151273915, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2076167037959536 + 9846573003649629 - 9475495315338918", + "ns_shortcut": "Near-cancellation: 9846573003649629 - 9475495315338918 = 371077688310711, offset is manageable.", + "shortcut_strength": 0.9623140265985769, + "pure_math": { + "question": "Evaluate: 2076167037959536 + 9846573003649629 - 9475495315338918", + "options": [ + "2447244726270237", + "2447244726270247", + "2447244726270267", + "2447244726270257" + ], + "answer": "2447244726270247", + "correct_value": 2447244726270247, + "explanation": "B-C = 371077688310711; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5436186151273914 + 6691128927295273 - 10849129447978382", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 5436186151273914 + 6691128927295273 - 10849129447978382", + "options": [ + "1278185630590795", + "1278185630590805", + "1278185630590815", + "1278185630590825" + ], + "answer": "1278185630590805", + "correct_value": 1278185630590805, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001390", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8806849288913202 + 4732742132459529 - 4732742132459531", + "ns_shortcut": "Near-cancellation: 4732742132459529 - 4732742132459531 = -2, so answer ≈ 8806849288913202 + -2.", + "shortcut_strength": 0.9999999999999996, + "pure_math": { + "question": "Evaluate: 8806849288913202 + 4732742132459529 - 4732742132459531", + "options": [ + "8806849288913200", + "8806849288913210", + "8806849288913220", + "8806849288913190" + ], + "answer": "8806849288913200", + "correct_value": 8806849288913200, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4670567744021902 + 2802608386909537 - 2906782871150999", + "ns_shortcut": "Near-cancellation: 2802608386909537 - 2906782871150999 = -104174484241462, offset is manageable.", + "shortcut_strength": 0.9628294539015719, + "pure_math": { + "question": "Evaluate: 4670567744021902 + 2802608386909537 - 2906782871150999", + "options": [ + "4566393259780430", + "4566393259780450", + "4566393259780460", + "4566393259780440" + ], + "answer": "4566393259780440", + "correct_value": 4566393259780440, + "explanation": "B-C = -104174484241462; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8806849288913202 + 4732742132459529 - 2020481848099117", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 8806849288913202 + 4732742132459529 - 2020481848099117", + "options": [ + "11519109573273634", + "11519109573273604", + "11519109573273624", + "11519109573273614" + ], + "answer": "11519109573273614", + "correct_value": 11519109573273614, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_001391", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1426592033336676 + 7877982591541203 - 7877982591541204", + "ns_shortcut": "Near-cancellation: 7877982591541203 - 7877982591541204 = -1, so answer ≈ 1426592033336676 + -1.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "Evaluate: 1426592033336676 + 7877982591541203 - 7877982591541204", + "options": [ + "1426592033336695", + "1426592033336665", + "1426592033336685", + "1426592033336675" + ], + "answer": "1426592033336675", + "correct_value": 1426592033336675, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3684355428677358 + 2619070755002426 - 2531731586987423", + "ns_shortcut": "Near-cancellation: 2619070755002426 - 2531731586987423 = 87339168015003, offset is manageable.", + "shortcut_strength": 0.9666526122487584, + "pure_math": { + "question": "Evaluate: 3684355428677358 + 2619070755002426 - 2531731586987423", + "options": [ + "3771694596692381", + "3771694596692371", + "3771694596692361", + "3771694596692351" + ], + "answer": "3771694596692361", + "correct_value": 3771694596692361, + "explanation": "B-C = 87339168015003; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1426592033336676 + 7877982591541203 - 3762610030923778", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 1426592033336676 + 7877982591541203 - 3762610030923778", + "options": [ + "5541964593954121", + "5541964593954101", + "5541964593954091", + "5541964593954111" + ], + "answer": "5541964593954101", + "correct_value": 5541964593954101, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001392", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8544238055458542 + 1692487206781134 - 1692487206781132", + "ns_shortcut": "Near-cancellation: 1692487206781134 - 1692487206781132 = 2, so answer ≈ 8544238055458542 + 2.", + "shortcut_strength": 0.9999999999999988, + "pure_math": { + "question": "Evaluate: 8544238055458542 + 1692487206781134 - 1692487206781132", + "options": [ + "8544238055458534", + "8544238055458544", + "8544238055458554", + "8544238055458564" + ], + "answer": "8544238055458544", + "correct_value": 8544238055458544, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5710521770186968 + 5931135809373146 - 6179126848634044", + "ns_shortcut": "Near-cancellation: 5931135809373146 - 6179126848634044 = -247991039260898, offset is manageable.", + "shortcut_strength": 0.9581882716512762, + "pure_math": { + "question": "Evaluate: 5710521770186968 + 5931135809373146 - 6179126848634044", + "options": [ + "5462530730926060", + "5462530730926090", + "5462530730926080", + "5462530730926070" + ], + "answer": "5462530730926070", + "correct_value": 5462530730926070, + "explanation": "B-C = -247991039260898; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8544238055458542 + 1692487206781134 - 2384142462066265", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 8544238055458542 + 1692487206781134 - 2384142462066265", + "options": [ + "7852582800173411", + "7852582800173401", + "7852582800173421", + "7852582800173431" + ], + "answer": "7852582800173411", + "correct_value": 7852582800173411, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001393", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3025787912279006 + 7258602041716913 - 7258602041716911", + "ns_shortcut": "Near-cancellation: 7258602041716913 - 7258602041716911 = 2, so answer ≈ 3025787912279006 + 2.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "Evaluate: 3025787912279006 + 7258602041716913 - 7258602041716911", + "options": [ + "3025787912278998", + "3025787912279008", + "3025787912279028", + "3025787912279018" + ], + "answer": "3025787912279008", + "correct_value": 3025787912279008, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4015822108784906 + 2084277139593633 - 2015595330536300", + "ns_shortcut": "Near-cancellation: 2084277139593633 - 2015595330536300 = 68681809057333, offset is manageable.", + "shortcut_strength": 0.9670476599523978, + "pure_math": { + "question": "Evaluate: 4015822108784906 + 2084277139593633 - 2015595330536300", + "options": [ + "4084503917842239", + "4084503917842259", + "4084503917842249", + "4084503917842229" + ], + "answer": "4084503917842239", + "correct_value": 4084503917842239, + "explanation": "B-C = 68681809057333; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3025787912279006 + 7258602041716913 - 10091453591651899", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 3025787912279006 + 7258602041716913 - 10091453591651899", + "options": [ + "192936362344010", + "192936362344030", + "192936362344020", + "192936362344040" + ], + "answer": "192936362344020", + "correct_value": 192936362344020, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001394", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3452923720689472 + 8508642515024003 - 8508642515024004", + "ns_shortcut": "Near-cancellation: 8508642515024003 - 8508642515024004 = -1, so answer ≈ 3452923720689472 + -1.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "Evaluate: 3452923720689472 + 8508642515024003 - 8508642515024004", + "options": [ + "3452923720689481", + "3452923720689471", + "3452923720689461", + "3452923720689491" + ], + "answer": "3452923720689471", + "correct_value": 3452923720689471, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3192378564063252 + 9840946500686358 - 9512174475010171", + "ns_shortcut": "Near-cancellation: 9840946500686358 - 9512174475010171 = 328772025676187, offset is manageable.", + "shortcut_strength": 0.9665914223135694, + "pure_math": { + "question": "Evaluate: 3192378564063252 + 9840946500686358 - 9512174475010171", + "options": [ + "3521150589739429", + "3521150589739439", + "3521150589739459", + "3521150589739449" + ], + "answer": "3521150589739439", + "correct_value": 3521150589739439, + "explanation": "B-C = 328772025676187; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3452923720689472 + 8508642515024003 - 5135504885239440", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 3452923720689472 + 8508642515024003 - 5135504885239440", + "options": [ + "6826061350474045", + "6826061350474055", + "6826061350474035", + "6826061350474025" + ], + "answer": "6826061350474035", + "correct_value": 6826061350474035, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001395", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5881903337016386 + 3101116878293009 - 3101116878293008", + "ns_shortcut": "Near-cancellation: 3101116878293009 - 3101116878293008 = 1, so answer ≈ 5881903337016386 + 1.", + "shortcut_strength": 0.9999999999999997, + "pure_math": { + "question": "Evaluate: 5881903337016386 + 3101116878293009 - 3101116878293008", + "options": [ + "5881903337016377", + "5881903337016387", + "5881903337016397", + "5881903337016407" + ], + "answer": "5881903337016387", + "correct_value": 5881903337016387, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1801255933184228 + 5260298652573074 - 5070102609629308", + "ns_shortcut": "Near-cancellation: 5260298652573074 - 5070102609629308 = 190196042943766, offset is manageable.", + "shortcut_strength": 0.9638431093925187, + "pure_math": { + "question": "Evaluate: 1801255933184228 + 5260298652573074 - 5070102609629308", + "options": [ + "1991451976128014", + "1991451976127984", + "1991451976127994", + "1991451976128004" + ], + "answer": "1991451976127994", + "correct_value": 1991451976127994, + "explanation": "B-C = 190196042943766; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5881903337016386 + 3101116878293009 - 4384928153178447", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 5881903337016386 + 3101116878293009 - 4384928153178447", + "options": [ + "4598092062130938", + "4598092062130968", + "4598092062130948", + "4598092062130958" + ], + "answer": "4598092062130948", + "correct_value": 4598092062130948, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001396", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9059244175206189 + 8214586299514995 - 8214586299514993", + "ns_shortcut": "Near-cancellation: 8214586299514995 - 8214586299514993 = 2, so answer ≈ 9059244175206189 + 2.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "Evaluate: 9059244175206189 + 8214586299514995 - 8214586299514993", + "options": [ + "9059244175206191", + "9059244175206201", + "9059244175206211", + "9059244175206181" + ], + "answer": "9059244175206191", + "correct_value": 9059244175206191, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7135798708326236 + 1827220320163975 - 1887770499010697", + "ns_shortcut": "Near-cancellation: 1827220320163975 - 1887770499010697 = -60550178846722, offset is manageable.", + "shortcut_strength": 0.9668621357925309, + "pure_math": { + "question": "Evaluate: 7135798708326236 + 1827220320163975 - 1887770499010697", + "options": [ + "7075248529479534", + "7075248529479504", + "7075248529479514", + "7075248529479524" + ], + "answer": "7075248529479514", + "correct_value": 7075248529479514, + "explanation": "B-C = -60550178846722; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9059244175206189 + 8214586299514995 - 13287874739398554", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 9059244175206189 + 8214586299514995 - 13287874739398554", + "options": [ + "3985955735322630", + "3985955735322620", + "3985955735322650", + "3985955735322640" + ], + "answer": "3985955735322630", + "correct_value": 3985955735322630, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001397", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7569671803769991 + 6065677840344046 - 6065677840344047", + "ns_shortcut": "Near-cancellation: 6065677840344046 - 6065677840344047 = -1, so answer ≈ 7569671803769991 + -1.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "Evaluate: 7569671803769991 + 6065677840344046 - 6065677840344047", + "options": [ + "7569671803769980", + "7569671803770010", + "7569671803770000", + "7569671803769990" + ], + "answer": "7569671803769990", + "correct_value": 7569671803769990, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5235300124491457 + 1838748042025660 - 1924907235642080", + "ns_shortcut": "Near-cancellation: 1838748042025660 - 1924907235642080 = -86159193616420, offset is manageable.", + "shortcut_strength": 0.9531424688716446, + "pure_math": { + "question": "Evaluate: 5235300124491457 + 1838748042025660 - 1924907235642080", + "options": [ + "5149140930875057", + "5149140930875027", + "5149140930875047", + "5149140930875037" + ], + "answer": "5149140930875037", + "correct_value": 5149140930875037, + "explanation": "B-C = -86159193616420; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7569671803769991 + 6065677840344046 - 4101021454170077", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 7569671803769991 + 6065677840344046 - 4101021454170077", + "options": [ + "9534328189943980", + "9534328189943950", + "9534328189943960", + "9534328189943970" + ], + "answer": "9534328189943960", + "correct_value": 9534328189943960, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001398", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9415473845540360 + 5078653109786016 - 5078653109786018", + "ns_shortcut": "Near-cancellation: 5078653109786016 - 5078653109786018 = -2, so answer ≈ 9415473845540360 + -2.", + "shortcut_strength": 0.9999999999999996, + "pure_math": { + "question": "Evaluate: 9415473845540360 + 5078653109786016 - 5078653109786018", + "options": [ + "9415473845540368", + "9415473845540358", + "9415473845540378", + "9415473845540348" + ], + "answer": "9415473845540358", + "correct_value": 9415473845540358, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7115339003600441 + 1668580966784889 - 1634801212842571", + "ns_shortcut": "Near-cancellation: 1668580966784889 - 1634801212842571 = 33779753942318, offset is manageable.", + "shortcut_strength": 0.9797554001785082, + "pure_math": { + "question": "Evaluate: 7115339003600441 + 1668580966784889 - 1634801212842571", + "options": [ + "7149118757542759", + "7149118757542769", + "7149118757542749", + "7149118757542779" + ], + "answer": "7149118757542759", + "correct_value": 7149118757542759, + "explanation": "B-C = 33779753942318; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9415473845540360 + 5078653109786016 - 3529824352988296", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 9415473845540360 + 5078653109786016 - 3529824352988296", + "options": [ + "10964302602338090", + "10964302602338070", + "10964302602338080", + "10964302602338100" + ], + "answer": "10964302602338080", + "correct_value": 10964302602338080, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001399", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2537854471044720 + 8664982181762846 - 8664982181762845", + "ns_shortcut": "Near-cancellation: 8664982181762846 - 8664982181762845 = 1, so answer ≈ 2537854471044720 + 1.", + "shortcut_strength": 0.9999999999999999, + "pure_math": { + "question": "Evaluate: 2537854471044720 + 8664982181762846 - 8664982181762845", + "options": [ + "2537854471044731", + "2537854471044711", + "2537854471044741", + "2537854471044721" + ], + "answer": "2537854471044721", + "correct_value": 2537854471044721, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7489377885647361 + 5312625832658185 - 5148436741115562", + "ns_shortcut": "Near-cancellation: 5312625832658185 - 5148436741115562 = 164189091542623, offset is manageable.", + "shortcut_strength": 0.9690945500936078, + "pure_math": { + "question": "Evaluate: 7489377885647361 + 5312625832658185 - 5148436741115562", + "options": [ + "7653566977189994", + "7653566977189974", + "7653566977190004", + "7653566977189984" + ], + "answer": "7653566977189984", + "correct_value": 7653566977189984, + "explanation": "B-C = 164189091542623; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2537854471044720 + 8664982181762846 - 12672622382489264", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 2537854471044720 + 8664982181762846 - 12672622382489264", + "options": [ + "-1469785729681692", + "-1469785729681682", + "-1469785729681712", + "-1469785729681698" + ], + "answer": "-1469785729681698", + "correct_value": -1469785729681698, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_001400", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8123820227372139 + 3613793873750583 - 3613793873750582", + "ns_shortcut": "Near-cancellation: 3613793873750583 - 3613793873750582 = 1, so answer ≈ 8123820227372139 + 1.", + "shortcut_strength": 0.9999999999999998, + "pure_math": { + "question": "Evaluate: 8123820227372139 + 3613793873750583 - 3613793873750582", + "options": [ + "8123820227372140", + "8123820227372160", + "8123820227372130", + "8123820227372150" + ], + "answer": "8123820227372140", + "correct_value": 8123820227372140, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9850062723915098 + 5958186192025091 - 6101045492791627", + "ns_shortcut": "Near-cancellation: 5958186192025091 - 6101045492791627 = -142859300766536, offset is manageable.", + "shortcut_strength": 0.9760230217448138, + "pure_math": { + "question": "Evaluate: 9850062723915098 + 5958186192025091 - 6101045492791627", + "options": [ + "9707203423148552", + "9707203423148582", + "9707203423148572", + "9707203423148562" + ], + "answer": "9707203423148562", + "correct_value": 9707203423148562, + "explanation": "B-C = -142859300766536; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8123820227372139 + 3613793873750583 - 4714421995253350", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 8123820227372139 + 3613793873750583 - 4714421995253350", + "options": [ + "7023192105869372", + "7023192105869392", + "7023192105869382", + "7023192105869362" + ], + "answer": "7023192105869372", + "correct_value": 7023192105869372, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_001401", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9056451127371004 + 1818410046845831 - 1818410046845833", + "ns_shortcut": "Near-cancellation: 1818410046845831 - 1818410046845833 = -2, so answer ≈ 9056451127371004 + -2.", + "shortcut_strength": 0.9999999999999989, + "pure_math": { + "question": "Evaluate: 9056451127371004 + 1818410046845831 - 1818410046845833", + "options": [ + "9056451127371012", + "9056451127371002", + "9056451127371022", + "9056451127370992" + ], + "answer": "9056451127371002", + "correct_value": 9056451127371002, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1759057509254454 + 6080357899437250 - 5946796702449351", + "ns_shortcut": "Near-cancellation: 6080357899437250 - 5946796702449351 = 133561196987899, offset is manageable.", + "shortcut_strength": 0.9780339908938156, + "pure_math": { + "question": "Evaluate: 1759057509254454 + 6080357899437250 - 5946796702449351", + "options": [ + "1892618706242373", + "1892618706242353", + "1892618706242343", + "1892618706242363" + ], + "answer": "1892618706242353", + "correct_value": 1892618706242353, + "explanation": "B-C = 133561196987899; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9056451127371004 + 1818410046845831 - 2711752840028592", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 9056451127371004 + 1818410046845831 - 2711752840028592", + "options": [ + "8163108334188253", + "8163108334188233", + "8163108334188243", + "8163108334188263" + ], + "answer": "8163108334188243", + "correct_value": 8163108334188243, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001402", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5004 * 21", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9746, + "pure_math": { + "question": "Estimate 5004 × 21.", + "options": [ + "105084", + "105104", + "105074", + "105094" + ], + "answer": "105084", + "correct_value": 105084, + "explanation": "Round to compatible pair 5000×20 = 100000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4913 * 14", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8412999999999999, + "pure_math": { + "question": "Estimate 4913 × 14.", + "options": [ + "68802", + "68782", + "68772", + "68792" + ], + "answer": "68782", + "correct_value": 68782, + "explanation": "One number rounds to 5000, other is farther from 20." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6183 * 26", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 6183 × 26.", + "options": [ + "160768", + "160748", + "160778", + "160758" + ], + "answer": "160758", + "correct_value": 160758, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_001403", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1260 * 82", + "ns_shortcut": "Round to 1250×80 for easy mental computation.", + "shortcut_strength": 0.9835, + "pure_math": { + "question": "Estimate 1260 × 82.", + "options": [ + "103310", + "103340", + "103320", + "103330" + ], + "answer": "103320", + "correct_value": 103320, + "explanation": "Round to compatible pair 1250×80 = 100000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1235 * 102", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8565, + "pure_math": { + "question": "Estimate 1235 × 102.", + "options": [ + "125970", + "125980", + "125990", + "125960" + ], + "answer": "125970", + "correct_value": 125970, + "explanation": "One number rounds to 1250, other is farther from 80." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1528 * 108", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1528 × 108.", + "options": [ + "165024", + "165014", + "165044", + "165034" + ], + "answer": "165024", + "correct_value": 165024, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001404", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4932 * 20", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9932, + "pure_math": { + "question": "Estimate 4932 × 20.", + "options": [ + "98650", + "98630", + "98640", + "98660" + ], + "answer": "98640", + "correct_value": 98640, + "explanation": "Round to compatible pair 5000×20 = 100000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4923 * 14", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8423, + "pure_math": { + "question": "Estimate 4923 × 14.", + "options": [ + "68942", + "68922", + "68912", + "68932" + ], + "answer": "68922", + "correct_value": 68922, + "explanation": "One number rounds to 5000, other is farther from 20." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5787 * 27", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5787 × 27.", + "options": [ + "156239", + "156259", + "156249", + "156269" + ], + "answer": "156249", + "correct_value": 156249, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001405", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25075 * 3813", + "ns_shortcut": "Round to 25000×4000 for easy mental computation.", + "shortcut_strength": 0.975125, + "pure_math": { + "question": "Estimate 25075 × 3813.", + "options": [ + "95610975", + "95610995", + "95610965", + "95610985" + ], + "answer": "95610975", + "correct_value": 95610975, + "explanation": "Round to compatible pair 25000×4000 = 100000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "24585 * 5236", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8371999999999999, + "pure_math": { + "question": "Estimate 24585 × 5236.", + "options": [ + "128727060", + "128727080", + "128727050", + "128727070" + ], + "answer": "128727060", + "correct_value": 128727060, + "explanation": "One number rounds to 25000, other is farther from 4000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8099664881692863 * 8848130205665746", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 8099664881692863 × 8848130205665746.", + "options": [ + "71666889495476692139554411770788", + "71666889495476692139554411770818", + "71666889495476692139554411770798", + "71666889495476692139554411770808" + ], + "answer": "71666889495476692139554411770798", + "correct_value": 71666889495476692139554411770798, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001406", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2549 * 42", + "ns_shortcut": "Round to 2500×40 for easy mental computation.", + "shortcut_strength": 0.9652000000000001, + "pure_math": { + "question": "Estimate 2549 × 42.", + "options": [ + "107078", + "107058", + "107048", + "107068" + ], + "answer": "107058", + "correct_value": 107058, + "explanation": "Round to compatible pair 2500×40 = 100000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2476 * 28", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8452, + "pure_math": { + "question": "Estimate 2476 × 28.", + "options": [ + "69318", + "69328", + "69338", + "69348" + ], + "answer": "69328", + "correct_value": 69328, + "explanation": "One number rounds to 2500, other is farther from 40." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2980 * 57", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 2980 × 57.", + "options": [ + "169880", + "169860", + "169850", + "169870" + ], + "answer": "169860", + "correct_value": 169860, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001407", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1226 * 84", + "ns_shortcut": "Round to 1250×80 for easy mental computation.", + "shortcut_strength": 0.9654, + "pure_math": { + "question": "Estimate 1226 × 84.", + "options": [ + "102994", + "102984", + "103004", + "102974" + ], + "answer": "102984", + "correct_value": 102984, + "explanation": "Round to compatible pair 1250×80 = 100000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1232 * 105", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.83655, + "pure_math": { + "question": "Estimate 1232 × 105.", + "options": [ + "129360", + "129380", + "129370", + "129350" + ], + "answer": "129360", + "correct_value": 129360, + "explanation": "One number rounds to 1250, other is farther from 80." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1428 * 111", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1428 × 111.", + "options": [ + "158498", + "158528", + "158518", + "158508" + ], + "answer": "158508", + "correct_value": 158508, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_001408", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1242 * 81", + "ns_shortcut": "Round to 1250×80 for easy mental computation.", + "shortcut_strength": 0.99055, + "pure_math": { + "question": "Estimate 1242 × 81.", + "options": [ + "100592", + "100622", + "100602", + "100612" + ], + "answer": "100602", + "correct_value": 100602, + "explanation": "Round to compatible pair 1250×80 = 100000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1258 * 56", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8468, + "pure_math": { + "question": "Estimate 1258 × 56.", + "options": [ + "70468", + "70458", + "70448", + "70438" + ], + "answer": "70448", + "correct_value": 70448, + "explanation": "One number rounds to 1250, other is farther from 80." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1447 * 117", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1447 × 117.", + "options": [ + "169299", + "169309", + "169319", + "169289" + ], + "answer": "169299", + "correct_value": 169299, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001409", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25172 * 3789", + "ns_shortcut": "Round to 25000×4000 for easy mental computation.", + "shortcut_strength": 0.970185, + "pure_math": { + "question": "Estimate 25172 × 3789.", + "options": [ + "95376728", + "95376698", + "95376708", + "95376718" + ], + "answer": "95376708", + "correct_value": 95376708, + "explanation": "Round to compatible pair 25000×4000 = 100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "25076 * 5304", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.83548, + "pure_math": { + "question": "Estimate 25076 × 5304.", + "options": [ + "133003104", + "133003124", + "133003094", + "133003114" + ], + "answer": "133003104", + "correct_value": 133003104, + "explanation": "One number rounds to 25000, other is farther from 4000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6472790500899456 * 7451613880107531", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 6472790500899456 × 7451613880107531.", + "options": [ + "48232735539530564449401499403156", + "48232735539530564449401499403136", + "48232735539530564449401499403126", + "48232735539530564449401499403146" + ], + "answer": "48232735539530564449401499403136", + "correct_value": 48232735539530564449401499403136, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001410", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5091 * 20", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9909, + "pure_math": { + "question": "Estimate 5091 × 20.", + "options": [ + "101820", + "101810", + "101830", + "101840" + ], + "answer": "101820", + "correct_value": 101820, + "explanation": "Round to compatible pair 5000×20 = 100000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5053 * 26", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8447, + "pure_math": { + "question": "Estimate 5053 × 26.", + "options": [ + "131368", + "131388", + "131378", + "131398" + ], + "answer": "131378", + "correct_value": 131378, + "explanation": "One number rounds to 5000, other is farther from 20." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5578 * 30", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5578 × 30.", + "options": [ + "167350", + "167360", + "167340", + "167330" + ], + "answer": "167340", + "correct_value": 167340, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001411", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25194 * 4220", + "ns_shortcut": "Round to 25000×4000 for easy mental computation.", + "shortcut_strength": 0.96862, + "pure_math": { + "question": "Estimate 25194 × 4220.", + "options": [ + "106318690", + "106318700", + "106318670", + "106318680" + ], + "answer": "106318680", + "correct_value": 106318680, + "explanation": "Round to compatible pair 25000×4000 = 100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "25164 * 5315", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.832345, + "pure_math": { + "question": "Estimate 25164 × 5315.", + "options": [ + "133746660", + "133746670", + "133746650", + "133746680" + ], + "answer": "133746660", + "correct_value": 133746660, + "explanation": "One number rounds to 25000, other is farther from 4000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3773823270149475 * 2856702546260156", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 3773823270149475 × 2856702546260156.", + "options": [ + "10780690544971833799732356818120", + "10780690544971833799732356818100", + "10780690544971833799732356818110", + "10780690544971833799732356818090" + ], + "answer": "10780690544971833799732356818100", + "correct_value": 10780690544971833799732356818100, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001412", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5043 * 20", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9957, + "pure_math": { + "question": "Estimate 5043 × 20.", + "options": [ + "100860", + "100850", + "100880", + "100870" + ], + "answer": "100860", + "correct_value": 100860, + "explanation": "Round to compatible pair 5000×20 = 100000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4929 * 24", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8929, + "pure_math": { + "question": "Estimate 4929 × 24.", + "options": [ + "118306", + "118296", + "118316", + "118286" + ], + "answer": "118296", + "correct_value": 118296, + "explanation": "One number rounds to 5000, other is farther from 20." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5839 * 29", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5839 × 29.", + "options": [ + "169351", + "169321", + "169341", + "169331" + ], + "answer": "169331", + "correct_value": 169331, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_001413", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5007 * 18", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9493, + "pure_math": { + "question": "Estimate 5007 × 18.", + "options": [ + "90126", + "90136", + "90116", + "90146" + ], + "answer": "90126", + "correct_value": 90126, + "explanation": "Round to compatible pair 5000×20 = 100000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5095 * 15", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8654999999999999, + "pure_math": { + "question": "Estimate 5095 × 15.", + "options": [ + "76435", + "76425", + "76415", + "76445" + ], + "answer": "76425", + "correct_value": 76425, + "explanation": "One number rounds to 5000, other is farther from 20." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6144 * 29", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 6144 × 29.", + "options": [ + "178176", + "178166", + "178196", + "178186" + ], + "answer": "178176", + "correct_value": 178176, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001414", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5071 * 18", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9429, + "pure_math": { + "question": "Estimate 5071 × 18.", + "options": [ + "91298", + "91288", + "91278", + "91268" + ], + "answer": "91278", + "correct_value": 91278, + "explanation": "Round to compatible pair 5000×20 = 100000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5028 * 14", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8472, + "pure_math": { + "question": "Estimate 5028 × 14.", + "options": [ + "70402", + "70412", + "70382", + "70392" + ], + "answer": "70392", + "correct_value": 70392, + "explanation": "One number rounds to 5000, other is farther from 20." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5644 * 29", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5644 × 29.", + "options": [ + "163676", + "163686", + "163696", + "163666" + ], + "answer": "163676", + "correct_value": 163676, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001415", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2500 * 44", + "ns_shortcut": "Round to 2500×40 for easy mental computation.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Estimate 2500 × 44.", + "options": [ + "109990", + "110010", + "110000", + "110020" + ], + "answer": "110000", + "correct_value": 110000, + "explanation": "Round to compatible pair 2500×40 = 100000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2522 * 51", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8581, + "pure_math": { + "question": "Estimate 2522 × 51.", + "options": [ + "128642", + "128632", + "128612", + "128622" + ], + "answer": "128622", + "correct_value": 128622, + "explanation": "One number rounds to 2500, other is farther from 40." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2917 * 59", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 2917 × 59.", + "options": [ + "172093", + "172123", + "172103", + "172113" + ], + "answer": "172103", + "correct_value": 172103, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001416", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12320 * 8126", + "ns_shortcut": "Round to 12500×8000 for easy mental computation.", + "shortcut_strength": 0.984925, + "pure_math": { + "question": "Estimate 12320 × 8126.", + "options": [ + "100112330", + "100112320", + "100112340", + "100112310" + ], + "answer": "100112320", + "correct_value": 100112320, + "explanation": "Round to compatible pair 12500×8000 = 100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "12451 * 10594", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.835915, + "pure_math": { + "question": "Estimate 12451 × 10594.", + "options": [ + "131905914", + "131905894", + "131905904", + "131905884" + ], + "answer": "131905894", + "correct_value": 131905894, + "explanation": "One number rounds to 12500, other is farther from 8000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9659902900757571 * 8347009167623358", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 9659902900757571 × 8347009167623358.", + "options": [ + "80631298070974914134063794943428", + "80631298070974914134063794943418", + "80631298070974914134063794943408", + "80631298070974914134063794943438" + ], + "answer": "80631298070974914134063794943418", + "correct_value": 80631298070974914134063794943418, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001417", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12306 * 8178", + "ns_shortcut": "Round to 12500×8000 for easy mental computation.", + "shortcut_strength": 0.981115, + "pure_math": { + "question": "Estimate 12306 × 8178.", + "options": [ + "100638488", + "100638468", + "100638458", + "100638478" + ], + "answer": "100638468", + "correct_value": 100638468, + "explanation": "Round to compatible pair 12500×8000 = 100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "12319 * 6129", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8758225, + "pure_math": { + "question": "Estimate 12319 × 6129.", + "options": [ + "75503151", + "75503171", + "75503161", + "75503141" + ], + "answer": "75503151", + "correct_value": 75503151, + "explanation": "One number rounds to 12500, other is farther from 8000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1174774348535227 * 3143675754891268", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1174774348535227 × 3143675754891268.", + "options": [ + "3693109636958377318856452697836", + "3693109636958377318856452697856", + "3693109636958377318856452697846", + "3693109636958377318856452697826" + ], + "answer": "3693109636958377318856452697836", + "correct_value": 3693109636958377318856452697836, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001418", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12367 * 7555", + "ns_shortcut": "Round to 12500×8000 for easy mental computation.", + "shortcut_strength": 0.9668675, + "pure_math": { + "question": "Estimate 12367 × 7555.", + "options": [ + "93432685", + "93432675", + "93432695", + "93432705" + ], + "answer": "93432685", + "correct_value": 93432685, + "explanation": "Round to compatible pair 12500×8000 = 100000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "12624 * 6211", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8832275, + "pure_math": { + "question": "Estimate 12624 × 6211.", + "options": [ + "78407664", + "78407684", + "78407674", + "78407654" + ], + "answer": "78407664", + "correct_value": 78407664, + "explanation": "One number rounds to 12500, other is farther from 8000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5466174183547032 * 1855605852007037", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5466174183547032 × 1855605852007037.", + "options": [ + "10143064803079660164160884464204", + "10143064803079660164160884464184", + "10143064803079660164160884464194", + "10143064803079660164160884464174" + ], + "answer": "10143064803079660164160884464184", + "correct_value": 10143064803079660164160884464184, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001419", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5095 * 21", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9655, + "pure_math": { + "question": "Estimate 5095 × 21.", + "options": [ + "107015", + "107005", + "106985", + "106995" + ], + "answer": "106995", + "correct_value": 106995, + "explanation": "Round to compatible pair 5000×20 = 100000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5059 * 26", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8441000000000001, + "pure_math": { + "question": "Estimate 5059 × 26.", + "options": [ + "131554", + "131534", + "131544", + "131524" + ], + "answer": "131534", + "correct_value": 131534, + "explanation": "One number rounds to 5000, other is farther from 20." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5771 * 27", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5771 × 27.", + "options": [ + "155807", + "155837", + "155827", + "155817" + ], + "answer": "155817", + "correct_value": 155817, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_001420", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12278 * 7286", + "ns_shortcut": "Round to 12500×8000 for easy mental computation.", + "shortcut_strength": 0.946495, + "pure_math": { + "question": "Estimate 12278 × 7286.", + "options": [ + "89457528", + "89457498", + "89457508", + "89457518" + ], + "answer": "89457508", + "correct_value": 89457508, + "explanation": "Round to compatible pair 12500×8000 = 100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "12383 * 10057", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8667575000000001, + "pure_math": { + "question": "Estimate 12383 × 10057.", + "options": [ + "124535831", + "124535851", + "124535841", + "124535821" + ], + "answer": "124535831", + "correct_value": 124535831, + "explanation": "One number rounds to 12500, other is farther from 8000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8919221252986174 * 4752771007203461", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 8919221252986174 × 4752771007203461.", + "options": [ + "42391016178025613634411037948224", + "42391016178025613634411037948234", + "42391016178025613634411037948214", + "42391016178025613634411037948204" + ], + "answer": "42391016178025613634411037948214", + "correct_value": 42391016178025613634411037948214, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001421", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1225 * 85", + "ns_shortcut": "Round to 1250×80 for easy mental computation.", + "shortcut_strength": 0.95875, + "pure_math": { + "question": "Estimate 1225 × 85.", + "options": [ + "104145", + "104125", + "104135", + "104115" + ], + "answer": "104125", + "correct_value": 104125, + "explanation": "Round to compatible pair 1250×80 = 100000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1227 * 100", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8658, + "pure_math": { + "question": "Estimate 1227 × 100.", + "options": [ + "122690", + "122710", + "122700", + "122720" + ], + "answer": "122700", + "correct_value": 122700, + "explanation": "One number rounds to 1250, other is farther from 80." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1542 * 116", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1542 × 116.", + "options": [ + "178892", + "178872", + "178882", + "178862" + ], + "answer": "178872", + "correct_value": 178872, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001422", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1244 * 74", + "ns_shortcut": "Round to 1250×80 for easy mental computation.", + "shortcut_strength": 0.9601, + "pure_math": { + "question": "Estimate 1244 × 74.", + "options": [ + "92066", + "92076", + "92056", + "92046" + ], + "answer": "92056", + "correct_value": 92056, + "explanation": "Round to compatible pair 1250×80 = 100000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1242 * 96", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8968, + "pure_math": { + "question": "Estimate 1242 × 96.", + "options": [ + "119242", + "119252", + "119222", + "119232" + ], + "answer": "119232", + "correct_value": 119232, + "explanation": "One number rounds to 1250, other is farther from 80." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1437 * 119", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1437 × 119.", + "options": [ + "171003", + "171013", + "170993", + "171023" + ], + "answer": "171003", + "correct_value": 171003, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001423", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12539 * 8265", + "ns_shortcut": "Round to 12500×8000 for easy mental computation.", + "shortcut_strength": 0.9818775, + "pure_math": { + "question": "Estimate 12539 × 8265.", + "options": [ + "103634825", + "103634845", + "103634835", + "103634855" + ], + "answer": "103634835", + "correct_value": 103634835, + "explanation": "Round to compatible pair 12500×8000 = 100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "12454 * 9933", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8773475, + "pure_math": { + "question": "Estimate 12454 × 9933.", + "options": [ + "123705572", + "123705592", + "123705582", + "123705602" + ], + "answer": "123705582", + "correct_value": 123705582, + "explanation": "One number rounds to 12500, other is farther from 8000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3378032915225661 * 1485563249512574", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 3378032915225661 × 1485563249512574.", + "options": [ + "5018281554503066366821466961424", + "5018281554503066366821466961404", + "5018281554503066366821466961414", + "5018281554503066366821466961434" + ], + "answer": "5018281554503066366821466961414", + "correct_value": 5018281554503066366821466961414, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001424", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12265 * 8114", + "ns_shortcut": "Round to 12500×8000 for easy mental computation.", + "shortcut_strength": 0.983475, + "pure_math": { + "question": "Estimate 12265 × 8114.", + "options": [ + "99518220", + "99518200", + "99518230", + "99518210" + ], + "answer": "99518210", + "correct_value": 99518210, + "explanation": "Round to compatible pair 12500×8000 = 100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "12281 * 6128", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.87424, + "pure_math": { + "question": "Estimate 12281 × 6128.", + "options": [ + "75257958", + "75257978", + "75257968", + "75257988" + ], + "answer": "75257968", + "correct_value": 75257968, + "explanation": "One number rounds to 12500, other is farther from 8000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1499948280203443 * 9484714131858546", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1499948280203443 × 9484714131858546.", + "options": [ + "14226580650302517973128578173888", + "14226580650302517973128578173878", + "14226580650302517973128578173868", + "14226580650302517973128578173898" + ], + "answer": "14226580650302517973128578173878", + "correct_value": 14226580650302517973128578173878, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001425", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1234 * 76", + "ns_shortcut": "Round to 1250×80 for easy mental computation.", + "shortcut_strength": 0.9686, + "pure_math": { + "question": "Estimate 1234 × 76.", + "options": [ + "93774", + "93784", + "93804", + "93794" + ], + "answer": "93784", + "correct_value": 93784, + "explanation": "Round to compatible pair 1250×80 = 100000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1250 * 55", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.84375, + "pure_math": { + "question": "Estimate 1250 × 55.", + "options": [ + "68750", + "68770", + "68740", + "68760" + ], + "answer": "68750", + "correct_value": 68750, + "explanation": "One number rounds to 1250, other is farther from 80." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1448 * 119", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1448 × 119.", + "options": [ + "172302", + "172332", + "172322", + "172312" + ], + "answer": "172312", + "correct_value": 172312, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_001426", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12631 * 7301", + "ns_shortcut": "Round to 12500×8000 for easy mental computation.", + "shortcut_strength": 0.9510725, + "pure_math": { + "question": "Estimate 12631 × 7301.", + "options": [ + "92218921", + "92218931", + "92218941", + "92218951" + ], + "answer": "92218931", + "correct_value": 92218931, + "explanation": "Round to compatible pair 12500×8000 = 100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "12678 * 5849", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8584425, + "pure_math": { + "question": "Estimate 12678 × 5849.", + "options": [ + "74153612", + "74153622", + "74153642", + "74153632" + ], + "answer": "74153622", + "correct_value": 74153622, + "explanation": "One number rounds to 12500, other is farther from 8000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5231253370538667 * 9052636122378665", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5231253370538667 × 9052636122378665.", + "options": [ + "47356633227453480039484398339565", + "47356633227453480039484398339545", + "47356633227453480039484398339555", + "47356633227453480039484398339575" + ], + "answer": "47356633227453480039484398339555", + "correct_value": 47356633227453480039484398339555, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001427", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12384 * 7642", + "ns_shortcut": "Round to 12500×8000 for easy mental computation.", + "shortcut_strength": 0.972985, + "pure_math": { + "question": "Estimate 12384 × 7642.", + "options": [ + "94638528", + "94638538", + "94638518", + "94638548" + ], + "answer": "94638528", + "correct_value": 94638528, + "explanation": "Round to compatible pair 12500×8000 = 100000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "12307 * 5480", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.83478, + "pure_math": { + "question": "Estimate 12307 × 5480.", + "options": [ + "67442370", + "67442350", + "67442360", + "67442380" + ], + "answer": "67442360", + "correct_value": 67442360, + "explanation": "One number rounds to 12500, other is farther from 8000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3671576375942932 * 6648265023707863", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 3671576375942932 × 6648265023707863.", + "options": [ + "24409612802053466527871527674326", + "24409612802053466527871527674316", + "24409612802053466527871527674336", + "24409612802053466527871527674306" + ], + "answer": "24409612802053466527871527674316", + "correct_value": 24409612802053466527871527674316, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001428", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2454 * 36", + "ns_shortcut": "Round to 2500×40 for easy mental computation.", + "shortcut_strength": 0.9408, + "pure_math": { + "question": "Estimate 2454 × 36.", + "options": [ + "88354", + "88344", + "88364", + "88334" + ], + "answer": "88344", + "correct_value": 88344, + "explanation": "Round to compatible pair 2500×40 = 100000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2505 * 51", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8614999999999999, + "pure_math": { + "question": "Estimate 2505 × 51.", + "options": [ + "127755", + "127745", + "127765", + "127775" + ], + "answer": "127755", + "correct_value": 127755, + "explanation": "One number rounds to 2500, other is farther from 40." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2904 * 54", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 2904 × 54.", + "options": [ + "156806", + "156836", + "156816", + "156826" + ], + "answer": "156816", + "correct_value": 156816, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001429", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1234 * 80", + "ns_shortcut": "Round to 1250×80 for easy mental computation.", + "shortcut_strength": 0.9936, + "pure_math": { + "question": "Estimate 1234 × 80.", + "options": [ + "98720", + "98740", + "98710", + "98730" + ], + "answer": "98720", + "correct_value": 98720, + "explanation": "Round to compatible pair 1250×80 = 100000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1242 * 101", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.86555, + "pure_math": { + "question": "Estimate 1242 × 101.", + "options": [ + "125462", + "125432", + "125452", + "125442" + ], + "answer": "125442", + "correct_value": 125442, + "explanation": "One number rounds to 1250, other is farther from 80." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1534 * 120", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1534 × 120.", + "options": [ + "184090", + "184080", + "184070", + "184100" + ], + "answer": "184080", + "correct_value": 184080, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001430", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2535 * 42", + "ns_shortcut": "Round to 2500×40 for easy mental computation.", + "shortcut_strength": 0.968, + "pure_math": { + "question": "Estimate 2535 × 42.", + "options": [ + "106460", + "106480", + "106490", + "106470" + ], + "answer": "106470", + "correct_value": 106470, + "explanation": "Round to compatible pair 2500×40 = 100000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2490 * 50", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.873, + "pure_math": { + "question": "Estimate 2490 × 50.", + "options": [ + "124510", + "124500", + "124490", + "124520" + ], + "answer": "124500", + "correct_value": 124500, + "explanation": "One number rounds to 2500, other is farther from 40." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2942 * 60", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 2942 × 60.", + "options": [ + "176530", + "176520", + "176510", + "176540" + ], + "answer": "176520", + "correct_value": 176520, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001431", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4988 * 19", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9738, + "pure_math": { + "question": "Estimate 4988 × 19.", + "options": [ + "94782", + "94762", + "94792", + "94772" + ], + "answer": "94772", + "correct_value": 94772, + "explanation": "Round to compatible pair 5000×20 = 100000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4975 * 15", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8725, + "pure_math": { + "question": "Estimate 4975 × 15.", + "options": [ + "74645", + "74635", + "74625", + "74615" + ], + "answer": "74625", + "correct_value": 74625, + "explanation": "One number rounds to 5000, other is farther from 20." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5630 * 30", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5630 × 30.", + "options": [ + "168920", + "168900", + "168890", + "168910" + ], + "answer": "168900", + "correct_value": 168900, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001432", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2536 * 41", + "ns_shortcut": "Round to 2500×40 for easy mental computation.", + "shortcut_strength": 0.9803, + "pure_math": { + "question": "Estimate 2536 × 41.", + "options": [ + "103986", + "103966", + "103996", + "103976" + ], + "answer": "103976", + "correct_value": 103976, + "explanation": "Round to compatible pair 2500×40 = 100000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2496 * 49", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8867, + "pure_math": { + "question": "Estimate 2496 × 49.", + "options": [ + "122324", + "122314", + "122304", + "122294" + ], + "answer": "122304", + "correct_value": 122304, + "explanation": "One number rounds to 2500, other is farther from 40." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2796 * 54", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 2796 × 54.", + "options": [ + "150984", + "150994", + "150974", + "151004" + ], + "answer": "150984", + "correct_value": 150984, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001433", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2485 * 38", + "ns_shortcut": "Round to 2500×40 for easy mental computation.", + "shortcut_strength": 0.972, + "pure_math": { + "question": "Estimate 2485 × 38.", + "options": [ + "94420", + "94430", + "94440", + "94450" + ], + "answer": "94430", + "correct_value": 94430, + "explanation": "Round to compatible pair 2500×40 = 100000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2496 * 49", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8867, + "pure_math": { + "question": "Estimate 2496 × 49.", + "options": [ + "122314", + "122324", + "122294", + "122304" + ], + "answer": "122304", + "correct_value": 122304, + "explanation": "One number rounds to 2500, other is farther from 40." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2907 * 57", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 2907 × 57.", + "options": [ + "165689", + "165719", + "165709", + "165699" + ], + "answer": "165699", + "correct_value": 165699, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_001434", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2497 * 42", + "ns_shortcut": "Round to 2500×40 for easy mental computation.", + "shortcut_strength": 0.9744, + "pure_math": { + "question": "Estimate 2497 × 42.", + "options": [ + "104874", + "104884", + "104864", + "104894" + ], + "answer": "104874", + "correct_value": 104874, + "explanation": "Round to compatible pair 2500×40 = 100000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2529 * 53", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8317, + "pure_math": { + "question": "Estimate 2529 × 53.", + "options": [ + "134057", + "134047", + "134037", + "134027" + ], + "answer": "134037", + "correct_value": 134037, + "explanation": "One number rounds to 2500, other is farther from 40." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2916 * 57", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 2916 × 57.", + "options": [ + "166222", + "166202", + "166232", + "166212" + ], + "answer": "166212", + "correct_value": 166212, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_001435", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2474 * 42", + "ns_shortcut": "Round to 2500×40 for easy mental computation.", + "shortcut_strength": 0.9698, + "pure_math": { + "question": "Estimate 2474 × 42.", + "options": [ + "103928", + "103898", + "103908", + "103918" + ], + "answer": "103908", + "correct_value": 103908, + "explanation": "Round to compatible pair 2500×40 = 100000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2543 * 28", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8414, + "pure_math": { + "question": "Estimate 2543 × 28.", + "options": [ + "71224", + "71214", + "71194", + "71204" + ], + "answer": "71204", + "correct_value": 71204, + "explanation": "One number rounds to 2500, other is farther from 40." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2803 * 59", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 2803 × 59.", + "options": [ + "165387", + "165377", + "165397", + "165367" + ], + "answer": "165377", + "correct_value": 165377, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001436", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2452 * 39", + "ns_shortcut": "Round to 2500×40 for easy mental computation.", + "shortcut_strength": 0.9779, + "pure_math": { + "question": "Estimate 2452 × 39.", + "options": [ + "95638", + "95618", + "95628", + "95648" + ], + "answer": "95628", + "correct_value": 95628, + "explanation": "Round to compatible pair 2500×40 = 100000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2534 * 29", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8557, + "pure_math": { + "question": "Estimate 2534 × 29.", + "options": [ + "73476", + "73506", + "73486", + "73496" + ], + "answer": "73486", + "correct_value": 73486, + "explanation": "One number rounds to 2500, other is farther from 40." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3074 * 60", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 3074 × 60.", + "options": [ + "184440", + "184450", + "184430", + "184460" + ], + "answer": "184440", + "correct_value": 184440, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001437", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5079 * 20", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9921, + "pure_math": { + "question": "Estimate 5079 × 20.", + "options": [ + "101590", + "101570", + "101600", + "101580" + ], + "answer": "101580", + "correct_value": 101580, + "explanation": "Round to compatible pair 5000×20 = 100000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5063 * 14", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8437, + "pure_math": { + "question": "Estimate 5063 × 14.", + "options": [ + "70902", + "70892", + "70872", + "70882" + ], + "answer": "70882", + "correct_value": 70882, + "explanation": "One number rounds to 5000, other is farther from 20." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6082 * 27", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 6082 × 27.", + "options": [ + "164214", + "164204", + "164234", + "164224" + ], + "answer": "164214", + "correct_value": 164214, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001438", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25115 * 3718", + "ns_shortcut": "Round to 25000×4000 for easy mental computation.", + "shortcut_strength": 0.96245, + "pure_math": { + "question": "Estimate 25115 × 3718.", + "options": [ + "93377560", + "93377570", + "93377580", + "93377590" + ], + "answer": "93377570", + "correct_value": 93377570, + "explanation": "Round to compatible pair 25000×4000 = 100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "25083 * 2897", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.860465, + "pure_math": { + "question": "Estimate 25083 × 2897.", + "options": [ + "72665461", + "72665451", + "72665441", + "72665471" + ], + "answer": "72665451", + "correct_value": 72665451, + "explanation": "One number rounds to 25000, other is farther from 4000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9629849444815274 * 9393404118088864", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 9629849444815274 × 9393404118088864.", + "options": [ + "90457067431503555481962396508756", + "90457067431503555481962396508736", + "90457067431503555481962396508726", + "90457067431503555481962396508746" + ], + "answer": "90457067431503555481962396508736", + "correct_value": 90457067431503555481962396508736, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001439", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12580 * 8354", + "ns_shortcut": "Round to 12500×8000 for easy mental computation.", + "shortcut_strength": 0.974675, + "pure_math": { + "question": "Estimate 12580 × 8354.", + "options": [ + "105093340", + "105093320", + "105093310", + "105093330" + ], + "answer": "105093320", + "correct_value": 105093320, + "explanation": "Round to compatible pair 12500×8000 = 100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "12630 * 6310", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.889175, + "pure_math": { + "question": "Estimate 12630 × 6310.", + "options": [ + "79695290", + "79695300", + "79695320", + "79695310" + ], + "answer": "79695300", + "correct_value": 79695300, + "explanation": "One number rounds to 12500, other is farther from 8000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6887467545148327 * 6152846126004433", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 6887467545148327 × 6152846126004433.", + "options": [ + "42377528003147146020958844533601", + "42377528003147146020958844533591", + "42377528003147146020958844533611", + "42377528003147146020958844533581" + ], + "answer": "42377528003147146020958844533591", + "correct_value": 42377528003147146020958844533591, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001440", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1237 * 84", + "ns_shortcut": "Round to 1250×80 for easy mental computation.", + "shortcut_strength": 0.9698, + "pure_math": { + "question": "Estimate 1237 × 84.", + "options": [ + "103908", + "103928", + "103918", + "103898" + ], + "answer": "103908", + "correct_value": 103908, + "explanation": "Round to compatible pair 1250×80 = 100000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1272 * 96", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8912, + "pure_math": { + "question": "Estimate 1272 × 96.", + "options": [ + "122132", + "122102", + "122122", + "122112" + ], + "answer": "122112", + "correct_value": 122112, + "explanation": "One number rounds to 1250, other is farther from 80." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1537 * 114", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1537 × 114.", + "options": [ + "175218", + "175208", + "175228", + "175238" + ], + "answer": "175218", + "correct_value": 175218, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001441", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1249 * 74", + "ns_shortcut": "Round to 1250×80 for easy mental computation.", + "shortcut_strength": 0.9621, + "pure_math": { + "question": "Estimate 1249 × 74.", + "options": [ + "92426", + "92436", + "92416", + "92446" + ], + "answer": "92426", + "correct_value": 92426, + "explanation": "Round to compatible pair 1250×80 = 100000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1275 * 61", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.87125, + "pure_math": { + "question": "Estimate 1275 × 61.", + "options": [ + "77775", + "77785", + "77795", + "77765" + ], + "answer": "77775", + "correct_value": 77775, + "explanation": "One number rounds to 1250, other is farther from 80." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1561 * 120", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1561 × 120.", + "options": [ + "187320", + "187340", + "187330", + "187310" + ], + "answer": "187320", + "correct_value": 187320, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001442", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4974 * 22", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9474, + "pure_math": { + "question": "Estimate 4974 × 22.", + "options": [ + "109448", + "109418", + "109438", + "109428" + ], + "answer": "109428", + "correct_value": 109428, + "explanation": "Round to compatible pair 5000×20 = 100000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4925 * 16", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8925, + "pure_math": { + "question": "Estimate 4925 × 16.", + "options": [ + "78810", + "78790", + "78820", + "78800" + ], + "answer": "78800", + "correct_value": 78800, + "explanation": "One number rounds to 5000, other is farther from 20." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6069 * 29", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 6069 × 29.", + "options": [ + "175991", + "176021", + "176011", + "176001" + ], + "answer": "176001", + "correct_value": 176001, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_001443", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1252 * 77", + "ns_shortcut": "Round to 1250×80 for easy mental computation.", + "shortcut_strength": 0.98045, + "pure_math": { + "question": "Estimate 1252 × 77.", + "options": [ + "96424", + "96414", + "96394", + "96404" + ], + "answer": "96404", + "correct_value": 96404, + "explanation": "Round to compatible pair 1250×80 = 100000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1226 * 104", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8404, + "pure_math": { + "question": "Estimate 1226 × 104.", + "options": [ + "127514", + "127504", + "127494", + "127524" + ], + "answer": "127504", + "correct_value": 127504, + "explanation": "One number rounds to 1250, other is farther from 80." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1518 * 108", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1518 × 108.", + "options": [ + "163934", + "163944", + "163954", + "163964" + ], + "answer": "163944", + "correct_value": 163944, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001444", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2529 * 38", + "ns_shortcut": "Round to 2500×40 for easy mental computation.", + "shortcut_strength": 0.9692, + "pure_math": { + "question": "Estimate 2529 × 38.", + "options": [ + "96102", + "96122", + "96092", + "96112" + ], + "answer": "96102", + "correct_value": 96102, + "explanation": "Round to compatible pair 2500×40 = 100000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2465 * 28", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.843, + "pure_math": { + "question": "Estimate 2465 × 28.", + "options": [ + "69010", + "69030", + "69020", + "69040" + ], + "answer": "69020", + "correct_value": 69020, + "explanation": "One number rounds to 2500, other is farther from 40." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2992 * 56", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 2992 × 56.", + "options": [ + "167542", + "167562", + "167572", + "167552" + ], + "answer": "167552", + "correct_value": 167552, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_001445", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4916 * 21", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9666, + "pure_math": { + "question": "Estimate 4916 × 21.", + "options": [ + "103226", + "103246", + "103256", + "103236" + ], + "answer": "103236", + "correct_value": 103236, + "explanation": "Round to compatible pair 5000×20 = 100000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5080 * 25", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.867, + "pure_math": { + "question": "Estimate 5080 × 25.", + "options": [ + "127010", + "127000", + "127020", + "126990" + ], + "answer": "127000", + "correct_value": 127000, + "explanation": "One number rounds to 5000, other is farther from 20." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6107 * 27", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 6107 × 27.", + "options": [ + "164899", + "164889", + "164879", + "164909" + ], + "answer": "164889", + "correct_value": 164889, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001446", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4909 * 20", + "ns_shortcut": "Round to 5000×20 for easy mental computation.", + "shortcut_strength": 0.9909, + "pure_math": { + "question": "Estimate 4909 × 20.", + "options": [ + "98170", + "98200", + "98180", + "98190" + ], + "answer": "98180", + "correct_value": 98180, + "explanation": "Round to compatible pair 5000×20 = 100000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5026 * 15", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8724000000000001, + "pure_math": { + "question": "Estimate 5026 × 15.", + "options": [ + "75390", + "75380", + "75400", + "75410" + ], + "answer": "75390", + "correct_value": 75390, + "explanation": "One number rounds to 5000, other is farther from 20." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6197 * 29", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 6197 × 29.", + "options": [ + "179713", + "179733", + "179723", + "179703" + ], + "answer": "179713", + "correct_value": 179713, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001447", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2517 * 44", + "ns_shortcut": "Round to 2500×40 for easy mental computation.", + "shortcut_strength": 0.9466, + "pure_math": { + "question": "Estimate 2517 × 44.", + "options": [ + "110748", + "110768", + "110738", + "110758" + ], + "answer": "110748", + "correct_value": 110748, + "explanation": "Round to compatible pair 2500×40 = 100000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2536 * 52", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8428, + "pure_math": { + "question": "Estimate 2536 × 52.", + "options": [ + "131892", + "131872", + "131882", + "131862" + ], + "answer": "131872", + "correct_value": 131872, + "explanation": "One number rounds to 2500, other is farther from 40." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2790 * 56", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 2790 × 56.", + "options": [ + "156250", + "156240", + "156230", + "156260" + ], + "answer": "156240", + "correct_value": 156240, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001448", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2520 * 42", + "ns_shortcut": "Round to 2500×40 for easy mental computation.", + "shortcut_strength": 0.971, + "pure_math": { + "question": "Estimate 2520 × 42.", + "options": [ + "105840", + "105830", + "105860", + "105850" + ], + "answer": "105840", + "correct_value": 105840, + "explanation": "Round to compatible pair 2500×40 = 100000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2534 * 50", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8682, + "pure_math": { + "question": "Estimate 2534 × 50.", + "options": [ + "126710", + "126690", + "126720", + "126700" + ], + "answer": "126700", + "correct_value": 126700, + "explanation": "One number rounds to 2500, other is farther from 40." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3111 * 57", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 3111 × 57.", + "options": [ + "177337", + "177327", + "177347", + "177317" + ], + "answer": "177327", + "correct_value": 177327, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001449", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12604 * 8724", + "ns_shortcut": "Round to 12500×8000 for easy mental computation.", + "shortcut_strength": 0.95059, + "pure_math": { + "question": "Estimate 12604 × 8724.", + "options": [ + "109957286", + "109957296", + "109957306", + "109957316" + ], + "answer": "109957296", + "correct_value": 109957296, + "explanation": "Round to compatible pair 12500×8000 = 100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "12707 * 5559", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8391575, + "pure_math": { + "question": "Estimate 12707 × 5559.", + "options": [ + "70638213", + "70638203", + "70638233", + "70638223" + ], + "answer": "70638213", + "correct_value": 70638213, + "explanation": "One number rounds to 12500, other is farther from 8000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3183236220829731 * 9057157894685061", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 3183236220829731 × 9057157894685061.", + "options": [ + "28831073068135436345224350348581", + "28831073068135436345224350348591", + "28831073068135436345224350348611", + "28831073068135436345224350348601" + ], + "answer": "28831073068135436345224350348591", + "correct_value": 28831073068135436345224350348591, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001450", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "50726 * 2178", + "ns_shortcut": "Round to 50000×2000 for easy mental computation.", + "shortcut_strength": 0.94824, + "pure_math": { + "question": "Estimate 50726 × 2178.", + "options": [ + "110481238", + "110481218", + "110481228", + "110481248" + ], + "answer": "110481228", + "correct_value": 110481228, + "explanation": "Round to compatible pair 50000×2000 = 100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "49978 * 2500", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.87478, + "pure_math": { + "question": "Estimate 49978 × 2500.", + "options": [ + "124945000", + "124945010", + "124945020", + "124944990" + ], + "answer": "124945000", + "correct_value": 124945000, + "explanation": "One number rounds to 50000, other is farther from 2000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1720962281842074 * 1674691518836428", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1720962281842074 × 1674691518836428.", + "options": [ + "2882080937638307782804934271662", + "2882080937638307782804934271692", + "2882080937638307782804934271672", + "2882080937638307782804934271682" + ], + "answer": "2882080937638307782804934271672", + "correct_value": 2882080937638307782804934271672, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_001451", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12731 * 8051", + "ns_shortcut": "Round to 12500×8000 for easy mental computation.", + "shortcut_strength": 0.9875725, + "pure_math": { + "question": "Estimate 12731 × 8051.", + "options": [ + "102497301", + "102497281", + "102497271", + "102497291" + ], + "answer": "102497281", + "correct_value": 102497281, + "explanation": "Round to compatible pair 12500×8000 = 100000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "12677 * 10324", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.84767, + "pure_math": { + "question": "Estimate 12677 × 10324.", + "options": [ + "130877358", + "130877338", + "130877368", + "130877348" + ], + "answer": "130877348", + "correct_value": 130877348, + "explanation": "One number rounds to 12500, other is farther from 8000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3630342636561734 * 7839339991393261", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 3630342636561734 × 7839339991393261.", + "options": [ + "28459490213258452262101298074564", + "28459490213258452262101298074584", + "28459490213258452262101298074574", + "28459490213258452262101298074594" + ], + "answer": "28459490213258452262101298074574", + "correct_value": 28459490213258452262101298074574, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "landmark_comparison_001452", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "50% of 17459110716362590 vs 8050826308646471", + "ns_shortcut": "Use 50% as landmark: 50% of 17459110716362590 = 8729555358181295.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 50% of 17459110716362590 greater than 8050826308646471?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 8729555358181295.0, + "explanation": "50% ≈ 50%. 50% of 17459110716362590 = 8729555358181295. Compare to 8050826308646471." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "46% of 17459110716362590 vs 8050826308646471", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 46% of 17459110716362590 greater than 8050826308646471?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 8031190929526791.0, + "explanation": "46% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 17459110716362590 vs 8050826308646471", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 17459110716362590 greater than 8050826308646471?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 1.1697604179962936e+16, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001453", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "52% of 10451081240165040 vs 3592814225867691", + "ns_shortcut": "Use 50% as landmark: 50% of 10451081240165040 = 5225540620082520.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 52% of 10451081240165040 greater than 3592814225867691?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 5434562244885821.0, + "explanation": "52% ≈ 50%. 50% of 10451081240165040 = 5225540620082520. Compare to 3592814225867691." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "45% of 10451081240165040 vs 3592814225867691", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 45% of 10451081240165040 greater than 3592814225867691?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 4702986558074268.0, + "explanation": "45% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 10451081240165040 vs 3592814225867691", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 10451081240165040 greater than 3592814225867691?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 8674397429336983.0, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001454", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "24% of 46050855933712470 vs 22664602861215469", + "ns_shortcut": "Use 25% as landmark: 25% of 46050855933712470 = 11512713983428117.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 24% of 46050855933712470 greater than 22664602861215469?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 1.1052205424090992e+16, + "explanation": "24% ≈ 25%. 25% of 46050855933712470 = 11512713983428117. Compare to 22664602861215469." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "30% of 46050855933712470 vs 22664602861215469", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 30% of 46050855933712470 greater than 22664602861215469?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 1.381525678011374e+16, + "explanation": "30% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 46050855933712470 vs 22664602861215469", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 46050855933712470 greater than 22664602861215469?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 3.0854073475587356e+16, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001455", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "49% of 89331027387217860 vs 30297949728752351", + "ns_shortcut": "Use 50% as landmark: 50% of 89331027387217860 = 44665513693608930.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 49% of 89331027387217860 greater than 30297949728752351?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 4.377220341973675e+16, + "explanation": "49% ≈ 50%. 50% of 89331027387217860 = 44665513693608930. Compare to 30297949728752351." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "43% of 89331027387217860 vs 30297949728752351", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 43% of 89331027387217860 greater than 30297949728752351?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 3.841234177650368e+16, + "explanation": "43% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 89331027387217860 vs 30297949728752351", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 89331027387217860 greater than 30297949728752351?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 6.521164999266904e+16, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001456", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9% of 87082915564319140 vs 31932610460138434", + "ns_shortcut": "Use 10% as landmark: 10% of 87082915564319140 = 8708291556431914.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 9% of 87082915564319140 greater than 31932610460138434?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 7837462400788723.0, + "explanation": "9% ≈ 10%. 10% of 87082915564319140 = 8708291556431914. Compare to 31932610460138434." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7% of 87082915564319140 vs 31932610460138434", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 7% of 87082915564319140 greater than 31932610460138434?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 6095804089502340.0, + "explanation": "7% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 87082915564319140 vs 31932610460138434", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 87082915564319140 greater than 31932610460138434?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 5.834555342809382e+16, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001457", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12% of 28573790056161910 vs 14048027054512305", + "ns_shortcut": "Use 10% as landmark: 10% of 28573790056161910 = 2857379005616191.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 12% of 28573790056161910 greater than 14048027054512305?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 3428854806739429.0, + "explanation": "12% ≈ 10%. 10% of 28573790056161910 = 2857379005616191. Compare to 14048027054512305." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7% of 28573790056161910 vs 14048027054512305", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 7% of 28573790056161910 greater than 14048027054512305?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 2000165303931333.8, + "explanation": "7% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 28573790056161910 vs 14048027054512305", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 28573790056161910 greater than 14048027054512305?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 2.3716245746614384e+16, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001458", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "26% of 23795163979851260 vs 9539295793704942", + "ns_shortcut": "Use 25% as landmark: 25% of 23795163979851260 = 5948790994962815.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 26% of 23795163979851260 greater than 9539295793704942?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 6186742634761328.0, + "explanation": "26% ≈ 25%. 25% of 23795163979851260 = 5948790994962815. Compare to 9539295793704942." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "18% of 23795163979851260 vs 9539295793704942", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 18% of 23795163979851260 greater than 9539295793704942?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 4283129516373227.0, + "explanation": "18% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 23795163979851260 vs 9539295793704942", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 23795163979851260 greater than 9539295793704942?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 1.5942759866500344e+16, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001459", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "11% of 27011507141015730 vs 10554357249151569", + "ns_shortcut": "Use 10% as landmark: 10% of 27011507141015730 = 2701150714101573.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 11% of 27011507141015730 greater than 10554357249151569?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 2971265785511730.5, + "explanation": "11% ≈ 10%. 10% of 27011507141015730 = 2701150714101573. Compare to 10554357249151569." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6% of 27011507141015730 vs 10554357249151569", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 6% of 27011507141015730 greater than 10554357249151569?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 1620690428460943.8, + "explanation": "6% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "87% of 27011507141015730 vs 10554357249151569", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 87% of 27011507141015730 greater than 10554357249151569?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 2.3500011212683684e+16, + "explanation": "87% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001460", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 38429370135837590 vs 11686612358710279", + "ns_shortcut": "Use 10% as landmark: 10% of 38429370135837590 = 3842937013583759.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 38429370135837590 greater than 11686612358710279?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 3842937013583759.0, + "explanation": "10% ≈ 10%. 10% of 38429370135837590 = 3842937013583759. Compare to 11686612358710279." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6% of 38429370135837590 vs 11686612358710279", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 6% of 38429370135837590 greater than 11686612358710279?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 2305762208150255.5, + "explanation": "6% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 38429370135837590 vs 11686612358710279", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 38429370135837590 greater than 11686612358710279?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 1.6524629158410164e+16, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001461", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "51% of 26428481246050850 vs 7625961294369026", + "ns_shortcut": "Use 50% as landmark: 50% of 26428481246050850 = 13214240623025425.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 51% of 26428481246050850 greater than 7625961294369026?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 1.3478525435485934e+16, + "explanation": "51% ≈ 50%. 50% of 26428481246050850 = 13214240623025425. Compare to 7625961294369026." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "57% of 26428481246050850 vs 7625961294369026", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 57% of 26428481246050850 greater than 7625961294369026?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 1.5064234310248984e+16, + "explanation": "57% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 26428481246050850 vs 7625961294369026", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 26428481246050850 greater than 7625961294369026?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 1.770708243485407e+16, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001462", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "24% of 77408836909998320 vs 15572835271909931", + "ns_shortcut": "Use 25% as landmark: 25% of 77408836909998320 = 19352209227499580.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 24% of 77408836909998320 greater than 15572835271909931?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 1.8578120858399596e+16, + "explanation": "24% ≈ 25%. 25% of 77408836909998320 = 19352209227499580. Compare to 15572835271909931." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "28% of 77408836909998320 vs 15572835271909931", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 28% of 77408836909998320 greater than 15572835271909931?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 2.167447433479953e+16, + "explanation": "28% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 77408836909998320 vs 15572835271909931", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 77408836909998320 greater than 15572835271909931?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 6.424933463529861e+16, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001463", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9% of 61297616079390110 vs 15864383491083749", + "ns_shortcut": "Use 10% as landmark: 10% of 61297616079390110 = 6129761607939011.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 9% of 61297616079390110 greater than 15864383491083749?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 5516785447145110.0, + "explanation": "9% ≈ 10%. 10% of 61297616079390110 = 6129761607939011. Compare to 15864383491083749." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "15% of 61297616079390110 vs 15864383491083749", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 15% of 61297616079390110 greater than 15864383491083749?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 9194642411908516.0, + "explanation": "15% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 61297616079390110 vs 15864383491083749", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 61297616079390110 greater than 15864383491083749?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 4.106940277319138e+16, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001464", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "11% of 23796898186616470 vs 10862534919147394", + "ns_shortcut": "Use 10% as landmark: 10% of 23796898186616470 = 2379689818661647.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 11% of 23796898186616470 greater than 10862534919147394?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 2617658800527811.5, + "explanation": "11% ≈ 10%. 10% of 23796898186616470 = 2379689818661647. Compare to 10862534919147394." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5% of 23796898186616470 vs 10862534919147394", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 5% of 23796898186616470 greater than 10862534919147394?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 1189844909330823.5, + "explanation": "5% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 23796898186616470 vs 10862534919147394", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 23796898186616470 greater than 10862534919147394?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 1.975142549489167e+16, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001465", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9% of 24880935572536320 vs 6831747586587914", + "ns_shortcut": "Use 10% as landmark: 10% of 24880935572536320 = 2488093557253632.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 9% of 24880935572536320 greater than 6831747586587914?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 2239284201528268.8, + "explanation": "9% ≈ 10%. 10% of 24880935572536320 = 2488093557253632. Compare to 6831747586587914." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "17% of 24880935572536320 vs 6831747586587914", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 17% of 24880935572536320 greater than 6831747586587914?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 4229759047331174.5, + "explanation": "17% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 24880935572536320 vs 6831747586587914", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 24880935572536320 greater than 6831747586587914?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 2.0651176525205144e+16, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001466", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "52% of 56838644635808480 vs 23490767007714413", + "ns_shortcut": "Use 50% as landmark: 50% of 56838644635808480 = 28419322317904240.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 52% of 56838644635808480 greater than 23490767007714413?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 2.955609521062041e+16, + "explanation": "52% ≈ 50%. 50% of 56838644635808480 = 28419322317904240. Compare to 23490767007714413." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "55% of 56838644635808480 vs 23490767007714413", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 55% of 56838644635808480 greater than 23490767007714413?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 3.1261254549694664e+16, + "explanation": "55% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 56838644635808480 vs 23490767007714413", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 56838644635808480 greater than 23490767007714413?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 4.149221058414019e+16, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001467", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "24% of 49124851894302840 vs 16656667124369784", + "ns_shortcut": "Use 25% as landmark: 25% of 49124851894302840 = 12281212973575710.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 24% of 49124851894302840 greater than 16656667124369784?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 1.1789964454632682e+16, + "explanation": "24% ≈ 25%. 25% of 49124851894302840 = 12281212973575710. Compare to 16656667124369784." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "20% of 49124851894302840 vs 16656667124369784", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 20% of 49124851894302840 greater than 16656667124369784?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 9824970378860568.0, + "explanation": "20% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 49124851894302840 vs 16656667124369784", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 49124851894302840 greater than 16656667124369784?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 3.2913650769182904e+16, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001468", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8% of 75327797025445290 vs 24552805491308206", + "ns_shortcut": "Use 10% as landmark: 10% of 75327797025445290 = 7532779702544529.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 8% of 75327797025445290 greater than 24552805491308206?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 6026223762035623.0, + "explanation": "8% ≈ 10%. 10% of 75327797025445290 = 7532779702544529. Compare to 24552805491308206." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "14% of 75327797025445290 vs 24552805491308206", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 14% of 75327797025445290 greater than 24552805491308206?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 1.054589158356234e+16, + "explanation": "14% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 75327797025445290 vs 24552805491308206", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 75327797025445290 greater than 24552805491308206?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 5.4989291828575064e+16, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001469", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "51% of 52401443450506650 vs 19354707558755145", + "ns_shortcut": "Use 50% as landmark: 50% of 52401443450506650 = 26200721725253325.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 51% of 52401443450506650 greater than 19354707558755145?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 2.672473615975839e+16, + "explanation": "51% ≈ 50%. 50% of 52401443450506650 = 26200721725253325. Compare to 19354707558755145." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "46% of 52401443450506650 vs 19354707558755145", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 46% of 52401443450506650 greater than 19354707558755145?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 2.410466398723306e+16, + "explanation": "46% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 52401443450506650 vs 19354707558755145", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 52401443450506650 greater than 19354707558755145?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 1.938853407668746e+16, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001470", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "27% of 62445580654912800 vs 14426322035310753", + "ns_shortcut": "Use 25% as landmark: 25% of 62445580654912800 = 15611395163728200.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 27% of 62445580654912800 greater than 14426322035310753?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 1.6860306776826456e+16, + "explanation": "27% ≈ 25%. 25% of 62445580654912800 = 15611395163728200. Compare to 14426322035310753." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "20% of 62445580654912800 vs 14426322035310753", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 20% of 62445580654912800 greater than 14426322035310753?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 1.248911613098256e+16, + "explanation": "20% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 62445580654912800 vs 14426322035310753", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 62445580654912800 greater than 14426322035310753?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 2.3104864842317736e+16, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001471", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25% of 28960964022867870 vs 11140145893571161", + "ns_shortcut": "Use 25% as landmark: 25% of 28960964022867870 = 7240241005716967.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 25% of 28960964022867870 greater than 11140145893571161?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 7240241005716968.0, + "explanation": "25% ≈ 25%. 25% of 28960964022867870 = 7240241005716967. Compare to 11140145893571161." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "22% of 28960964022867870 vs 11140145893571161", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 22% of 28960964022867870 greater than 11140145893571161?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 6371412085030931.0, + "explanation": "22% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 28960964022867870 vs 11140145893571161", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 28960964022867870 greater than 11140145893571161?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 1.0715556688461112e+16, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001472", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "26% of 95545574749616180 vs 42291807950807293", + "ns_shortcut": "Use 25% as landmark: 25% of 95545574749616180 = 23886393687404045.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 26% of 95545574749616180 greater than 42291807950807293?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 2.484184943490021e+16, + "explanation": "26% ≈ 25%. 25% of 95545574749616180 = 23886393687404045. Compare to 42291807950807293." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "20% of 95545574749616180 vs 42291807950807293", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 20% of 95545574749616180 greater than 42291807950807293?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 1.9109114949923236e+16, + "explanation": "20% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 95545574749616180 vs 42291807950807293", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 95545574749616180 greater than 42291807950807293?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 7.930282704218142e+16, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001473", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "49% of 33673686259299930 vs 15449322689230523", + "ns_shortcut": "Use 50% as landmark: 50% of 33673686259299930 = 16836843129649965.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 49% of 33673686259299930 greater than 15449322689230523?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 1.6500106267056966e+16, + "explanation": "49% ≈ 50%. 50% of 33673686259299930 = 16836843129649965. Compare to 15449322689230523." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "56% of 33673686259299930 vs 15449322689230523", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 56% of 33673686259299930 greater than 15449322689230523?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 1.885726430520796e+16, + "explanation": "56% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 33673686259299930 vs 15449322689230523", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 33673686259299930 greater than 15449322689230523?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 2.794915959521894e+16, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001474", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "51% of 46145041887937950 vs 11100026074081148", + "ns_shortcut": "Use 50% as landmark: 50% of 46145041887937950 = 23072520943968975.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 51% of 46145041887937950 greater than 11100026074081148?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 2.3533971362848356e+16, + "explanation": "51% ≈ 50%. 50% of 46145041887937950 = 23072520943968975. Compare to 11100026074081148." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "55% of 46145041887937950 vs 11100026074081148", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 55% of 46145041887937950 greater than 11100026074081148?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 2.537977303836587e+16, + "explanation": "55% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 46145041887937950 vs 11100026074081148", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 46145041887937950 greater than 11100026074081148?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 1.7073665498537042e+16, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001475", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "11% of 58095490178373490 vs 27391357525101115", + "ns_shortcut": "Use 10% as landmark: 10% of 58095490178373490 = 5809549017837349.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 11% of 58095490178373490 greater than 27391357525101115?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 6390503919621084.0, + "explanation": "11% ≈ 10%. 10% of 58095490178373490 = 5809549017837349. Compare to 27391357525101115." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3% of 58095490178373490 vs 27391357525101115", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 3% of 58095490178373490 greater than 27391357525101115?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 1742864705351204.8, + "explanation": "3% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 58095490178373490 vs 27391357525101115", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 58095490178373490 greater than 27391357525101115?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 3.892397841951024e+16, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001476", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "11% of 31417977626260880 vs 8887891296580495", + "ns_shortcut": "Use 10% as landmark: 10% of 31417977626260880 = 3141797762626088.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 11% of 31417977626260880 greater than 8887891296580495?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 3455977538888697.0, + "explanation": "11% ≈ 10%. 10% of 31417977626260880 = 3141797762626088. Compare to 8887891296580495." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "17% of 31417977626260880 vs 8887891296580495", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 17% of 31417977626260880 greater than 8887891296580495?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 5341056196464350.0, + "explanation": "17% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 31417977626260880 vs 8887891296580495", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 31417977626260880 greater than 8887891296580495?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 1.1624651721716526e+16, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001477", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "11% of 79828043614688640 vs 33475040815007993", + "ns_shortcut": "Use 10% as landmark: 10% of 79828043614688640 = 7982804361468864.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 11% of 79828043614688640 greater than 33475040815007993?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 8781084797615750.0, + "explanation": "11% ≈ 10%. 10% of 79828043614688640 = 7982804361468864. Compare to 33475040815007993." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "17% of 79828043614688640 vs 33475040815007993", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 17% of 79828043614688640 greater than 33475040815007993?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 1.3570767414497068e+16, + "explanation": "17% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 79828043614688640 vs 33475040815007993", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 79828043614688640 greater than 33475040815007993?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 2.9536376137434796e+16, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001478", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "52% of 46820009607307570 vs 10162244796777552", + "ns_shortcut": "Use 50% as landmark: 50% of 46820009607307570 = 23410004803653785.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 52% of 46820009607307570 greater than 10162244796777552?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 2.4346404995799936e+16, + "explanation": "52% ≈ 50%. 50% of 46820009607307570 = 23410004803653785. Compare to 10162244796777552." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "46% of 46820009607307570 vs 10162244796777552", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 46% of 46820009607307570 greater than 10162244796777552?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 2.1537204419361484e+16, + "explanation": "46% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 46820009607307570 vs 10162244796777552", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 46820009607307570 greater than 10162244796777552?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 3.417860701333453e+16, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001479", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "24% of 31467413386844740 vs 11732962714053500", + "ns_shortcut": "Use 25% as landmark: 25% of 31467413386844740 = 7866853346711185.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 24% of 31467413386844740 greater than 11732962714053500?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 7552179212842738.0, + "explanation": "24% ≈ 25%. 25% of 31467413386844740 = 7866853346711185. Compare to 11732962714053500." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "19% of 31467413386844740 vs 11732962714053500", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 19% of 31467413386844740 greater than 11732962714053500?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 5978808543500501.0, + "explanation": "19% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 31467413386844740 vs 11732962714053500", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 31467413386844740 greater than 11732962714053500?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 1.1642942953132554e+16, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001480", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "24% of 46069060259573650 vs 19894604613570500", + "ns_shortcut": "Use 25% as landmark: 25% of 46069060259573650 = 11517265064893412.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 24% of 46069060259573650 greater than 19894604613570500?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 1.1056574462297676e+16, + "explanation": "24% ≈ 25%. 25% of 46069060259573650 = 11517265064893412. Compare to 19894604613570500." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "32% of 46069060259573650 vs 19894604613570500", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 32% of 46069060259573650 greater than 19894604613570500?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 1.4742099283063568e+16, + "explanation": "32% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 46069060259573650 vs 19894604613570500", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 46069060259573650 greater than 19894604613570500?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 3.3630413989488764e+16, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001481", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "23% of 82900443589700560 vs 25427208930332104", + "ns_shortcut": "Use 25% as landmark: 25% of 82900443589700560 = 20725110897425140.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 23% of 82900443589700560 greater than 25427208930332104?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 1.906710202563113e+16, + "explanation": "23% ≈ 25%. 25% of 82900443589700560 = 20725110897425140. Compare to 25427208930332104." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "30% of 82900443589700560 vs 25427208930332104", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 30% of 82900443589700560 greater than 25427208930332104?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 2.487013307691017e+16, + "explanation": "30% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 82900443589700560 vs 25427208930332104", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 82900443589700560 greater than 25427208930332104?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 6.8807368179451464e+16, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001482", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9% of 99594063191560240 vs 38588988408656215", + "ns_shortcut": "Use 10% as landmark: 10% of 99594063191560240 = 9959406319156024.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 9% of 99594063191560240 greater than 38588988408656215?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 8963465687240422.0, + "explanation": "9% ≈ 10%. 10% of 99594063191560240 = 9959406319156024. Compare to 38588988408656215." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "15% of 99594063191560240 vs 38588988408656215", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 15% of 99594063191560240 greater than 38588988408656215?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 1.4939109478734036e+16, + "explanation": "15% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 99594063191560240 vs 38588988408656215", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 99594063191560240 greater than 38588988408656215?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 7.270366612983898e+16, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001483", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "50% of 21048332515114690 vs 4438435623116392", + "ns_shortcut": "Use 50% as landmark: 50% of 21048332515114690 = 10524166257557345.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 50% of 21048332515114690 greater than 4438435623116392?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 1.0524166257557344e+16, + "explanation": "50% ≈ 50%. 50% of 21048332515114690 = 10524166257557345. Compare to 4438435623116392." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "47% of 21048332515114690 vs 4438435623116392", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 47% of 21048332515114690 greater than 4438435623116392?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 9892716282103904.0, + "explanation": "47% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 21048332515114690 vs 4438435623116392", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 21048332515114690 greater than 4438435623116392?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 1.5365282736033724e+16, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001484", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 65464062963051550 vs 14218346043413672", + "ns_shortcut": "Use 10% as landmark: 10% of 65464062963051550 = 6546406296305155.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 65464062963051550 greater than 14218346043413672?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 6546406296305155.0, + "explanation": "10% ≈ 10%. 10% of 65464062963051550 = 6546406296305155. Compare to 14218346043413672." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "13% of 65464062963051550 vs 14218346043413672", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 13% of 65464062963051550 greater than 14218346043413672?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 8510328185196702.0, + "explanation": "13% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 65464062963051550 vs 14218346043413672", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 65464062963051550 greater than 14218346043413672?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 4.778876596302763e+16, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001485", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12% of 61591295765932360 vs 29932057616674953", + "ns_shortcut": "Use 10% as landmark: 10% of 61591295765932360 = 6159129576593236.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 12% of 61591295765932360 greater than 29932057616674953?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 7390955491911883.0, + "explanation": "12% ≈ 10%. 10% of 61591295765932360 = 6159129576593236. Compare to 29932057616674953." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3% of 61591295765932360 vs 29932057616674953", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 3% of 61591295765932360 greater than 29932057616674953?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 1847738872977970.8, + "explanation": "3% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 61591295765932360 vs 29932057616674953", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 61591295765932360 greater than 29932057616674953?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 4.126616816317468e+16, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001486", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "50% of 14938538847356380 vs 3181972412748525", + "ns_shortcut": "Use 50% as landmark: 50% of 14938538847356380 = 7469269423678190.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 50% of 14938538847356380 greater than 3181972412748525?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 7469269423678190.0, + "explanation": "50% ≈ 50%. 50% of 14938538847356380 = 7469269423678190. Compare to 3181972412748525." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "44% of 14938538847356380 vs 3181972412748525", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 44% of 14938538847356380 greater than 3181972412748525?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 6572957092836807.0, + "explanation": "44% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "63% of 14938538847356380 vs 3181972412748525", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 63% of 14938538847356380 greater than 3181972412748525?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 9411279473834520.0, + "explanation": "63% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001487", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "11% of 59510521746277510 vs 26159801354059813", + "ns_shortcut": "Use 10% as landmark: 10% of 59510521746277510 = 5951052174627751.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 11% of 59510521746277510 greater than 26159801354059813?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 6546157392090526.0, + "explanation": "11% ≈ 10%. 10% of 59510521746277510 = 5951052174627751. Compare to 26159801354059813." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "14% of 59510521746277510 vs 26159801354059813", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 14% of 59510521746277510 greater than 26159801354059813?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 8331473044478851.0, + "explanation": "14% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 59510521746277510 vs 26159801354059813", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 59510521746277510 greater than 26159801354059813?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 3.987204957000593e+16, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001488", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "24% of 27829053414098080 vs 9424825013950520", + "ns_shortcut": "Use 25% as landmark: 25% of 27829053414098080 = 6957263353524520.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 24% of 27829053414098080 greater than 9424825013950520?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 6678972819383539.0, + "explanation": "24% ≈ 25%. 25% of 27829053414098080 = 6957263353524520. Compare to 9424825013950520." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "32% of 27829053414098080 vs 9424825013950520", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 32% of 27829053414098080 greater than 9424825013950520?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 8905297092511386.0, + "explanation": "32% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 27829053414098080 vs 9424825013950520", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 27829053414098080 greater than 9424825013950520?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 2.309811433370141e+16, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001489", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25% of 14096041913667020 vs 3079485501323762", + "ns_shortcut": "Use 25% as landmark: 25% of 14096041913667020 = 3524010478416755.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 25% of 14096041913667020 greater than 3079485501323762?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 3524010478416755.0, + "explanation": "25% ≈ 25%. 25% of 14096041913667020 = 3524010478416755. Compare to 3079485501323762." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "20% of 14096041913667020 vs 3079485501323762", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 20% of 14096041913667020 greater than 3079485501323762?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 2819208382733404.0, + "explanation": "20% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 14096041913667020 vs 3079485501323762", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 14096041913667020 greater than 3079485501323762?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 5215535508056797.0, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001490", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25% of 29242225164514270 vs 13061817826652028", + "ns_shortcut": "Use 25% as landmark: 25% of 29242225164514270 = 7310556291128567.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 25% of 29242225164514270 greater than 13061817826652028?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 7310556291128568.0, + "explanation": "25% ≈ 25%. 25% of 29242225164514270 = 7310556291128567. Compare to 13061817826652028." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "22% of 29242225164514270 vs 13061817826652028", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 22% of 29242225164514270 greater than 13061817826652028?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 6433289536193139.0, + "explanation": "22% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 29242225164514270 vs 13061817826652028", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 29242225164514270 greater than 13061817826652028?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 1.959229086022456e+16, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001491", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8% of 78340976882500670 vs 23153379741504276", + "ns_shortcut": "Use 10% as landmark: 10% of 78340976882500670 = 7834097688250067.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 8% of 78340976882500670 greater than 23153379741504276?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 6267278150600054.0, + "explanation": "8% ≈ 10%. 10% of 78340976882500670 = 7834097688250067. Compare to 23153379741504276." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7% of 78340976882500670 vs 23153379741504276", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 7% of 78340976882500670 greater than 23153379741504276?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 5483868381775047.0, + "explanation": "7% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 78340976882500670 vs 23153379741504276", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 78340976882500670 greater than 23153379741504276?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 3.368662005947529e+16, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001492", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "51% of 46496159563794750 vs 21411610278461941", + "ns_shortcut": "Use 50% as landmark: 50% of 46496159563794750 = 23248079781897375.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 51% of 46496159563794750 greater than 21411610278461941?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 2.3713041377535324e+16, + "explanation": "51% ≈ 50%. 50% of 46496159563794750 = 23248079781897375. Compare to 21411610278461941." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "56% of 46496159563794750 vs 21411610278461941", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 56% of 46496159563794750 greater than 21411610278461941?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 2.603784935572506e+16, + "explanation": "56% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 46496159563794750 vs 21411610278461941", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 46496159563794750 greater than 21411610278461941?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 1.7203579038604058e+16, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001493", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "48% of 72555380504779420 vs 33251814860914235", + "ns_shortcut": "Use 50% as landmark: 50% of 72555380504779420 = 36277690252389710.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 48% of 72555380504779420 greater than 33251814860914235?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 3.482658264229412e+16, + "explanation": "48% ≈ 50%. 50% of 72555380504779420 = 36277690252389710. Compare to 33251814860914235." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "47% of 72555380504779420 vs 33251814860914235", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 47% of 72555380504779420 greater than 33251814860914235?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 3.410102883724633e+16, + "explanation": "47% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 72555380504779420 vs 33251814860914235", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 72555380504779420 greater than 33251814860914235?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 5.296542776848898e+16, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001494", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "24% of 85617785986819640 vs 39210145274455251", + "ns_shortcut": "Use 25% as landmark: 25% of 85617785986819640 = 21404446496704910.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 24% of 85617785986819640 greater than 39210145274455251?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 2.054826863683671e+16, + "explanation": "24% ≈ 25%. 25% of 85617785986819640 = 21404446496704910. Compare to 39210145274455251." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "28% of 85617785986819640 vs 39210145274455251", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 28% of 85617785986819640 greater than 39210145274455251?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 2.39729800763095e+16, + "explanation": "28% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 85617785986819640 vs 39210145274455251", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 85617785986819640 greater than 39210145274455251?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 3.1678580815123268e+16, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001495", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 86543273958337710 vs 23846970877621619", + "ns_shortcut": "Use 10% as landmark: 10% of 86543273958337710 = 8654327395833771.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 86543273958337710 greater than 23846970877621619?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 8654327395833771.0, + "explanation": "10% ≈ 10%. 10% of 86543273958337710 = 8654327395833771. Compare to 23846970877621619." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "17% of 86543273958337710 vs 23846970877621619", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 17% of 86543273958337710 greater than 23846970877621619?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 1.471235657291741e+16, + "explanation": "17% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 86543273958337710 vs 23846970877621619", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 86543273958337710 greater than 23846970877621619?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 5.7983993552086264e+16, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001496", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "27% of 25414068387281590 vs 6636124686799575", + "ns_shortcut": "Use 25% as landmark: 25% of 25414068387281590 = 6353517096820397.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 27% of 25414068387281590 greater than 6636124686799575?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 6861798464566029.0, + "explanation": "27% ≈ 25%. 25% of 25414068387281590 = 6353517096820397. Compare to 6636124686799575." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "32% of 25414068387281590 vs 6636124686799575", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 32% of 25414068387281590 greater than 6636124686799575?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 8132501883930109.0, + "explanation": "32% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 25414068387281590 vs 6636124686799575", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 25414068387281590 greater than 6636124686799575?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 1.0928049406531084e+16, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001497", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25% of 99113393492527500 vs 27131593840766456", + "ns_shortcut": "Use 25% as landmark: 25% of 99113393492527500 = 24778348373131875.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 25% of 99113393492527500 greater than 27131593840766456?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 2.4778348373131876e+16, + "explanation": "25% ≈ 25%. 25% of 99113393492527500 = 24778348373131875. Compare to 27131593840766456." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "22% of 99113393492527500 vs 27131593840766456", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 22% of 99113393492527500 greater than 27131593840766456?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 2.180494656835605e+16, + "explanation": "22% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 99113393492527500 vs 27131593840766456", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 99113393492527500 greater than 27131593840766456?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 4.2618759201786824e+16, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001498", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "24% of 44012181243688880 vs 19351323069051348", + "ns_shortcut": "Use 25% as landmark: 25% of 44012181243688880 = 11003045310922220.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 24% of 44012181243688880 greater than 19351323069051348?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 1.0562923498485332e+16, + "explanation": "24% ≈ 25%. 25% of 44012181243688880 = 11003045310922220. Compare to 19351323069051348." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "31% of 44012181243688880 vs 19351323069051348", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 31% of 44012181243688880 greater than 19351323069051348?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 1.3643776185543552e+16, + "explanation": "31% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 44012181243688880 vs 19351323069051348", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 44012181243688880 greater than 19351323069051348?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 2.9488161433271548e+16, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001499", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "50% of 99093244084081020 vs 35340469217099591", + "ns_shortcut": "Use 50% as landmark: 50% of 99093244084081020 = 49546622042040510.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 50% of 99093244084081020 greater than 35340469217099591?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 4.954662204204051e+16, + "explanation": "50% ≈ 50%. 50% of 99093244084081020 = 49546622042040510. Compare to 35340469217099591." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "57% of 99093244084081020 vs 35340469217099591", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 57% of 99093244084081020 greater than 35340469217099591?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 5.6483149127926184e+16, + "explanation": "57% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 99093244084081020 vs 35340469217099591", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 99093244084081020 greater than 35340469217099591?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 3.6664500311109976e+16, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001500", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25% of 71829076799906750 vs 25787867789323824", + "ns_shortcut": "Use 25% as landmark: 25% of 71829076799906750 = 17957269199976687.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 25% of 71829076799906750 greater than 25787867789323824?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 1.7957269199976688e+16, + "explanation": "25% ≈ 25%. 25% of 71829076799906750 = 17957269199976687. Compare to 25787867789323824." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "18% of 71829076799906750 vs 25787867789323824", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 18% of 71829076799906750 greater than 25787867789323824?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 1.2929233823983216e+16, + "explanation": "18% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 71829076799906750 vs 25787867789323824", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 71829076799906750 greater than 25787867789323824?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 2.6576758415965496e+16, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_001501", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 79577516596178380 vs 35094476574023310", + "ns_shortcut": "Use 10% as landmark: 10% of 79577516596178380 = 7957751659617838.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 79577516596178380 greater than 35094476574023310?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 7957751659617838.0, + "explanation": "10% ≈ 10%. 10% of 79577516596178380 = 7957751659617838. Compare to 35094476574023310." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4% of 79577516596178380 vs 35094476574023310", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 4% of 79577516596178380 greater than 35094476574023310?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 3183100663847135.0, + "explanation": "4% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 79577516596178380 vs 35094476574023310", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 79577516596178380 greater than 35094476574023310?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 5.8091587115210216e+16, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "equation_reasoning_001502", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9439839984087080 + 5680357738248731 = ___ + 6721737474255938", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 9439839984087080 + 5680357738248731 = ___ + 6721737474255938", + "options": [ + "8398460248079893", + "8398460248079873", + "8398460248079883", + "8398460248079863" + ], + "answer": "8398460248079873", + "correct_value": 8398460248079873, + "explanation": "Blank = A + B − C = 9439839984087080 + 5680357738248731 − 6721737474255938 = 8398460248079873." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4647263882519947 + 5531351252492803 + ___ = 5531351252492803 + 5770025176328175 + 9003434594886877", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 4647263882519947 + 5531351252492803 + ___ = 5531351252492803 + 5770025176328175 + 9003434594886877", + "options": [ + "10126195888695105", + "10126195888695095", + "10126195888695115", + "10126195888695125" + ], + "answer": "10126195888695105", + "correct_value": 10126195888695105, + "explanation": "B cancels; blank = C + D − A = 5770025176328175 + 9003434594886877 − 4647263882519947 = 10126195888695105." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9809306984370131 + 5392916904357432 + ___ = 1949456289674931 + 2946559008092851", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 9809306984370131 + 5392916904357432 + ___ = 1949456289674931 + 2946559008092851", + "options": [ + "-10306208590959781", + "-10306208590959769", + "-10306208590959779", + "-10306208590959799" + ], + "answer": "-10306208590959781", + "correct_value": -10306208590959781, + "explanation": "No matching terms; blank = 1949456289674931 + 2946559008092851 − 9809306984370131 − 5392916904357432 = -10306208590959781." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001503", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7909921596135286 × 6968059258509573 × 5076712796416482 = ___ × 5076712796416482 × 7909921596135286", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 7909921596135286 × 6968059258509573 × 5076712796416482 = ___ × 5076712796416482 × 7909921596135286", + "options": [ + "6968059258509573", + "6968059258509593", + "6968059258509563", + "6968059258509583" + ], + "answer": "6968059258509573", + "correct_value": 6968059258509573, + "explanation": "A and C appear on both sides; blank = B = 6968059258509573." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2731919332502536 + 6344046269483549 + ___ = 6344046269483549 + 7824767275565529 + 6959515523447174", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 2731919332502536 + 6344046269483549 + ___ = 6344046269483549 + 7824767275565529 + 6959515523447174", + "options": [ + "12052363466510167", + "12052363466510157", + "12052363466510177", + "12052363466510187" + ], + "answer": "12052363466510167", + "correct_value": 12052363466510167, + "explanation": "B cancels; blank = C + D − A = 7824767275565529 + 6959515523447174 − 2731919332502536 = 12052363466510167." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7977782355592264 + 3269479168870098 + ___ = 9559212115258965 + 3074702339819025", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 7977782355592264 + 3269479168870098 + ___ = 9559212115258965 + 3074702339819025", + "options": [ + "1386652930615648", + "1386652930615618", + "1386652930615628", + "1386652930615638" + ], + "answer": "1386652930615628", + "correct_value": 1386652930615628, + "explanation": "No matching terms; blank = 9559212115258965 + 3074702339819025 − 7977782355592264 − 3269479168870098 = 1386652930615628." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001504", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1224754964184304 + 2704385503800682 = ___ + 6654494801878172", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 1224754964184304 + 2704385503800682 = ___ + 6654494801878172", + "options": [ + "-2725354333893186", + "-2725354333893204", + "-2725354333893174", + "-2725354333893184" + ], + "answer": "-2725354333893186", + "correct_value": -2725354333893186, + "explanation": "Blank = A + B − C = 1224754964184304 + 2704385503800682 − 6654494801878172 = -2725354333893186." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7229121268105070 + 8747354521770206 + ___ = 8747354521770206 + 7277272901012433 + 7373730950440976", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 7229121268105070 + 8747354521770206 + ___ = 8747354521770206 + 7277272901012433 + 7373730950440976", + "options": [ + "7421882583348349", + "7421882583348339", + "7421882583348329", + "7421882583348359" + ], + "answer": "7421882583348339", + "correct_value": 7421882583348339, + "explanation": "B cancels; blank = C + D − A = 7277272901012433 + 7373730950440976 − 7229121268105070 = 7421882583348339." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8657611627518062 + 6051155890340580 + ___ = 6990492710298970 + 4633377579646802", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 8657611627518062 + 6051155890340580 + ___ = 6990492710298970 + 4633377579646802", + "options": [ + "-3084897227912870", + "-3084897227912850", + "-3084897227912880", + "-3084897227912860" + ], + "answer": "-3084897227912870", + "correct_value": -3084897227912870, + "explanation": "No matching terms; blank = 6990492710298970 + 4633377579646802 − 8657611627518062 − 6051155890340580 = -3084897227912870." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001505", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8612658183754669 × (9293786595441622 + 8136416571009627) = 8612658183754669 × ___ + 8612658183754669 × 8136416571009627", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 8612658183754669 × (9293786595441622 + 8136416571009627) = 8612658183754669 × ___ + 8612658183754669 × 8136416571009627", + "options": [ + "9293786595441622", + "9293786595441642", + "9293786595441632", + "9293786595441612" + ], + "answer": "9293786595441622", + "correct_value": 9293786595441622, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 9293786595441622." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5775209737122446 + 2846262099739843 + ___ = 2846262099739843 + 8350638490566745 + 9140569559694007", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 5775209737122446 + 2846262099739843 + ___ = 2846262099739843 + 8350638490566745 + 9140569559694007", + "options": [ + "11715998313138296", + "11715998313138326", + "11715998313138316", + "11715998313138306" + ], + "answer": "11715998313138306", + "correct_value": 11715998313138306, + "explanation": "B cancels; blank = C + D − A = 8350638490566745 + 9140569559694007 − 5775209737122446 = 11715998313138306." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3549641158719565 + 7314660177521054 + ___ = 1747721233648055 + 2828101486056783", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 3549641158719565 + 7314660177521054 + ___ = 1747721233648055 + 2828101486056783", + "options": [ + "-6288478616535769", + "-6288478616535781", + "-6288478616535799", + "-6288478616535779" + ], + "answer": "-6288478616535781", + "correct_value": -6288478616535781, + "explanation": "No matching terms; blank = 1747721233648055 + 2828101486056783 − 3549641158719565 − 7314660177521054 = -6288478616535781." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001506", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5974521391895561 + 8205556615168986 = ___ + 3502022703234047", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 5974521391895561 + 8205556615168986 = ___ + 3502022703234047", + "options": [ + "10678055303830500", + "10678055303830510", + "10678055303830520", + "10678055303830490" + ], + "answer": "10678055303830500", + "correct_value": 10678055303830500, + "explanation": "Blank = A + B − C = 5974521391895561 + 8205556615168986 − 3502022703234047 = 10678055303830500." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6265103165085395 + 9684307404438930 + ___ = 9684307404438930 + 1631162435031096 + 2280586721861644", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 6265103165085395 + 9684307404438930 + ___ = 9684307404438930 + 1631162435031096 + 2280586721861644", + "options": [ + "-2353354008192635", + "-2353354008192645", + "-2353354008192665", + "-2353354008192655" + ], + "answer": "-2353354008192655", + "correct_value": -2353354008192655, + "explanation": "B cancels; blank = C + D − A = 1631162435031096 + 2280586721861644 − 6265103165085395 = -2353354008192655." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5811932851708041 + 6247120149805131 + ___ = 3149219295718641 + 1842384905083632", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 5811932851708041 + 6247120149805131 + ___ = 3149219295718641 + 1842384905083632", + "options": [ + "-7067448800710891", + "-7067448800710911", + "-7067448800710899", + "-7067448800710881" + ], + "answer": "-7067448800710899", + "correct_value": -7067448800710899, + "explanation": "No matching terms; blank = 3149219295718641 + 1842384905083632 − 5811932851708041 − 6247120149805131 = -7067448800710899." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001507", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5179102363628298 × 2466728830248527 × 5907368591580582 = ___ × 5907368591580582 × 5179102363628298", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 5179102363628298 × 2466728830248527 × 5907368591580582 = ___ × 5907368591580582 × 5179102363628298", + "options": [ + "2466728830248527", + "2466728830248537", + "2466728830248517", + "2466728830248547" + ], + "answer": "2466728830248527", + "correct_value": 2466728830248527, + "explanation": "A and C appear on both sides; blank = B = 2466728830248527." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8007220611479638 + 7442815449765966 + ___ = 7442815449765966 + 6837654827511075 + 7185271377355697", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 8007220611479638 + 7442815449765966 + ___ = 7442815449765966 + 6837654827511075 + 7185271377355697", + "options": [ + "6015705593387154", + "6015705593387124", + "6015705593387134", + "6015705593387144" + ], + "answer": "6015705593387134", + "correct_value": 6015705593387134, + "explanation": "B cancels; blank = C + D − A = 6837654827511075 + 7185271377355697 − 8007220611479638 = 6015705593387134." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4781287458112570 + 1343359791635911 + ___ = 6117269957183821 + 8360353002955122", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 4781287458112570 + 1343359791635911 + ___ = 6117269957183821 + 8360353002955122", + "options": [ + "8352975710390452", + "8352975710390472", + "8352975710390482", + "8352975710390462" + ], + "answer": "8352975710390462", + "correct_value": 8352975710390462, + "explanation": "No matching terms; blank = 6117269957183821 + 8360353002955122 − 4781287458112570 − 1343359791635911 = 8352975710390462." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_001508", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3503444304661885 × 2217833446729142 × 4487150853184415 = ___ × 4487150853184415 × 3503444304661885", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 3503444304661885 × 2217833446729142 × 4487150853184415 = ___ × 4487150853184415 × 3503444304661885", + "options": [ + "2217833446729132", + "2217833446729142", + "2217833446729152", + "2217833446729162" + ], + "answer": "2217833446729142", + "correct_value": 2217833446729142, + "explanation": "A and C appear on both sides; blank = B = 2217833446729142." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1249360224389294 + 4492262323170989 + ___ = 4492262323170989 + 2197302396419487 + 1301488027998843", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 1249360224389294 + 4492262323170989 + ___ = 4492262323170989 + 2197302396419487 + 1301488027998843", + "options": [ + "2249430200029056", + "2249430200029026", + "2249430200029036", + "2249430200029046" + ], + "answer": "2249430200029036", + "correct_value": 2249430200029036, + "explanation": "B cancels; blank = C + D − A = 2197302396419487 + 1301488027998843 − 1249360224389294 = 2249430200029036." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2141995407863297 + 6004066561484669 + ___ = 8888578569874782 + 9870768075143887", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 2141995407863297 + 6004066561484669 + ___ = 8888578569874782 + 9870768075143887", + "options": [ + "10613284675670693", + "10613284675670713", + "10613284675670703", + "10613284675670723" + ], + "answer": "10613284675670703", + "correct_value": 10613284675670703, + "explanation": "No matching terms; blank = 8888578569874782 + 9870768075143887 − 2141995407863297 − 6004066561484669 = 10613284675670703." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001509", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6167470022133045 + 1945830225624289 = ___ + 6671291864107536", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 6167470022133045 + 1945830225624289 = ___ + 6671291864107536", + "options": [ + "1442008383649818", + "1442008383649798", + "1442008383649808", + "1442008383649788" + ], + "answer": "1442008383649798", + "correct_value": 1442008383649798, + "explanation": "Blank = A + B − C = 6167470022133045 + 1945830225624289 − 6671291864107536 = 1442008383649798." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6690536995196885 + 8697590516018011 + ___ = 8697590516018011 + 5608649569405207 + 1352768725591595", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 6690536995196885 + 8697590516018011 + ___ = 8697590516018011 + 5608649569405207 + 1352768725591595", + "options": [ + "270881299799937", + "270881299799927", + "270881299799917", + "270881299799907" + ], + "answer": "270881299799917", + "correct_value": 270881299799917, + "explanation": "B cancels; blank = C + D − A = 5608649569405207 + 1352768725591595 − 6690536995196885 = 270881299799917." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5760739749337140 + 4484505648916694 + ___ = 7079121924670356 + 7680979884042997", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 5760739749337140 + 4484505648916694 + ___ = 7079121924670356 + 7680979884042997", + "options": [ + "4514856410459529", + "4514856410459539", + "4514856410459519", + "4514856410459509" + ], + "answer": "4514856410459519", + "correct_value": 4514856410459519, + "explanation": "No matching terms; blank = 7079121924670356 + 7680979884042997 − 5760739749337140 − 4484505648916694 = 4514856410459519." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001510", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2154647614665123 × 2297658492912723 × 9528783858458853 = ___ × 9528783858458853 × 2154647614665123", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 2154647614665123 × 2297658492912723 × 9528783858458853 = ___ × 9528783858458853 × 2154647614665123", + "options": [ + "2297658492912713", + "2297658492912723", + "2297658492912743", + "2297658492912733" + ], + "answer": "2297658492912723", + "correct_value": 2297658492912723, + "explanation": "A and C appear on both sides; blank = B = 2297658492912723." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7000811481908131 + 5856310734043744 + ___ = 5856310734043744 + 5922648152861091 + 8509379635428475", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 7000811481908131 + 5856310734043744 + ___ = 5856310734043744 + 5922648152861091 + 8509379635428475", + "options": [ + "7431216306381455", + "7431216306381445", + "7431216306381425", + "7431216306381435" + ], + "answer": "7431216306381435", + "correct_value": 7431216306381435, + "explanation": "B cancels; blank = C + D − A = 5922648152861091 + 8509379635428475 − 7000811481908131 = 7431216306381435." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7857392149871659 + 2875405980562751 + ___ = 9924427766561797 + 1201689030062336", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 7857392149871659 + 2875405980562751 + ___ = 9924427766561797 + 1201689030062336", + "options": [ + "393318666189713", + "393318666189723", + "393318666189733", + "393318666189743" + ], + "answer": "393318666189723", + "correct_value": 393318666189723, + "explanation": "No matching terms; blank = 9924427766561797 + 1201689030062336 − 7857392149871659 − 2875405980562751 = 393318666189723." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001511", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1352932065089713 + 7922599369077142 = ___ + 3918108682507707", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 1352932065089713 + 7922599369077142 = ___ + 3918108682507707", + "options": [ + "5357422751659138", + "5357422751659148", + "5357422751659168", + "5357422751659158" + ], + "answer": "5357422751659148", + "correct_value": 5357422751659148, + "explanation": "Blank = A + B − C = 1352932065089713 + 7922599369077142 − 3918108682507707 = 5357422751659148." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8904700791745449 + 9936763267108657 + ___ = 9936763267108657 + 3968370350681155 + 9291612054874840", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 8904700791745449 + 9936763267108657 + ___ = 9936763267108657 + 3968370350681155 + 9291612054874840", + "options": [ + "4355281613810556", + "4355281613810566", + "4355281613810546", + "4355281613810536" + ], + "answer": "4355281613810546", + "correct_value": 4355281613810546, + "explanation": "B cancels; blank = C + D − A = 3968370350681155 + 9291612054874840 − 8904700791745449 = 4355281613810546." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4366283419240008 + 6165962382764854 + ___ = 4266458290910921 + 9816298188983972", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 4366283419240008 + 6165962382764854 + ___ = 4266458290910921 + 9816298188983972", + "options": [ + "3550510677890031", + "3550510677890051", + "3550510677890021", + "3550510677890041" + ], + "answer": "3550510677890031", + "correct_value": 3550510677890031, + "explanation": "No matching terms; blank = 4266458290910921 + 9816298188983972 − 4366283419240008 − 6165962382764854 = 3550510677890031." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001512", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1289538925669141 + 2685669920518577 + ___ = 2685669920518577 + 9455848728341831 + 1289538925669141", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 1289538925669141 + 2685669920518577 + ___ = 2685669920518577 + 9455848728341831 + 1289538925669141", + "options": [ + "9455848728341831", + "9455848728341841", + "9455848728341851", + "9455848728341821" + ], + "answer": "9455848728341831", + "correct_value": 9455848728341831, + "explanation": "A and B appear on both sides and cancel; blank = C = 9455848728341831." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9700712202697092 + 7711748503208479 + ___ = 7711748503208479 + 6318697878099063 + 6425313924595461", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 9700712202697092 + 7711748503208479 + ___ = 7711748503208479 + 6318697878099063 + 6425313924595461", + "options": [ + "3043299599997452", + "3043299599997442", + "3043299599997432", + "3043299599997422" + ], + "answer": "3043299599997432", + "correct_value": 3043299599997432, + "explanation": "B cancels; blank = C + D − A = 6318697878099063 + 6425313924595461 − 9700712202697092 = 3043299599997432." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2484252941877543 + 6953708637481571 + ___ = 3563259901808394 + 6238864316542788", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 2484252941877543 + 6953708637481571 + ___ = 3563259901808394 + 6238864316542788", + "options": [ + "364162638992068", + "364162638992078", + "364162638992088", + "364162638992058" + ], + "answer": "364162638992068", + "correct_value": 364162638992068, + "explanation": "No matching terms; blank = 3563259901808394 + 6238864316542788 − 2484252941877543 − 6953708637481571 = 364162638992068." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001513", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8040019777565046 + 9342123878230022 + ___ = 9342123878230022 + 3066258182699380 + 8040019777565046", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 8040019777565046 + 9342123878230022 + ___ = 9342123878230022 + 3066258182699380 + 8040019777565046", + "options": [ + "3066258182699390", + "3066258182699380", + "3066258182699370", + "3066258182699400" + ], + "answer": "3066258182699380", + "correct_value": 3066258182699380, + "explanation": "A and B appear on both sides and cancel; blank = C = 3066258182699380." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1666645147493674 + 7524344790140717 + ___ = 7524344790140717 + 7591128516309375 + 5119185510908635", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 1666645147493674 + 7524344790140717 + ___ = 7524344790140717 + 7591128516309375 + 5119185510908635", + "options": [ + "11043668879724326", + "11043668879724336", + "11043668879724356", + "11043668879724346" + ], + "answer": "11043668879724336", + "correct_value": 11043668879724336, + "explanation": "B cancels; blank = C + D − A = 7591128516309375 + 5119185510908635 − 1666645147493674 = 11043668879724336." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1857939131804874 + 3209797707082725 + ___ = 8412858250339722 + 9379659681320141", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 1857939131804874 + 3209797707082725 + ___ = 8412858250339722 + 9379659681320141", + "options": [ + "12724781092772274", + "12724781092772284", + "12724781092772264", + "12724781092772254" + ], + "answer": "12724781092772264", + "correct_value": 12724781092772264, + "explanation": "No matching terms; blank = 8412858250339722 + 9379659681320141 − 1857939131804874 − 3209797707082725 = 12724781092772264." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001514", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5109907754692016 + 2594650108515777 + ___ = 2594650108515777 + 5777013124315858 + 5109907754692016", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 5109907754692016 + 2594650108515777 + ___ = 2594650108515777 + 5777013124315858 + 5109907754692016", + "options": [ + "5777013124315848", + "5777013124315878", + "5777013124315868", + "5777013124315858" + ], + "answer": "5777013124315858", + "correct_value": 5777013124315858, + "explanation": "A and B appear on both sides and cancel; blank = C = 5777013124315858." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3626867301443028 + 8790300678105402 + ___ = 8790300678105402 + 2412902667882084 + 4111331848260626", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 3626867301443028 + 8790300678105402 + ___ = 8790300678105402 + 2412902667882084 + 4111331848260626", + "options": [ + "2897367214699682", + "2897367214699702", + "2897367214699672", + "2897367214699692" + ], + "answer": "2897367214699682", + "correct_value": 2897367214699682, + "explanation": "B cancels; blank = C + D − A = 2412902667882084 + 4111331848260626 − 3626867301443028 = 2897367214699682." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2904924619645161 + 2473102519052394 + ___ = 2978298170996849 + 5476957687659638", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 2904924619645161 + 2473102519052394 + ___ = 2978298170996849 + 5476957687659638", + "options": [ + "3077228719958922", + "3077228719958942", + "3077228719958952", + "3077228719958932" + ], + "answer": "3077228719958932", + "correct_value": 3077228719958932, + "explanation": "No matching terms; blank = 2978298170996849 + 5476957687659638 − 2904924619645161 − 2473102519052394 = 3077228719958932." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_001515", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2985677485819603 × (5646076198799709 + 3130145396661229) = 2985677485819603 × ___ + 2985677485819603 × 3130145396661229", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 2985677485819603 × (5646076198799709 + 3130145396661229) = 2985677485819603 × ___ + 2985677485819603 × 3130145396661229", + "options": [ + "5646076198799699", + "5646076198799709", + "5646076198799719", + "5646076198799729" + ], + "answer": "5646076198799709", + "correct_value": 5646076198799709, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 5646076198799709." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4502429267327816 + 8972237822316753 + ___ = 8972237822316753 + 8318961080764666 + 4826110909076862", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 4502429267327816 + 8972237822316753 + ___ = 8972237822316753 + 8318961080764666 + 4826110909076862", + "options": [ + "8642642722513722", + "8642642722513702", + "8642642722513732", + "8642642722513712" + ], + "answer": "8642642722513712", + "correct_value": 8642642722513712, + "explanation": "B cancels; blank = C + D − A = 8318961080764666 + 4826110909076862 − 4502429267327816 = 8642642722513712." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5375864411024493 + 9224696592961774 + ___ = 2383676791322264 + 8303919470119732", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 5375864411024493 + 9224696592961774 + ___ = 2383676791322264 + 8303919470119732", + "options": [ + "-3912964742544259", + "-3912964742544289", + "-3912964742544269", + "-3912964742544271" + ], + "answer": "-3912964742544271", + "correct_value": -3912964742544271, + "explanation": "No matching terms; blank = 2383676791322264 + 8303919470119732 − 5375864411024493 − 9224696592961774 = -3912964742544271." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_001516", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8150906495039786 × 1923367957727910 × 3835687087073904 = ___ × 3835687087073904 × 8150906495039786", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 8150906495039786 × 1923367957727910 × 3835687087073904 = ___ × 3835687087073904 × 8150906495039786", + "options": [ + "1923367957727920", + "1923367957727930", + "1923367957727900", + "1923367957727910" + ], + "answer": "1923367957727910", + "correct_value": 1923367957727910, + "explanation": "A and C appear on both sides; blank = B = 1923367957727910." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8900636316856695 + 6558344635305247 + ___ = 6558344635305247 + 7339461356284370 + 5399672464654445", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 8900636316856695 + 6558344635305247 + ___ = 6558344635305247 + 7339461356284370 + 5399672464654445", + "options": [ + "3838497504082140", + "3838497504082110", + "3838497504082130", + "3838497504082120" + ], + "answer": "3838497504082120", + "correct_value": 3838497504082120, + "explanation": "B cancels; blank = C + D − A = 7339461356284370 + 5399672464654445 − 8900636316856695 = 3838497504082120." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8421641457724341 + 2167394213787652 + ___ = 9896436579820751 + 4081880727477021", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 8421641457724341 + 2167394213787652 + ___ = 9896436579820751 + 4081880727477021", + "options": [ + "3389281635785769", + "3389281635785789", + "3389281635785799", + "3389281635785779" + ], + "answer": "3389281635785779", + "correct_value": 3389281635785779, + "explanation": "No matching terms; blank = 9896436579820751 + 4081880727477021 − 8421641457724341 − 2167394213787652 = 3389281635785779." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_001517", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1177407098922862 + 7018385328557726 = ___ + 8650949767640421", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 1177407098922862 + 7018385328557726 = ___ + 8650949767640421", + "options": [ + "-455157340159847", + "-455157340159817", + "-455157340159833", + "-455157340159827" + ], + "answer": "-455157340159833", + "correct_value": -455157340159833, + "explanation": "Blank = A + B − C = 1177407098922862 + 7018385328557726 − 8650949767640421 = -455157340159833." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6346614295363390 + 1572340138898159 + ___ = 1572340138898159 + 7827761145344733 + 1793631559764931", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 6346614295363390 + 1572340138898159 + ___ = 1572340138898159 + 7827761145344733 + 1793631559764931", + "options": [ + "3274778409746284", + "3274778409746294", + "3274778409746274", + "3274778409746264" + ], + "answer": "3274778409746274", + "correct_value": 3274778409746274, + "explanation": "B cancels; blank = C + D − A = 7827761145344733 + 1793631559764931 − 6346614295363390 = 3274778409746274." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1600553480410154 + 1050171560575380 + ___ = 5110200140154331 + 4806303018954805", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 1600553480410154 + 1050171560575380 + ___ = 5110200140154331 + 4806303018954805", + "options": [ + "7265778118123602", + "7265778118123612", + "7265778118123622", + "7265778118123592" + ], + "answer": "7265778118123602", + "correct_value": 7265778118123602, + "explanation": "No matching terms; blank = 5110200140154331 + 4806303018954805 − 1600553480410154 − 1050171560575380 = 7265778118123602." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001518", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2112773581227864 + 1414579604105222 + ___ = 1414579604105222 + 6678690112241287 + 2112773581227864", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 2112773581227864 + 1414579604105222 + ___ = 1414579604105222 + 6678690112241287 + 2112773581227864", + "options": [ + "6678690112241307", + "6678690112241297", + "6678690112241287", + "6678690112241277" + ], + "answer": "6678690112241287", + "correct_value": 6678690112241287, + "explanation": "A and B appear on both sides and cancel; blank = C = 6678690112241287." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4803860560914905 + 3018065474221164 + ___ = 3018065474221164 + 7303677807736335 + 2127195074626803", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 4803860560914905 + 3018065474221164 + ___ = 3018065474221164 + 7303677807736335 + 2127195074626803", + "options": [ + "4627012321448233", + "4627012321448243", + "4627012321448223", + "4627012321448253" + ], + "answer": "4627012321448233", + "correct_value": 4627012321448233, + "explanation": "B cancels; blank = C + D − A = 7303677807736335 + 2127195074626803 − 4803860560914905 = 4627012321448233." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8242404637826093 + 4366852062762029 + ___ = 8143899447524841 + 2151552225614427", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 8242404637826093 + 4366852062762029 + ___ = 8143899447524841 + 2151552225614427", + "options": [ + "-2313805027448836", + "-2313805027448846", + "-2313805027448866", + "-2313805027448854" + ], + "answer": "-2313805027448854", + "correct_value": -2313805027448854, + "explanation": "No matching terms; blank = 8143899447524841 + 2151552225614427 − 8242404637826093 − 4366852062762029 = -2313805027448854." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_001519", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6623478785750054 + 8014713585438972 + ___ = 8014713585438972 + 7046952963218514 + 6623478785750054", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 6623478785750054 + 8014713585438972 + ___ = 8014713585438972 + 7046952963218514 + 6623478785750054", + "options": [ + "7046952963218524", + "7046952963218534", + "7046952963218504", + "7046952963218514" + ], + "answer": "7046952963218514", + "correct_value": 7046952963218514, + "explanation": "A and B appear on both sides and cancel; blank = C = 7046952963218514." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5490537961674600 + 6828358005395124 + ___ = 6828358005395124 + 5774022162915461 + 4894615404225851", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 5490537961674600 + 6828358005395124 + ___ = 6828358005395124 + 5774022162915461 + 4894615404225851", + "options": [ + "5178099605466712", + "5178099605466732", + "5178099605466702", + "5178099605466722" + ], + "answer": "5178099605466712", + "correct_value": 5178099605466712, + "explanation": "B cancels; blank = C + D − A = 5774022162915461 + 4894615404225851 − 5490537961674600 = 5178099605466712." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6009023038881706 + 7293337910089244 + ___ = 6600795267894023 + 7624449218669616", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 6009023038881706 + 7293337910089244 + ___ = 6600795267894023 + 7624449218669616", + "options": [ + "922883537592699", + "922883537592689", + "922883537592709", + "922883537592679" + ], + "answer": "922883537592689", + "correct_value": 922883537592689, + "explanation": "No matching terms; blank = 6600795267894023 + 7624449218669616 − 6009023038881706 − 7293337910089244 = 922883537592689." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001520", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3783232999236154 + 5540993599874260 = ___ + 8578080502857062", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 3783232999236154 + 5540993599874260 = ___ + 8578080502857062", + "options": [ + "746146096253342", + "746146096253362", + "746146096253352", + "746146096253372" + ], + "answer": "746146096253352", + "correct_value": 746146096253352, + "explanation": "Blank = A + B − C = 3783232999236154 + 5540993599874260 − 8578080502857062 = 746146096253352." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4496489777806444 + 8708018234559428 + ___ = 8708018234559428 + 2185876600708184 + 1726288414321106", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 4496489777806444 + 8708018234559428 + ___ = 8708018234559428 + 2185876600708184 + 1726288414321106", + "options": [ + "-584324762777146", + "-584324762777154", + "-584324762777166", + "-584324762777136" + ], + "answer": "-584324762777154", + "correct_value": -584324762777154, + "explanation": "B cancels; blank = C + D − A = 2185876600708184 + 1726288414321106 − 4496489777806444 = -584324762777154." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4400656012946353 + 6322755340699458 + ___ = 2618864962276194 + 6043779327385039", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 4400656012946353 + 6322755340699458 + ___ = 2618864962276194 + 6043779327385039", + "options": [ + "-2060767063984562", + "-2060767063984578", + "-2060767063984572", + "-2060767063984592" + ], + "answer": "-2060767063984578", + "correct_value": -2060767063984578, + "explanation": "No matching terms; blank = 2618864962276194 + 6043779327385039 − 4400656012946353 − 6322755340699458 = -2060767063984578." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001521", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9799822745904316 × (4955072442514443 + 2727017033639288) = 9799822745904316 × ___ + 9799822745904316 × 2727017033639288", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 9799822745904316 × (4955072442514443 + 2727017033639288) = 9799822745904316 × ___ + 9799822745904316 × 2727017033639288", + "options": [ + "4955072442514433", + "4955072442514443", + "4955072442514453", + "4955072442514463" + ], + "answer": "4955072442514443", + "correct_value": 4955072442514443, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 4955072442514443." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3744478940441406 + 9426503303185783 + ___ = 9426503303185783 + 9030784457070724 + 5147544767118164", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 3744478940441406 + 9426503303185783 + ___ = 9426503303185783 + 9030784457070724 + 5147544767118164", + "options": [ + "10433850283747492", + "10433850283747502", + "10433850283747482", + "10433850283747472" + ], + "answer": "10433850283747482", + "correct_value": 10433850283747482, + "explanation": "B cancels; blank = C + D − A = 9030784457070724 + 5147544767118164 − 3744478940441406 = 10433850283747482." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3218081288377565 + 1522931031786912 + ___ = 6228184581566130 + 1262176782801083", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 3218081288377565 + 1522931031786912 + ___ = 6228184581566130 + 1262176782801083", + "options": [ + "2749349044202726", + "2749349044202756", + "2749349044202736", + "2749349044202746" + ], + "answer": "2749349044202736", + "correct_value": 2749349044202736, + "explanation": "No matching terms; blank = 6228184581566130 + 1262176782801083 − 3218081288377565 − 1522931031786912 = 2749349044202736." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001522", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6177521085584930 × (8637858249115732 + 2500401195567394) = 6177521085584930 × ___ + 6177521085584930 × 2500401195567394", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 6177521085584930 × (8637858249115732 + 2500401195567394) = 6177521085584930 × ___ + 6177521085584930 × 2500401195567394", + "options": [ + "8637858249115732", + "8637858249115742", + "8637858249115752", + "8637858249115722" + ], + "answer": "8637858249115732", + "correct_value": 8637858249115732, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 8637858249115732." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7678157765547604 + 2257814850270630 + ___ = 2257814850270630 + 3552847933856027 + 8548343318885714", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 7678157765547604 + 2257814850270630 + ___ = 2257814850270630 + 3552847933856027 + 8548343318885714", + "options": [ + "4423033487194127", + "4423033487194147", + "4423033487194157", + "4423033487194137" + ], + "answer": "4423033487194137", + "correct_value": 4423033487194137, + "explanation": "B cancels; blank = C + D − A = 3552847933856027 + 8548343318885714 − 7678157765547604 = 4423033487194137." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6361739528366893 + 2311966976517363 + ___ = 1370107699456199 + 8438935617172475", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 6361739528366893 + 2311966976517363 + ___ = 1370107699456199 + 8438935617172475", + "options": [ + "1135336811744438", + "1135336811744408", + "1135336811744418", + "1135336811744428" + ], + "answer": "1135336811744418", + "correct_value": 1135336811744418, + "explanation": "No matching terms; blank = 1370107699456199 + 8438935617172475 − 6361739528366893 − 2311966976517363 = 1135336811744418." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001523", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5279848701229619 × 8067058568554400 × 6848080896658693 = ___ × 6848080896658693 × 5279848701229619", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 5279848701229619 × 8067058568554400 × 6848080896658693 = ___ × 6848080896658693 × 5279848701229619", + "options": [ + "8067058568554410", + "8067058568554390", + "8067058568554420", + "8067058568554400" + ], + "answer": "8067058568554400", + "correct_value": 8067058568554400, + "explanation": "A and C appear on both sides; blank = B = 8067058568554400." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6409571735225285 + 8819001476294742 + ___ = 8819001476294742 + 4509081856907372 + 2364434605004501", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 6409571735225285 + 8819001476294742 + ___ = 8819001476294742 + 4509081856907372 + 2364434605004501", + "options": [ + "463944726686578", + "463944726686598", + "463944726686588", + "463944726686608" + ], + "answer": "463944726686588", + "correct_value": 463944726686588, + "explanation": "B cancels; blank = C + D − A = 4509081856907372 + 2364434605004501 − 6409571735225285 = 463944726686588." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9665244905236245 + 5027905169318434 + ___ = 3376800092758929 + 8965168085357801", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 9665244905236245 + 5027905169318434 + ___ = 3376800092758929 + 8965168085357801", + "options": [ + "-2351181896437941", + "-2351181896437949", + "-2351181896437961", + "-2351181896437931" + ], + "answer": "-2351181896437949", + "correct_value": -2351181896437949, + "explanation": "No matching terms; blank = 3376800092758929 + 8965168085357801 − 9665244905236245 − 5027905169318434 = -2351181896437949." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001524", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3641883926565584 + 4987457468571662 = ___ + 2678808084509157", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 3641883926565584 + 4987457468571662 = ___ + 2678808084509157", + "options": [ + "5950533310628109", + "5950533310628079", + "5950533310628099", + "5950533310628089" + ], + "answer": "5950533310628089", + "correct_value": 5950533310628089, + "explanation": "Blank = A + B − C = 3641883926565584 + 4987457468571662 − 2678808084509157 = 5950533310628089." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4651905840594284 + 5022947496227180 + ___ = 5022947496227180 + 7641311254811318 + 8448416746792996", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 4651905840594284 + 5022947496227180 + ___ = 5022947496227180 + 7641311254811318 + 8448416746792996", + "options": [ + "11437822161010040", + "11437822161010020", + "11437822161010050", + "11437822161010030" + ], + "answer": "11437822161010030", + "correct_value": 11437822161010030, + "explanation": "B cancels; blank = C + D − A = 7641311254811318 + 8448416746792996 − 4651905840594284 = 11437822161010030." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8555603530971725 + 5728804618040901 + ___ = 6695559803388096 + 5238189805176004", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 8555603530971725 + 5728804618040901 + ___ = 6695559803388096 + 5238189805176004", + "options": [ + "-2350658540448526", + "-2350658540448524", + "-2350658540448544", + "-2350658540448514" + ], + "answer": "-2350658540448526", + "correct_value": -2350658540448526, + "explanation": "No matching terms; blank = 6695559803388096 + 5238189805176004 − 8555603530971725 − 5728804618040901 = -2350658540448526." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001525", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8220675246136888 × 5716956132596674 × 7882204116131020 = ___ × 7882204116131020 × 8220675246136888", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 8220675246136888 × 5716956132596674 × 7882204116131020 = ___ × 7882204116131020 × 8220675246136888", + "options": [ + "5716956132596684", + "5716956132596664", + "5716956132596694", + "5716956132596674" + ], + "answer": "5716956132596674", + "correct_value": 5716956132596674, + "explanation": "A and C appear on both sides; blank = B = 5716956132596674." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4238520678086216 + 5704118530477137 + ___ = 5704118530477137 + 6808587733194744 + 3065104510693303", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 4238520678086216 + 5704118530477137 + ___ = 5704118530477137 + 6808587733194744 + 3065104510693303", + "options": [ + "5635171565801851", + "5635171565801821", + "5635171565801841", + "5635171565801831" + ], + "answer": "5635171565801831", + "correct_value": 5635171565801831, + "explanation": "B cancels; blank = C + D − A = 6808587733194744 + 3065104510693303 − 4238520678086216 = 5635171565801831." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2477609906096385 + 9729057415790933 + ___ = 4792895096745303 + 6194809155064944", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 2477609906096385 + 9729057415790933 + ___ = 4792895096745303 + 6194809155064944", + "options": [ + "-1218963070077069", + "-1218963070077071", + "-1218963070077059", + "-1218963070077089" + ], + "answer": "-1218963070077071", + "correct_value": -1218963070077071, + "explanation": "No matching terms; blank = 4792895096745303 + 6194809155064944 − 2477609906096385 − 9729057415790933 = -1218963070077071." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001526", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1635940226141318 + 4528076738061711 + ___ = 4528076738061711 + 3025322630736057 + 1635940226141318", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 1635940226141318 + 4528076738061711 + ___ = 4528076738061711 + 3025322630736057 + 1635940226141318", + "options": [ + "3025322630736077", + "3025322630736067", + "3025322630736047", + "3025322630736057" + ], + "answer": "3025322630736057", + "correct_value": 3025322630736057, + "explanation": "A and B appear on both sides and cancel; blank = C = 3025322630736057." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7422552463665516 + 1702979937337549 + ___ = 1702979937337549 + 4556207391042188 + 1937014635407686", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 7422552463665516 + 1702979937337549 + ___ = 1702979937337549 + 4556207391042188 + 1937014635407686", + "options": [ + "-929330437215638", + "-929330437215628", + "-929330437215658", + "-929330437215642" + ], + "answer": "-929330437215642", + "correct_value": -929330437215642, + "explanation": "B cancels; blank = C + D − A = 4556207391042188 + 1937014635407686 − 7422552463665516 = -929330437215642." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8244252261533432 + 6345410395154468 + ___ = 1045173059623523 + 5500859413577684", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 8244252261533432 + 6345410395154468 + ___ = 1045173059623523 + 5500859413577684", + "options": [ + "-8043630183486707", + "-8043630183486677", + "-8043630183486687", + "-8043630183486693" + ], + "answer": "-8043630183486693", + "correct_value": -8043630183486693, + "explanation": "No matching terms; blank = 1045173059623523 + 5500859413577684 − 8244252261533432 − 6345410395154468 = -8043630183486693." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_001527", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2331554701290316 + 2750223032444389 = ___ + 2943991846229781", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 2331554701290316 + 2750223032444389 = ___ + 2943991846229781", + "options": [ + "2137785887504924", + "2137785887504934", + "2137785887504944", + "2137785887504914" + ], + "answer": "2137785887504924", + "correct_value": 2137785887504924, + "explanation": "Blank = A + B − C = 2331554701290316 + 2750223032444389 − 2943991846229781 = 2137785887504924." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2969667866174914 + 2835208904672195 + ___ = 2835208904672195 + 9411621780563791 + 8787938837962200", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 2969667866174914 + 2835208904672195 + ___ = 2835208904672195 + 9411621780563791 + 8787938837962200", + "options": [ + "15229892752351087", + "15229892752351097", + "15229892752351077", + "15229892752351067" + ], + "answer": "15229892752351077", + "correct_value": 15229892752351077, + "explanation": "B cancels; blank = C + D − A = 9411621780563791 + 8787938837962200 − 2969667866174914 = 15229892752351077." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1975576113516337 + 3160842760844015 + ___ = 1359145942987413 + 8641094422753294", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 1975576113516337 + 3160842760844015 + ___ = 1359145942987413 + 8641094422753294", + "options": [ + "4863821491380355", + "4863821491380375", + "4863821491380345", + "4863821491380365" + ], + "answer": "4863821491380355", + "correct_value": 4863821491380355, + "explanation": "No matching terms; blank = 1359145942987413 + 8641094422753294 − 1975576113516337 − 3160842760844015 = 4863821491380355." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001528", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1053178554533564 × 7227177900025980 × 4634679686962713 = ___ × 4634679686962713 × 1053178554533564", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 1053178554533564 × 7227177900025980 × 4634679686962713 = ___ × 4634679686962713 × 1053178554533564", + "options": [ + "7227177900025970", + "7227177900025980", + "7227177900025990", + "7227177900026000" + ], + "answer": "7227177900025980", + "correct_value": 7227177900025980, + "explanation": "A and C appear on both sides; blank = B = 7227177900025980." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1269926382503418 + 9150809689568890 + ___ = 9150809689568890 + 4479553786449994 + 2047706625547910", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 1269926382503418 + 9150809689568890 + ___ = 9150809689568890 + 4479553786449994 + 2047706625547910", + "options": [ + "5257334029494476", + "5257334029494486", + "5257334029494496", + "5257334029494506" + ], + "answer": "5257334029494486", + "correct_value": 5257334029494486, + "explanation": "B cancels; blank = C + D − A = 4479553786449994 + 2047706625547910 − 1269926382503418 = 5257334029494486." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4383358308500639 + 2043346310459568 + ___ = 2252358307425676 + 8426730381445619", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 4383358308500639 + 2043346310459568 + ___ = 2252358307425676 + 8426730381445619", + "options": [ + "4252384069911078", + "4252384069911088", + "4252384069911098", + "4252384069911108" + ], + "answer": "4252384069911088", + "correct_value": 4252384069911088, + "explanation": "No matching terms; blank = 2252358307425676 + 8426730381445619 − 4383358308500639 − 2043346310459568 = 4252384069911088." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001529", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3804984043419640 × (5109926858958150 + 1399339712696575) = 3804984043419640 × ___ + 3804984043419640 × 1399339712696575", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 3804984043419640 × (5109926858958150 + 1399339712696575) = 3804984043419640 × ___ + 3804984043419640 × 1399339712696575", + "options": [ + "5109926858958170", + "5109926858958160", + "5109926858958150", + "5109926858958140" + ], + "answer": "5109926858958150", + "correct_value": 5109926858958150, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 5109926858958150." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8993585900192683 + 3063606481066844 + ___ = 3063606481066844 + 5472040642910330 + 7807134294419628", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 8993585900192683 + 3063606481066844 + ___ = 3063606481066844 + 5472040642910330 + 7807134294419628", + "options": [ + "4285589037137265", + "4285589037137295", + "4285589037137285", + "4285589037137275" + ], + "answer": "4285589037137275", + "correct_value": 4285589037137275, + "explanation": "B cancels; blank = C + D − A = 5472040642910330 + 7807134294419628 − 8993585900192683 = 4285589037137275." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6989359793732632 + 6168817558749705 + ___ = 2993279725679857 + 7070223707602262", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 6989359793732632 + 6168817558749705 + ___ = 2993279725679857 + 7070223707602262", + "options": [ + "-3094673919200202", + "-3094673919200212", + "-3094673919200218", + "-3094673919200232" + ], + "answer": "-3094673919200218", + "correct_value": -3094673919200218, + "explanation": "No matching terms; blank = 2993279725679857 + 7070223707602262 − 6989359793732632 − 6168817558749705 = -3094673919200218." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001530", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1558248099715812 + 2493640720624898 + ___ = 2493640720624898 + 8297637911266294 + 1558248099715812", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 1558248099715812 + 2493640720624898 + ___ = 2493640720624898 + 8297637911266294 + 1558248099715812", + "options": [ + "8297637911266314", + "8297637911266304", + "8297637911266294", + "8297637911266284" + ], + "answer": "8297637911266294", + "correct_value": 8297637911266294, + "explanation": "A and B appear on both sides and cancel; blank = C = 8297637911266294." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1672408278892294 + 3681912780186421 + ___ = 3681912780186421 + 5630047012415380 + 1346210079936730", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 1672408278892294 + 3681912780186421 + ___ = 3681912780186421 + 5630047012415380 + 1346210079936730", + "options": [ + "5303848813459836", + "5303848813459806", + "5303848813459826", + "5303848813459816" + ], + "answer": "5303848813459816", + "correct_value": 5303848813459816, + "explanation": "B cancels; blank = C + D − A = 5630047012415380 + 1346210079936730 − 1672408278892294 = 5303848813459816." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5554919492266349 + 9230798831324212 + ___ = 1834549347427670 + 9764518853299343", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 5554919492266349 + 9230798831324212 + ___ = 1834549347427670 + 9764518853299343", + "options": [ + "-3186650122863542", + "-3186650122863562", + "-3186650122863548", + "-3186650122863532" + ], + "answer": "-3186650122863548", + "correct_value": -3186650122863548, + "explanation": "No matching terms; blank = 1834549347427670 + 9764518853299343 − 5554919492266349 − 9230798831324212 = -3186650122863548." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001531", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6527972947479958 + 5331973890368190 = ___ + 1213658118102539", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 6527972947479958 + 5331973890368190 = ___ + 1213658118102539", + "options": [ + "10646288719745599", + "10646288719745629", + "10646288719745609", + "10646288719745619" + ], + "answer": "10646288719745609", + "correct_value": 10646288719745609, + "explanation": "Blank = A + B − C = 6527972947479958 + 5331973890368190 − 1213658118102539 = 10646288719745609." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3957337784694626 + 7220197236470882 + ___ = 7220197236470882 + 1228007244388564 + 3058940027635768", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 3957337784694626 + 7220197236470882 + ___ = 7220197236470882 + 1228007244388564 + 3058940027635768", + "options": [ + "329609487329726", + "329609487329716", + "329609487329696", + "329609487329706" + ], + "answer": "329609487329706", + "correct_value": 329609487329706, + "explanation": "B cancels; blank = C + D − A = 1228007244388564 + 3058940027635768 − 3957337784694626 = 329609487329706." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7482360483237204 + 4884662767184543 + ___ = 5799027556998043 + 6863182341505315", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 7482360483237204 + 4884662767184543 + ___ = 5799027556998043 + 6863182341505315", + "options": [ + "295186648081631", + "295186648081621", + "295186648081601", + "295186648081611" + ], + "answer": "295186648081611", + "correct_value": 295186648081611, + "explanation": "No matching terms; blank = 5799027556998043 + 6863182341505315 − 7482360483237204 − 4884662767184543 = 295186648081611." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_001532", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5210909806817207 + 8687748902934265 + ___ = 8687748902934265 + 4766604526544077 + 5210909806817207", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 5210909806817207 + 8687748902934265 + ___ = 8687748902934265 + 4766604526544077 + 5210909806817207", + "options": [ + "4766604526544067", + "4766604526544097", + "4766604526544087", + "4766604526544077" + ], + "answer": "4766604526544077", + "correct_value": 4766604526544077, + "explanation": "A and B appear on both sides and cancel; blank = C = 4766604526544077." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2608821142028725 + 6551140325300192 + ___ = 6551140325300192 + 8441602010359219 + 1180593286726693", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 2608821142028725 + 6551140325300192 + ___ = 6551140325300192 + 8441602010359219 + 1180593286726693", + "options": [ + "7013374155057197", + "7013374155057207", + "7013374155057187", + "7013374155057177" + ], + "answer": "7013374155057187", + "correct_value": 7013374155057187, + "explanation": "B cancels; blank = C + D − A = 8441602010359219 + 1180593286726693 − 2608821142028725 = 7013374155057187." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9648695889988034 + 1811024757233040 + ___ = 4643537247595849 + 2744594432365468", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 9648695889988034 + 1811024757233040 + ___ = 4643537247595849 + 2744594432365468", + "options": [ + "-4071588967259773", + "-4071588967259757", + "-4071588967259743", + "-4071588967259753" + ], + "answer": "-4071588967259757", + "correct_value": -4071588967259757, + "explanation": "No matching terms; blank = 4643537247595849 + 2744594432365468 − 9648695889988034 − 1811024757233040 = -4071588967259757." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001533", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7362474948979208 × (3096403967180910 + 2868708106811189) = 7362474948979208 × ___ + 7362474948979208 × 2868708106811189", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 7362474948979208 × (3096403967180910 + 2868708106811189) = 7362474948979208 × ___ + 7362474948979208 × 2868708106811189", + "options": [ + "3096403967180900", + "3096403967180920", + "3096403967180930", + "3096403967180910" + ], + "answer": "3096403967180910", + "correct_value": 3096403967180910, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 3096403967180910." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9314607380742707 + 7751766541961534 + ___ = 7751766541961534 + 9269139053443230 + 1003982980841327", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 9314607380742707 + 7751766541961534 + ___ = 7751766541961534 + 9269139053443230 + 1003982980841327", + "options": [ + "958514653541850", + "958514653541840", + "958514653541870", + "958514653541860" + ], + "answer": "958514653541850", + "correct_value": 958514653541850, + "explanation": "B cancels; blank = C + D − A = 9269139053443230 + 1003982980841327 − 9314607380742707 = 958514653541850." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7372970343022899 + 1932788624818773 + ___ = 7647218364977824 + 3372934120545147", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 7372970343022899 + 1932788624818773 + ___ = 7647218364977824 + 3372934120545147", + "options": [ + "1714393517681319", + "1714393517681309", + "1714393517681299", + "1714393517681289" + ], + "answer": "1714393517681299", + "correct_value": 1714393517681299, + "explanation": "No matching terms; blank = 7647218364977824 + 3372934120545147 − 7372970343022899 − 1932788624818773 = 1714393517681299." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001534", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8602606603691833 × (1258085147733797 + 9069041956716424) = 8602606603691833 × ___ + 8602606603691833 × 9069041956716424", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 8602606603691833 × (1258085147733797 + 9069041956716424) = 8602606603691833 × ___ + 8602606603691833 × 9069041956716424", + "options": [ + "1258085147733797", + "1258085147733817", + "1258085147733807", + "1258085147733787" + ], + "answer": "1258085147733797", + "correct_value": 1258085147733797, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 1258085147733797." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2591932087263427 + 7708309892032582 + ___ = 7708309892032582 + 6997721657690295 + 3363939889587527", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 2591932087263427 + 7708309892032582 + ___ = 7708309892032582 + 6997721657690295 + 3363939889587527", + "options": [ + "7769729460014395", + "7769729460014385", + "7769729460014415", + "7769729460014405" + ], + "answer": "7769729460014395", + "correct_value": 7769729460014395, + "explanation": "B cancels; blank = C + D − A = 6997721657690295 + 3363939889587527 − 2591932087263427 = 7769729460014395." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2034089414628812 + 8964343016150275 + ___ = 1415296126326191 + 5804444176093503", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 2034089414628812 + 8964343016150275 + ___ = 1415296126326191 + 5804444176093503", + "options": [ + "-3778692128359393", + "-3778692128359407", + "-3778692128359377", + "-3778692128359387" + ], + "answer": "-3778692128359393", + "correct_value": -3778692128359393, + "explanation": "No matching terms; blank = 1415296126326191 + 5804444176093503 − 2034089414628812 − 8964343016150275 = -3778692128359393." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001535", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4121120999123190 + 2324915079786940 + ___ = 2324915079786940 + 4851750228294284 + 4121120999123190", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 4121120999123190 + 2324915079786940 + ___ = 2324915079786940 + 4851750228294284 + 4121120999123190", + "options": [ + "4851750228294294", + "4851750228294304", + "4851750228294284", + "4851750228294274" + ], + "answer": "4851750228294284", + "correct_value": 4851750228294284, + "explanation": "A and B appear on both sides and cancel; blank = C = 4851750228294284." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6363324370274417 + 9801247958144421 + ___ = 9801247958144421 + 7773067894115681 + 1396790104218304", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 6363324370274417 + 9801247958144421 + ___ = 9801247958144421 + 7773067894115681 + 1396790104218304", + "options": [ + "2806533628059588", + "2806533628059578", + "2806533628059558", + "2806533628059568" + ], + "answer": "2806533628059568", + "correct_value": 2806533628059568, + "explanation": "B cancels; blank = C + D − A = 7773067894115681 + 1396790104218304 − 6363324370274417 = 2806533628059568." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4473744137432268 + 8770445041402556 + ___ = 3866549158744246 + 2894603128571256", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 4473744137432268 + 8770445041402556 + ___ = 3866549158744246 + 2894603128571256", + "options": [ + "-6483036891519338", + "-6483036891519322", + "-6483036891519308", + "-6483036891519318" + ], + "answer": "-6483036891519322", + "correct_value": -6483036891519322, + "explanation": "No matching terms; blank = 3866549158744246 + 2894603128571256 − 4473744137432268 − 8770445041402556 = -6483036891519322." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001536", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7092693933349702 + 6549579805424133 = ___ + 5140582229494738", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 7092693933349702 + 6549579805424133 = ___ + 5140582229494738", + "options": [ + "8501691509279087", + "8501691509279097", + "8501691509279117", + "8501691509279107" + ], + "answer": "8501691509279097", + "correct_value": 8501691509279097, + "explanation": "Blank = A + B − C = 7092693933349702 + 6549579805424133 − 5140582229494738 = 8501691509279097." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5966648575683013 + 6773720296764147 + ___ = 6773720296764147 + 7052806545995782 + 7273360208528615", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 5966648575683013 + 6773720296764147 + ___ = 6773720296764147 + 7052806545995782 + 7273360208528615", + "options": [ + "8359518178841384", + "8359518178841404", + "8359518178841394", + "8359518178841374" + ], + "answer": "8359518178841384", + "correct_value": 8359518178841384, + "explanation": "B cancels; blank = C + D − A = 7052806545995782 + 7273360208528615 − 5966648575683013 = 8359518178841384." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4248551597586604 + 7819374017845253 + ___ = 7581676545272329 + 6307386984137290", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 4248551597586604 + 7819374017845253 + ___ = 7581676545272329 + 6307386984137290", + "options": [ + "1821137913977752", + "1821137913977762", + "1821137913977772", + "1821137913977782" + ], + "answer": "1821137913977762", + "correct_value": 1821137913977762, + "explanation": "No matching terms; blank = 7581676545272329 + 6307386984137290 − 4248551597586604 − 7819374017845253 = 1821137913977762." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001537", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9705526572272258 × (9682265322061810 + 8826504239402538) = 9705526572272258 × ___ + 9705526572272258 × 8826504239402538", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 9705526572272258 × (9682265322061810 + 8826504239402538) = 9705526572272258 × ___ + 9705526572272258 × 8826504239402538", + "options": [ + "9682265322061820", + "9682265322061800", + "9682265322061810", + "9682265322061830" + ], + "answer": "9682265322061810", + "correct_value": 9682265322061810, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 9682265322061810." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4486659640098485 + 3168176662677883 + ___ = 3168176662677883 + 3034901893688522 + 1239614113100124", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 4486659640098485 + 3168176662677883 + ___ = 3168176662677883 + 3034901893688522 + 1239614113100124", + "options": [ + "-212143633309839", + "-212143633309831", + "-212143633309851", + "-212143633309821" + ], + "answer": "-212143633309839", + "correct_value": -212143633309839, + "explanation": "B cancels; blank = C + D − A = 3034901893688522 + 1239614113100124 − 4486659640098485 = -212143633309839." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1660415726549050 + 8287736090551328 + ___ = 5548294621840376 + 6096120602703683", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 1660415726549050 + 8287736090551328 + ___ = 5548294621840376 + 6096120602703683", + "options": [ + "1696263407443701", + "1696263407443671", + "1696263407443691", + "1696263407443681" + ], + "answer": "1696263407443681", + "correct_value": 1696263407443681, + "explanation": "No matching terms; blank = 5548294621840376 + 6096120602703683 − 1660415726549050 − 8287736090551328 = 1696263407443681." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_001538", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8567110385018472 + 1284928631894407 = ___ + 7368772355517850", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 8567110385018472 + 1284928631894407 = ___ + 7368772355517850", + "options": [ + "2483266661395049", + "2483266661395029", + "2483266661395039", + "2483266661395019" + ], + "answer": "2483266661395029", + "correct_value": 2483266661395029, + "explanation": "Blank = A + B − C = 8567110385018472 + 1284928631894407 − 7368772355517850 = 2483266661395029." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4142055866604761 + 6961119553855947 + ___ = 6961119553855947 + 6198989687222077 + 4251662158579775", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 4142055866604761 + 6961119553855947 + ___ = 6961119553855947 + 6198989687222077 + 4251662158579775", + "options": [ + "6308595979197081", + "6308595979197101", + "6308595979197111", + "6308595979197091" + ], + "answer": "6308595979197091", + "correct_value": 6308595979197091, + "explanation": "B cancels; blank = C + D − A = 6198989687222077 + 4251662158579775 − 4142055866604761 = 6308595979197091." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2366439511432283 + 6736053632823127 + ___ = 6632272192440828 + 7197749211610348", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 2366439511432283 + 6736053632823127 + ___ = 6632272192440828 + 7197749211610348", + "options": [ + "4727528259795786", + "4727528259795776", + "4727528259795756", + "4727528259795766" + ], + "answer": "4727528259795766", + "correct_value": 4727528259795766, + "explanation": "No matching terms; blank = 6632272192440828 + 7197749211610348 − 2366439511432283 − 6736053632823127 = 4727528259795766." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_001539", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5546435633866435 + 1789414082090709 + ___ = 1789414082090709 + 8744302183355268 + 5546435633866435", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 5546435633866435 + 1789414082090709 + ___ = 1789414082090709 + 8744302183355268 + 5546435633866435", + "options": [ + "8744302183355288", + "8744302183355278", + "8744302183355258", + "8744302183355268" + ], + "answer": "8744302183355268", + "correct_value": 8744302183355268, + "explanation": "A and B appear on both sides and cancel; blank = C = 8744302183355268." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9177031969368272 + 3756324981116268 + ___ = 3756324981116268 + 5619686992164732 + 9841527275780593", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 9177031969368272 + 3756324981116268 + ___ = 3756324981116268 + 5619686992164732 + 9841527275780593", + "options": [ + "6284182298577063", + "6284182298577053", + "6284182298577073", + "6284182298577043" + ], + "answer": "6284182298577053", + "correct_value": 6284182298577053, + "explanation": "B cancels; blank = C + D − A = 5619686992164732 + 9841527275780593 − 9177031969368272 = 6284182298577053." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2623290847445194 + 9627740616700838 + ___ = 5473770064427568 + 3279583191885382", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 2623290847445194 + 9627740616700838 + ___ = 5473770064427568 + 3279583191885382", + "options": [ + "-3497678207833068", + "-3497678207833082", + "-3497678207833098", + "-3497678207833078" + ], + "answer": "-3497678207833082", + "correct_value": -3497678207833082, + "explanation": "No matching terms; blank = 5473770064427568 + 3279583191885382 − 2623290847445194 − 9627740616700838 = -3497678207833082." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001540", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5779586281922916 + 9649000890147332 + ___ = 9649000890147332 + 5445616426713288 + 5779586281922916", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 5779586281922916 + 9649000890147332 + ___ = 9649000890147332 + 5445616426713288 + 5779586281922916", + "options": [ + "5445616426713288", + "5445616426713298", + "5445616426713278", + "5445616426713308" + ], + "answer": "5445616426713288", + "correct_value": 5445616426713288, + "explanation": "A and B appear on both sides and cancel; blank = C = 5445616426713288." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9279154286019750 + 9376151788091536 + ___ = 9376151788091536 + 5892203765188854 + 5416298213014737", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 9279154286019750 + 9376151788091536 + ___ = 9376151788091536 + 5892203765188854 + 5416298213014737", + "options": [ + "2029347692183841", + "2029347692183851", + "2029347692183861", + "2029347692183831" + ], + "answer": "2029347692183841", + "correct_value": 2029347692183841, + "explanation": "B cancels; blank = C + D − A = 5892203765188854 + 5416298213014737 − 9279154286019750 = 2029347692183841." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9161078002344448 + 8072009496056966 + ___ = 5149091563954446 + 8229613880656996", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 9161078002344448 + 8072009496056966 + ___ = 5149091563954446 + 8229613880656996", + "options": [ + "-3854382053789968", + "-3854382053789972", + "-3854382053789988", + "-3854382053789958" + ], + "answer": "-3854382053789972", + "correct_value": -3854382053789972, + "explanation": "No matching terms; blank = 5149091563954446 + 8229613880656996 − 9161078002344448 − 8072009496056966 = -3854382053789972." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001541", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7930357640608270 × (2686770542842925 + 8828674763399836) = 7930357640608270 × ___ + 7930357640608270 × 8828674763399836", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 7930357640608270 × (2686770542842925 + 8828674763399836) = 7930357640608270 × ___ + 7930357640608270 × 8828674763399836", + "options": [ + "2686770542842945", + "2686770542842915", + "2686770542842935", + "2686770542842925" + ], + "answer": "2686770542842925", + "correct_value": 2686770542842925, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 2686770542842925." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6420697250042561 + 8566801919529005 + ___ = 8566801919529005 + 6738470214947001 + 2773095636631891", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 6420697250042561 + 8566801919529005 + ___ = 8566801919529005 + 6738470214947001 + 2773095636631891", + "options": [ + "3090868601536321", + "3090868601536341", + "3090868601536331", + "3090868601536351" + ], + "answer": "3090868601536331", + "correct_value": 3090868601536331, + "explanation": "B cancels; blank = C + D − A = 6738470214947001 + 2773095636631891 − 6420697250042561 = 3090868601536331." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4162435170458615 + 5144009478775509 + ___ = 3688980578438333 + 4256043172532636", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 4162435170458615 + 5144009478775509 + ___ = 3688980578438333 + 4256043172532636", + "options": [ + "-1361420898263135", + "-1361420898263165", + "-1361420898263145", + "-1361420898263155" + ], + "answer": "-1361420898263155", + "correct_value": -1361420898263155, + "explanation": "No matching terms; blank = 3688980578438333 + 4256043172532636 − 4162435170458615 − 5144009478775509 = -1361420898263155." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_001542", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4040350569059189 × (8101456112862280 + 7951044226508976) = 4040350569059189 × ___ + 4040350569059189 × 7951044226508976", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 4040350569059189 × (8101456112862280 + 7951044226508976) = 4040350569059189 × ___ + 4040350569059189 × 7951044226508976", + "options": [ + "8101456112862280", + "8101456112862300", + "8101456112862270", + "8101456112862290" + ], + "answer": "8101456112862280", + "correct_value": 8101456112862280, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 8101456112862280." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9381869575195913 + 3181148814224181 + ___ = 3181148814224181 + 9957683638004322 + 5326675778838333", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 9381869575195913 + 3181148814224181 + ___ = 3181148814224181 + 9957683638004322 + 5326675778838333", + "options": [ + "5902489841646762", + "5902489841646752", + "5902489841646742", + "5902489841646732" + ], + "answer": "5902489841646742", + "correct_value": 5902489841646742, + "explanation": "B cancels; blank = C + D − A = 9957683638004322 + 5326675778838333 − 9381869575195913 = 5902489841646742." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2764481616711720 + 2932945505346258 + ___ = 6100028888533813 + 3643025948340725", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 2764481616711720 + 2932945505346258 + ___ = 6100028888533813 + 3643025948340725", + "options": [ + "4045627714816560", + "4045627714816570", + "4045627714816550", + "4045627714816580" + ], + "answer": "4045627714816560", + "correct_value": 4045627714816560, + "explanation": "No matching terms; blank = 6100028888533813 + 3643025948340725 − 2764481616711720 − 2932945505346258 = 4045627714816560." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001543", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6434842961228522 + 2406174307339384 + ___ = 2406174307339384 + 7967513799762804 + 6434842961228522", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 6434842961228522 + 2406174307339384 + ___ = 2406174307339384 + 7967513799762804 + 6434842961228522", + "options": [ + "7967513799762824", + "7967513799762804", + "7967513799762814", + "7967513799762794" + ], + "answer": "7967513799762804", + "correct_value": 7967513799762804, + "explanation": "A and B appear on both sides and cancel; blank = C = 7967513799762804." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1759428135988573 + 2885624518936929 + ___ = 2885624518936929 + 4674916132093709 + 1443614856555700", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 1759428135988573 + 2885624518936929 + ___ = 2885624518936929 + 4674916132093709 + 1443614856555700", + "options": [ + "4359102852660826", + "4359102852660856", + "4359102852660836", + "4359102852660846" + ], + "answer": "4359102852660836", + "correct_value": 4359102852660836, + "explanation": "B cancels; blank = C + D − A = 4674916132093709 + 1443614856555700 − 1759428135988573 = 4359102852660836." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7684146107471134 + 3479922420634625 + ___ = 4710462794975655 + 1378621435304972", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 7684146107471134 + 3479922420634625 + ___ = 4710462794975655 + 1378621435304972", + "options": [ + "-5074984297825148", + "-5074984297825128", + "-5074984297825132", + "-5074984297825118" + ], + "answer": "-5074984297825132", + "correct_value": -5074984297825132, + "explanation": "No matching terms; blank = 4710462794975655 + 1378621435304972 − 7684146107471134 − 3479922420634625 = -5074984297825132." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001544", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4808451641380022 + 7159593992100287 + ___ = 7159593992100287 + 7413149497672321 + 4808451641380022", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 4808451641380022 + 7159593992100287 + ___ = 7159593992100287 + 7413149497672321 + 4808451641380022", + "options": [ + "7413149497672331", + "7413149497672311", + "7413149497672341", + "7413149497672321" + ], + "answer": "7413149497672321", + "correct_value": 7413149497672321, + "explanation": "A and B appear on both sides and cancel; blank = C = 7413149497672321." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4999612740312860 + 4339578254099992 + ___ = 4339578254099992 + 2083299081016873 + 7187169998390544", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 4999612740312860 + 4339578254099992 + ___ = 4339578254099992 + 2083299081016873 + 7187169998390544", + "options": [ + "4270856339094547", + "4270856339094577", + "4270856339094557", + "4270856339094567" + ], + "answer": "4270856339094557", + "correct_value": 4270856339094557, + "explanation": "B cancels; blank = C + D − A = 2083299081016873 + 7187169998390544 − 4999612740312860 = 4270856339094557." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2583488210897906 + 8223587992672437 + ___ = 3668419096589427 + 8281620012018478", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 2583488210897906 + 8223587992672437 + ___ = 3668419096589427 + 8281620012018478", + "options": [ + "1142962905037572", + "1142962905037562", + "1142962905037552", + "1142962905037582" + ], + "answer": "1142962905037562", + "correct_value": 1142962905037562, + "explanation": "No matching terms; blank = 3668419096589427 + 8281620012018478 − 2583488210897906 − 8223587992672437 = 1142962905037562." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001545", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7588589430236939 × (2786356109818763 + 3888250562473902) = 7588589430236939 × ___ + 7588589430236939 × 3888250562473902", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 7588589430236939 × (2786356109818763 + 3888250562473902) = 7588589430236939 × ___ + 7588589430236939 × 3888250562473902", + "options": [ + "2786356109818773", + "2786356109818753", + "2786356109818783", + "2786356109818763" + ], + "answer": "2786356109818763", + "correct_value": 2786356109818763, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 2786356109818763." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9592244723381687 + 1276780178492779 + ___ = 1276780178492779 + 1905660082081232 + 8622618628948106", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 9592244723381687 + 1276780178492779 + ___ = 1276780178492779 + 1905660082081232 + 8622618628948106", + "options": [ + "936033987647661", + "936033987647671", + "936033987647651", + "936033987647641" + ], + "answer": "936033987647651", + "correct_value": 936033987647651, + "explanation": "B cancels; blank = C + D − A = 1905660082081232 + 8622618628948106 − 9592244723381687 = 936033987647651." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2633186068596676 + 1276205797427340 + ___ = 5250573011182617 + 4538143596295017", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 2633186068596676 + 1276205797427340 + ___ = 5250573011182617 + 4538143596295017", + "options": [ + "5879324741453608", + "5879324741453638", + "5879324741453628", + "5879324741453618" + ], + "answer": "5879324741453618", + "correct_value": 5879324741453618, + "explanation": "No matching terms; blank = 5250573011182617 + 4538143596295017 − 2633186068596676 − 1276205797427340 = 5879324741453618." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_001546", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2179499626991260 + 4090609559610474 + ___ = 4090609559610474 + 7540453928018498 + 2179499626991260", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 2179499626991260 + 4090609559610474 + ___ = 4090609559610474 + 7540453928018498 + 2179499626991260", + "options": [ + "7540453928018518", + "7540453928018488", + "7540453928018508", + "7540453928018498" + ], + "answer": "7540453928018498", + "correct_value": 7540453928018498, + "explanation": "A and B appear on both sides and cancel; blank = C = 7540453928018498." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6974378193549744 + 8259878256643015 + ___ = 8259878256643015 + 3781521404367227 + 7421030228673248", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 6974378193549744 + 8259878256643015 + ___ = 8259878256643015 + 3781521404367227 + 7421030228673248", + "options": [ + "4228173439490721", + "4228173439490741", + "4228173439490751", + "4228173439490731" + ], + "answer": "4228173439490731", + "correct_value": 4228173439490731, + "explanation": "B cancels; blank = C + D − A = 3781521404367227 + 7421030228673248 − 6974378193549744 = 4228173439490731." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4904884027639950 + 8581618191050997 + ___ = 2588523606788874 + 1994917565483293", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 4904884027639950 + 8581618191050997 + ___ = 2588523606788874 + 1994917565483293", + "options": [ + "-8903061046418780", + "-8903061046418760", + "-8903061046418770", + "-8903061046418790" + ], + "answer": "-8903061046418780", + "correct_value": -8903061046418780, + "explanation": "No matching terms; blank = 2588523606788874 + 1994917565483293 − 4904884027639950 − 8581618191050997 = -8903061046418780." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001547", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8271184018436236 × 1961806206927511 × 6660562264930959 = ___ × 6660562264930959 × 8271184018436236", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 8271184018436236 × 1961806206927511 × 6660562264930959 = ___ × 6660562264930959 × 8271184018436236", + "options": [ + "1961806206927531", + "1961806206927511", + "1961806206927501", + "1961806206927521" + ], + "answer": "1961806206927511", + "correct_value": 1961806206927511, + "explanation": "A and C appear on both sides; blank = B = 1961806206927511." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4302516795408079 + 6059218749366943 + ___ = 6059218749366943 + 5310895210652928 + 5971568305995865", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 4302516795408079 + 6059218749366943 + ___ = 6059218749366943 + 5310895210652928 + 5971568305995865", + "options": [ + "6979946721240724", + "6979946721240714", + "6979946721240734", + "6979946721240704" + ], + "answer": "6979946721240714", + "correct_value": 6979946721240714, + "explanation": "B cancels; blank = C + D − A = 5310895210652928 + 5971568305995865 − 4302516795408079 = 6979946721240714." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3574711165269986 + 2070305504398164 + ___ = 7970841844637652 + 2075064048686238", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 3574711165269986 + 2070305504398164 + ___ = 7970841844637652 + 2075064048686238", + "options": [ + "4400889223655730", + "4400889223655740", + "4400889223655760", + "4400889223655750" + ], + "answer": "4400889223655740", + "correct_value": 4400889223655740, + "explanation": "No matching terms; blank = 7970841844637652 + 2075064048686238 − 3574711165269986 − 2070305504398164 = 4400889223655740." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001548", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2450248408144842 × 6844204389213416 × 4189558827380758 = ___ × 4189558827380758 × 2450248408144842", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 2450248408144842 × 6844204389213416 × 4189558827380758 = ___ × 4189558827380758 × 2450248408144842", + "options": [ + "6844204389213416", + "6844204389213436", + "6844204389213426", + "6844204389213406" + ], + "answer": "6844204389213416", + "correct_value": 6844204389213416, + "explanation": "A and C appear on both sides; blank = B = 6844204389213416." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5675011729469962 + 4212894898055065 + ___ = 4212894898055065 + 3150277309359643 + 8373655772025434", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 5675011729469962 + 4212894898055065 + ___ = 4212894898055065 + 3150277309359643 + 8373655772025434", + "options": [ + "5848921351915115", + "5848921351915135", + "5848921351915105", + "5848921351915125" + ], + "answer": "5848921351915115", + "correct_value": 5848921351915115, + "explanation": "B cancels; blank = C + D − A = 3150277309359643 + 8373655772025434 − 5675011729469962 = 5848921351915115." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2227768180762713 + 4802803633413909 + ___ = 4065402958608648 + 4315068719599908", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 2227768180762713 + 4802803633413909 + ___ = 4065402958608648 + 4315068719599908", + "options": [ + "1349899864031954", + "1349899864031944", + "1349899864031924", + "1349899864031934" + ], + "answer": "1349899864031934", + "correct_value": 1349899864031934, + "explanation": "No matching terms; blank = 4065402958608648 + 4315068719599908 − 2227768180762713 − 4802803633413909 = 1349899864031934." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_001549", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1075520320366280 + 2258145426800559 + ___ = 2258145426800559 + 5625074371984145 + 1075520320366280", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 1075520320366280 + 2258145426800559 + ___ = 2258145426800559 + 5625074371984145 + 1075520320366280", + "options": [ + "5625074371984155", + "5625074371984165", + "5625074371984145", + "5625074371984135" + ], + "answer": "5625074371984145", + "correct_value": 5625074371984145, + "explanation": "A and B appear on both sides and cancel; blank = C = 5625074371984145." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1097727547347667 + 2238939376301181 + ___ = 2238939376301181 + 5699300420409497 + 4166752520351224", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 1097727547347667 + 2238939376301181 + ___ = 2238939376301181 + 5699300420409497 + 4166752520351224", + "options": [ + "8768325393413044", + "8768325393413054", + "8768325393413064", + "8768325393413074" + ], + "answer": "8768325393413054", + "correct_value": 8768325393413054, + "explanation": "B cancels; blank = C + D − A = 5699300420409497 + 4166752520351224 − 1097727547347667 = 8768325393413054." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5243751945272956 + 8825149157449464 + ___ = 2450480957709452 + 5479400679294102", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 5243751945272956 + 8825149157449464 + ___ = 2450480957709452 + 5479400679294102", + "options": [ + "-6139019465718866", + "-6139019465718884", + "-6139019465718854", + "-6139019465718864" + ], + "answer": "-6139019465718866", + "correct_value": -6139019465718866, + "explanation": "No matching terms; blank = 2450480957709452 + 5479400679294102 − 5243751945272956 − 8825149157449464 = -6139019465718866." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001550", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9701365724720143 + 9872498440745459 + ___ = 9872498440745459 + 1944242811036027 + 9701365724720143", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 9701365724720143 + 9872498440745459 + ___ = 9872498440745459 + 1944242811036027 + 9701365724720143", + "options": [ + "1944242811036047", + "1944242811036027", + "1944242811036037", + "1944242811036017" + ], + "answer": "1944242811036027", + "correct_value": 1944242811036027, + "explanation": "A and B appear on both sides and cancel; blank = C = 1944242811036027." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6456116102479139 + 1514001708300421 + ___ = 1514001708300421 + 8265966107208020 + 1697186552362223", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 6456116102479139 + 1514001708300421 + ___ = 1514001708300421 + 8265966107208020 + 1697186552362223", + "options": [ + "3507036557091104", + "3507036557091124", + "3507036557091114", + "3507036557091094" + ], + "answer": "3507036557091104", + "correct_value": 3507036557091104, + "explanation": "B cancels; blank = C + D − A = 8265966107208020 + 1697186552362223 − 6456116102479139 = 3507036557091104." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1335604668120552 + 7938165956294774 + ___ = 9158241693554797 + 6297339728436959", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 1335604668120552 + 7938165956294774 + ___ = 9158241693554797 + 6297339728436959", + "options": [ + "6181810797576440", + "6181810797576450", + "6181810797576430", + "6181810797576420" + ], + "answer": "6181810797576430", + "correct_value": 6181810797576430, + "explanation": "No matching terms; blank = 9158241693554797 + 6297339728436959 − 1335604668120552 − 7938165956294774 = 6181810797576430." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_001551", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3067540763036053 × 8656456881613739 × 1982939910026357 = ___ × 1982939910026357 × 3067540763036053", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 3067540763036053 × 8656456881613739 × 1982939910026357 = ___ × 1982939910026357 × 3067540763036053", + "options": [ + "8656456881613739", + "8656456881613749", + "8656456881613759", + "8656456881613729" + ], + "answer": "8656456881613739", + "correct_value": 8656456881613739, + "explanation": "A and C appear on both sides; blank = B = 8656456881613739." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7889855830438321 + 1218209599272062 + ___ = 1218209599272062 + 4748615990751375 + 1394874416459976", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 7889855830438321 + 1218209599272062 + ___ = 1218209599272062 + 4748615990751375 + 1394874416459976", + "options": [ + "-1746365423226950", + "-1746365423226980", + "-1746365423226960", + "-1746365423226970" + ], + "answer": "-1746365423226970", + "correct_value": -1746365423226970, + "explanation": "B cancels; blank = C + D − A = 4748615990751375 + 1394874416459976 − 7889855830438321 = -1746365423226970." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6944630308971588 + 4770037120131624 + ___ = 8726397552817575 + 9816875847622960", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 6944630308971588 + 4770037120131624 + ___ = 8726397552817575 + 9816875847622960", + "options": [ + "6828605971337313", + "6828605971337343", + "6828605971337333", + "6828605971337323" + ], + "answer": "6828605971337323", + "correct_value": 6828605971337323, + "explanation": "No matching terms; blank = 8726397552817575 + 9816875847622960 − 6944630308971588 − 4770037120131624 = 6828605971337323." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "option_elimination_001552", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9701770548774270 × 3786889370837892", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 9701770548774270 × 3786889370837892?", + "options": [ + "36739531769461385521163470638840", + "3673953176946138552116347063884076", + "367395317694613855211634706388475", + "3673953176946138552116347063884" + ], + "answer": "36739531769461385521163470638840", + "correct_value": 36739531769461385521163470638840, + "explanation": "Product has 32 digits; only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5796549157068825 × 4495584738272522", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 5796549157068825 × 4495584738272522?", + "options": [ + "26058877925165061654975560326650", + "26058877925165061654975560326651", + "26058877925165061654975560326649", + "26058877925165061654975560326640" + ], + "answer": "26058877925165061654975560326650", + "correct_value": 26058877925165061654975560326650, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1548919619890571 × 9564396928689351", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 1548919619890571 × 9564396928689351?", + "options": [ + "17142751318182894257457073009421", + "14814482055268054257457073009421", + "15556831448477464257457073009421", + "15593714292370214257457073009421" + ], + "answer": "14814482055268054257457073009421", + "correct_value": 14814482055268054257457073009421, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001553", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9701770548774270 × 3786889370837892", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 9701770548774270 × 3786889370837892?", + "options": [ + "36739531769461385521163470638840", + "3673953176946138552116347063884076", + "367395317694613855211634706388475", + "3673953176946138552116347063884" + ], + "answer": "36739531769461385521163470638840", + "correct_value": 36739531769461385521163470638840, + "explanation": "Product has 32 digits; only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5796549157068825 × 4495584738272522", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 5796549157068825 × 4495584738272522?", + "options": [ + "26058877925165061654975560326650", + "26058877925165061654975560326651", + "26058877925165061654975560326649", + "26058877925165061654975560326640" + ], + "answer": "26058877925165061654975560326650", + "correct_value": 26058877925165061654975560326650, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1548919619890571 × 9564396928689351", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 1548919619890571 × 9564396928689351?", + "options": [ + "17142751318182894257457073009421", + "14814482055268054257457073009421", + "15556831448477464257457073009421", + "15593714292370214257457073009421" + ], + "answer": "14814482055268054257457073009421", + "correct_value": 14814482055268054257457073009421, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001554", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9894722443785397 × 1999149853124014", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 9894722443785397 × 1999149853124014?", + "options": [ + "19781032920196461285240243223558", + "1978103292019646128524024322355814", + "1978103292019646128524024322354", + "3956206584039292257048048644710" + ], + "answer": "19781032920196461285240243223558", + "correct_value": 19781032920196461285240243223558, + "explanation": "Product has 32 digits; only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2305857800177520 × 8222437279440358", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 2305857800177520 × 8222437279440358?", + "options": [ + "18959771137267976194938652352160", + "18959771137267976194938652352159", + "18959771137267976194938652352161", + "18959771137267976194938652352140" + ], + "answer": "18959771137267976194938652352160", + "correct_value": 18959771137267976194938652352160, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2634734013801253 × 2714068522933927", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 2634734013801253 × 2714068522933927?", + "options": [ + "7150848653161343564765428810531", + "8459075884082677564765428810531", + "6258097218066191564765428810531", + "5979526457831248564765428810531" + ], + "answer": "7150848653161343564765428810531", + "correct_value": 7150848653161343564765428810531, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001555", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6678355599032622 × 6328880242283253", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 6678355599032622 × 6328880242283253?", + "options": [ + "422665128016592999477473112793701", + "422665128016592999477473112793724", + "42266512801659299947747311279366", + "4226651280165929994774731127933" + ], + "answer": "42266512801659299947747311279366", + "correct_value": 42266512801659299947747311279366, + "explanation": "Product has 32 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8879760874513620 × 1460092885435774", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 8879760874513620 × 1460092885435774?", + "options": [ + "12965275677248283312923998241881", + "12965275677248283312923998241860", + "12965275677248283312923998241880", + "12965275677248283312923998241879" + ], + "answer": "12965275677248283312923998241880", + "correct_value": 12965275677248283312923998241880, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9731543961643649 × 8367365962109257", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 9731543961643649 × 8367365962109257?", + "options": [ + "67551866688432111514692638158793", + "81427389703426941514692638158793", + "85635906098207471514692638158793", + "87127750795425281514692638158793" + ], + "answer": "81427389703426941514692638158793", + "correct_value": 81427389703426941514692638158793, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001556", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5332517376285870 × 3819109221392685", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 5332517376285870 × 3819109221392685?", + "options": [ + "20365466285010092434914086860951", + "20365466285010092434914086860952", + "20365466285010092434914086860949", + "20365466285010092434914086860950" + ], + "answer": "20365466285010092434914086860950", + "correct_value": 20365466285010092434914086860950, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5281644763579870 × 5699237064456672", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 5281644763579870 × 5699237064456672?", + "options": [ + "30101345597887891705775226392640", + "30101345597887891705775226392641", + "30101345597887891705775226392639", + "30101345597887891705775226392660" + ], + "answer": "30101345597887891705775226392640", + "correct_value": 30101345597887891705775226392640, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8889817708746851 × 4861702720971151", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 8889817708746851 × 4861702720971151?", + "options": [ + "43219650943552088655801933095501", + "34913488002996478655801933095501", + "42213904505954208655801933095501", + "35198387935107068655801933095501" + ], + "answer": "43219650943552088655801933095501", + "correct_value": 43219650943552088655801933095501, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001557", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9894722443785397 × 1999149853124014", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 9894722443785397 × 1999149853124014?", + "options": [ + "19781032920196461285240243223558", + "1978103292019646128524024322355814", + "1978103292019646128524024322354", + "3956206584039292257048048644710" + ], + "answer": "19781032920196461285240243223558", + "correct_value": 19781032920196461285240243223558, + "explanation": "Product has 32 digits; only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2305857800177520 × 8222437279440358", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 2305857800177520 × 8222437279440358?", + "options": [ + "18959771137267976194938652352160", + "18959771137267976194938652352159", + "18959771137267976194938652352161", + "18959771137267976194938652352140" + ], + "answer": "18959771137267976194938652352160", + "correct_value": 18959771137267976194938652352160, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2634734013801253 × 2714068522933927", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 2634734013801253 × 2714068522933927?", + "options": [ + "7150848653161343564765428810531", + "8459075884082677564765428810531", + "6258097218066191564765428810531", + "5979526457831248564765428810531" + ], + "answer": "7150848653161343564765428810531", + "correct_value": 7150848653161343564765428810531, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001558", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5591935194458535 × 7063918300464727", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 5591935194458535 × 7063918300464727?", + "options": [ + "39500973355148427244805631594944", + "39500973355148427244805631594948", + "39500973355148427244805631594945", + "39500973355148427244805631594946" + ], + "answer": "39500973355148427244805631594945", + "correct_value": 39500973355148427244805631594945, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2171165790781490 × 2005770742435570", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 2171165790781490 × 2005770742435570?", + "options": [ + "4354860820126500640656273599320", + "4354860820126500640656273599300", + "4354860820126500640656273599299", + "4354860820126500640656273599301" + ], + "answer": "4354860820126500640656273599300", + "correct_value": 4354860820126500640656273599300, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4904278846389561 × 5225199829061953", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4904278846389561 × 5225199829061953?", + "options": [ + "25200284730975606191955441472633", + "26648977144190646191955441472633", + "25625836989826886191955441472633", + "25178230381341876191955441472633" + ], + "answer": "25625836989826886191955441472633", + "correct_value": 25625836989826886191955441472633, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001559", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6678355599032622 × 6328880242283253", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 6678355599032622 × 6328880242283253?", + "options": [ + "422665128016592999477473112793701", + "422665128016592999477473112793724", + "42266512801659299947747311279366", + "4226651280165929994774731127933" + ], + "answer": "42266512801659299947747311279366", + "correct_value": 42266512801659299947747311279366, + "explanation": "Product has 32 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8879760874513620 × 1460092885435774", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 8879760874513620 × 1460092885435774?", + "options": [ + "12965275677248283312923998241881", + "12965275677248283312923998241860", + "12965275677248283312923998241880", + "12965275677248283312923998241879" + ], + "answer": "12965275677248283312923998241880", + "correct_value": 12965275677248283312923998241880, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9731543961643649 × 8367365962109257", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 9731543961643649 × 8367365962109257?", + "options": [ + "67551866688432111514692638158793", + "81427389703426941514692638158793", + "85635906098207471514692638158793", + "87127750795425281514692638158793" + ], + "answer": "81427389703426941514692638158793", + "correct_value": 81427389703426941514692638158793, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001560", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5332517376285870 × 3819109221392685", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 5332517376285870 × 3819109221392685?", + "options": [ + "20365466285010092434914086860951", + "20365466285010092434914086860952", + "20365466285010092434914086860949", + "20365466285010092434914086860950" + ], + "answer": "20365466285010092434914086860950", + "correct_value": 20365466285010092434914086860950, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5281644763579870 × 5699237064456672", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 5281644763579870 × 5699237064456672?", + "options": [ + "30101345597887891705775226392640", + "30101345597887891705775226392641", + "30101345597887891705775226392639", + "30101345597887891705775226392660" + ], + "answer": "30101345597887891705775226392640", + "correct_value": 30101345597887891705775226392640, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8889817708746851 × 4861702720971151", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 8889817708746851 × 4861702720971151?", + "options": [ + "43219650943552088655801933095501", + "34913488002996478655801933095501", + "42213904505954208655801933095501", + "35198387935107068655801933095501" + ], + "answer": "43219650943552088655801933095501", + "correct_value": 43219650943552088655801933095501, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001561", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5963386418730146 × 1941811390038431", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 5963386418730146 × 1941811390038431?", + "options": [ + "1157977167109068574262115824089", + "11579771671090685742621158240926", + "2315954334218137148524231648182", + "1157977167109068574262115824092671" + ], + "answer": "11579771671090685742621158240926", + "correct_value": 11579771671090685742621158240926, + "explanation": "Product has 32 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7104144620250465 × 1801487161267680", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 7104144620250465 × 1801487161267680?", + "options": [ + "12798025325170070733728509471199", + "12798025325170070733728509471190", + "12798025325170070733728509471201", + "12798025325170070733728509471200" + ], + "answer": "12798025325170070733728509471200", + "correct_value": 12798025325170070733728509471200, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2438730543144625 × 5046821684534037", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 2438730543144625 × 5046821684534037?", + "options": [ + "13387671963083243341117826101125", + "14523299660449533341117826101125", + "12307838187877763341117826101125", + "14373529610625103341117826101125" + ], + "answer": "12307838187877763341117826101125", + "correct_value": 12307838187877763341117826101125, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001562", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7213731379752666 × 3087612137773096", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 7213731379752666 × 3087612137773096?", + "options": [ + "4454640913351798894893701814784", + "2227320456675899447446850907390", + "22273204566758994474468509073936", + "222732045667589944744685090739402" + ], + "answer": "22273204566758994474468509073936", + "correct_value": 22273204566758994474468509073936, + "explanation": "Product has 32 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8239672695843940 × 5633478364354926", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 8239672695843940 × 5633478364354926?", + "options": [ + "46418017861402862781759266248439", + "46418017861402862781759266248441", + "46418017861402862781759266248420", + "46418017861402862781759266248440" + ], + "answer": "46418017861402862781759266248440", + "correct_value": 46418017861402862781759266248440, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9153190170892871 × 8012294981653827", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 9153190170892871 × 8012294981653827?", + "options": [ + "82214400917405455471844824167317", + "72759159757192865471844824167317", + "73338059672368085471844824167317", + "81720260669730595471844824167317" + ], + "answer": "73338059672368085471844824167317", + "correct_value": 73338059672368085471844824167317, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001563", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6014195717486709 × 7956601730014575", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 6014195717486709 × 7956601730014575?", + "options": [ + "47852560050400996983988938783674", + "47852560050400996983988938783675", + "47852560050400996983988938783676", + "47852560050400996983988938783677" + ], + "answer": "47852560050400996983988938783675", + "correct_value": 47852560050400996983988938783675, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3050834015351900 × 2874530445678702", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 3050834015351900 × 2874530445678702?", + "options": [ + "8769715261841241086482865233800", + "8769715261841241086482865233799", + "8769715261841241086482865233810", + "8769715261841241086482865233801" + ], + "answer": "8769715261841241086482865233800", + "correct_value": 8769715261841241086482865233800, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5314545849152773 × 9870476216872057", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5314545849152773 × 9870476216872057?", + "options": [ + "43767791207246466556521587764061", + "62644484343349236556521587764061", + "46356535312419606556521587764061", + "52457098407538556556521587764061" + ], + "answer": "52457098407538556556521587764061", + "correct_value": 52457098407538556556521587764061, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_001564", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5591935194458535 × 7063918300464727", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 5591935194458535 × 7063918300464727?", + "options": [ + "39500973355148427244805631594944", + "39500973355148427244805631594948", + "39500973355148427244805631594945", + "39500973355148427244805631594946" + ], + "answer": "39500973355148427244805631594945", + "correct_value": 39500973355148427244805631594945, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2171165790781490 × 2005770742435570", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 2171165790781490 × 2005770742435570?", + "options": [ + "4354860820126500640656273599320", + "4354860820126500640656273599300", + "4354860820126500640656273599299", + "4354860820126500640656273599301" + ], + "answer": "4354860820126500640656273599300", + "correct_value": 4354860820126500640656273599300, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4904278846389561 × 5225199829061953", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4904278846389561 × 5225199829061953?", + "options": [ + "25200284730975606191955441472633", + "26648977144190646191955441472633", + "25625836989826886191955441472633", + "25178230381341876191955441472633" + ], + "answer": "25625836989826886191955441472633", + "correct_value": 25625836989826886191955441472633, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001565", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2409644947927275 × 9114446616307930", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 2409644947927275 × 9114446616307930?", + "options": [ + "21962580242139249806666645790750", + "21962580242139249806666645790753", + "21962580242139249806666645790751", + "21962580242139249806666645790749" + ], + "answer": "21962580242139249806666645790750", + "correct_value": 21962580242139249806666645790750, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8136276330904985 × 8324273751164176", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 8136276330904985 × 8324273751164176?", + "options": [ + "67728591493570738013451391817361", + "67728591493570738013451391817350", + "67728591493570738013451391817359", + "67728591493570738013451391817360" + ], + "answer": "67728591493570738013451391817360", + "correct_value": 67728591493570738013451391817360, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6621053849287665 × 3193913882299449", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6621053849287665 × 3193913882299449?", + "options": [ + "20960412925696817008980871996585", + "21147075804692077008980871996585", + "17281837588610317008980871996585", + "19847006240986717008980871996585" + ], + "answer": "21147075804692077008980871996585", + "correct_value": 21147075804692077008980871996585, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001566", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5963386418730146 × 1941811390038431", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 5963386418730146 × 1941811390038431?", + "options": [ + "1157977167109068574262115824089", + "11579771671090685742621158240926", + "2315954334218137148524231648182", + "1157977167109068574262115824092671" + ], + "answer": "11579771671090685742621158240926", + "correct_value": 11579771671090685742621158240926, + "explanation": "Product has 32 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7104144620250465 × 1801487161267680", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 7104144620250465 × 1801487161267680?", + "options": [ + "12798025325170070733728509471199", + "12798025325170070733728509471190", + "12798025325170070733728509471201", + "12798025325170070733728509471200" + ], + "answer": "12798025325170070733728509471200", + "correct_value": 12798025325170070733728509471200, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2438730543144625 × 5046821684534037", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 2438730543144625 × 5046821684534037?", + "options": [ + "13387671963083243341117826101125", + "14523299660449533341117826101125", + "12307838187877763341117826101125", + "14373529610625103341117826101125" + ], + "answer": "12307838187877763341117826101125", + "correct_value": 12307838187877763341117826101125, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001567", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9008249277766075 × 4719231914903010", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 9008249277766075 × 4719231914903010?", + "options": [ + "42512017489035650946833093385753", + "42512017489035650946833093385751", + "42512017489035650946833093385750", + "42512017489035650946833093385749" + ], + "answer": "42512017489035650946833093385750", + "correct_value": 42512017489035650946833093385750, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3961231072948790 × 6920300038169768", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 3961231072948790 × 6920300038169768?", + "options": [ + "27412907545326782485846390180720", + "27412907545326782485846390180740", + "27412907545326782485846390180721", + "27412907545326782485846390180719" + ], + "answer": "27412907545326782485846390180720", + "correct_value": 27412907545326782485846390180720, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7594940461319255 × 3569544830163475", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 7594940461319255 × 3569544830163475?", + "options": [ + "24189195360583884556615815211125", + "27110480459101544556615815211125", + "22829190544194544556615815211125", + "23865571289585134556615815211125" + ], + "answer": "27110480459101544556615815211125", + "correct_value": 27110480459101544556615815211125, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001568", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2947716861205530 × 4360790708480534", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 2947716861205530 × 4360790708480534?", + "options": [ + "12854376299576479076397778153021", + "12854376299576479076397778153023", + "12854376299576479076397778153019", + "12854376299576479076397778153020" + ], + "answer": "12854376299576479076397778153020", + "correct_value": 12854376299576479076397778153020, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9710334381904345 × 1448691498429008", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 9710334381904345 × 1448691498429008?", + "options": [ + "14067278865967720783270829239760", + "14067278865967720783270829239761", + "14067278865967720783270829239770", + "14067278865967720783270829239759" + ], + "answer": "14067278865967720783270829239760", + "correct_value": 14067278865967720783270829239760, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7589618085519567 × 2413382906301361", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 7589618085519567 × 2413382906301361?", + "options": [ + "18069646613584524022191964230687", + "18316654552948584022191964230687", + "21900940213488564022191964230687", + "17813527205003504022191964230687" + ], + "answer": "18316654552948584022191964230687", + "correct_value": 18316654552948584022191964230687, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001569", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7257074709127995 × 4309839363906321", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 7257074709127995 × 4309839363906321?", + "options": [ + "31276826248208847463718778556394", + "31276826248208847463718778556395", + "31276826248208847463718778556396", + "31276826248208847463718778556398" + ], + "answer": "31276826248208847463718778556395", + "correct_value": 31276826248208847463718778556395, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8392765216632970 × 4481587829716726", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 8392765216632970 × 4481587829716726?", + "options": [ + "37612914452532179754778612056219", + "37612914452532179754778612056230", + "37612914452532179754778612056220", + "37612914452532179754778612056221" + ], + "answer": "37612914452532179754778612056220", + "correct_value": 37612914452532179754778612056220, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4752218979054061 × 7428595475812331", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4752218979054061 × 7428595475812331?", + "options": [ + "39404170481385712120447939426191", + "35302312407870492120447939426191", + "41161341525070742120447939426191", + "36253695873953652120447939426191" + ], + "answer": "35302312407870492120447939426191", + "correct_value": 35302312407870492120447939426191, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001570", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1186576085242495 × 1380148338252802", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 1186576085242495 × 1380148338252802?", + "options": [ + "1637651012257944608724783220990", + "1637651012257944608724783220991", + "1637651012257944608724783220993", + "1637651012257944608724783220989" + ], + "answer": "1637651012257944608724783220990", + "correct_value": 1637651012257944608724783220990, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9125956543514820 × 3923959478246032", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 9125956543514820 × 3923959478246032?", + "options": [ + "35809883676986374712777998194239", + "35809883676986374712777998194241", + "35809883676986374712777998194240", + "35809883676986374712777998194230" + ], + "answer": "35809883676986374712777998194240", + "correct_value": 35809883676986374712777998194240, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5840568221869651 × 9272421890100167", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5840568221869651 × 9272421890100167?", + "options": [ + "54156212631087560856140407331717", + "48081020052301500856140407331717", + "58389275508354000856140407331717", + "49054825372375680856140407331717" + ], + "answer": "54156212631087560856140407331717", + "correct_value": 54156212631087560856140407331717, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001571", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1550089299659688 × 2382079852886920", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 1550089299659688 × 2382079852886920?", + "options": [ + "3692436490894938443060346480960", + "36924364908949384430603464809664", + "369243649089493844306034648096", + "36924364908949384430603464809648" + ], + "answer": "3692436490894938443060346480960", + "correct_value": 3692436490894938443060346480960, + "explanation": "Product has 31 digits; only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7217036772319915 × 6729809887789582", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 7217036772319915 × 6729809887789582?", + "options": [ + "48569285430899574222761508125510", + "48569285430899574222761508125529", + "48569285430899574222761508125530", + "48569285430899574222761508125531" + ], + "answer": "48569285430899574222761508125530", + "correct_value": 48569285430899574222761508125530, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5108123157119671 × 7965649255735447", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5108123157119671 × 7965649255735447?", + "options": [ + "40689517424715309098529295677937", + "43564598793290839098529295677937", + "37020156783056139098529295677937", + "43709917511051019098529295677937" + ], + "answer": "40689517424715309098529295677937", + "correct_value": 40689517424715309098529295677937, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001572", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7598032523781675 × 6423468101874976", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 7598032523781675 × 6423468101874976?", + "options": [ + "48805719553520209356377569864800", + "48805719553520209356377569864799", + "48805719553520209356377569864803", + "48805719553520209356377569864801" + ], + "answer": "48805719553520209356377569864800", + "correct_value": 48805719553520209356377569864800, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2093862161165225 × 8790662106873714", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 2093862161165225 × 8790662106873714?", + "options": [ + "18406434757171844896744163395650", + "18406434757171844896744163395649", + "18406434757171844896744163395640", + "18406434757171844896744163395651" + ], + "answer": "18406434757171844896744163395650", + "correct_value": 18406434757171844896744163395650, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2016239973104857 × 4904643861346173", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 2016239973104857 × 4904643861346173?", + "options": [ + "8383117178187791834542504662261", + "9445789168474064834542504662261", + "9888939007089509834542504662261", + "9133602747545082834542504662261" + ], + "answer": "9888939007089509834542504662261", + "correct_value": 9888939007089509834542504662261, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001573", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6734630132140245 × 9161805020842608", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 6734630132140245 × 9161805020842608?", + "options": [ + "61701368158160413211412327558960", + "61701368158160413211412327558959", + "61701368158160413211412327558963", + "61701368158160413211412327558961" + ], + "answer": "61701368158160413211412327558960", + "correct_value": 61701368158160413211412327558960, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6276742937807470 × 2637771402884234", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 6276742937807470 × 2637771402884234?", + "options": [ + "16556613024604118462842190427980", + "16556613024604118462842190427981", + "16556613024604118462842190427960", + "16556613024604118462842190427979" + ], + "answer": "16556613024604118462842190427980", + "correct_value": 16556613024604118462842190427980, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5018426210631055 × 1371214757075541", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5018426210631055 × 1371214757075541?", + "options": [ + "7911612293585113832855915525755", + "6881340077311989832855915525755", + "6115207432099197832855915525755", + "6051905532906997832855915525755" + ], + "answer": "6881340077311989832855915525755", + "correct_value": 6881340077311989832855915525755, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001574", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6536595329094101 × 2212744390773225", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 6536595329094101 × 2212744390773225?", + "options": [ + "14463814649207434693182176245724", + "14463814649207434693182176245726", + "14463814649207434693182176245725", + "14463814649207434693182176245728" + ], + "answer": "14463814649207434693182176245725", + "correct_value": 14463814649207434693182176245725, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1916934649007425 × 3573014363382228", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 1916934649007425 × 3573014363382228?", + "options": [ + "6849235034568599315666085042901", + "6849235034568599315666085042910", + "6849235034568599315666085042899", + "6849235034568599315666085042900" + ], + "answer": "6849235034568599315666085042900", + "correct_value": 6849235034568599315666085042900, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6602980250671767 × 6957653357974253", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6602980250671767 × 6957653357974253?", + "options": [ + "50550949703013174490829540015051", + "54663245437716554490829540015051", + "43143119127407944490829540015051", + "45941247713724094490829540015051" + ], + "answer": "45941247713724094490829540015051", + "correct_value": 45941247713724094490829540015051, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_001575", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4210958519718385 × 9675937841090658", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 4210958519718385 × 9675937841090658?", + "options": [ + "40744972888206223162387414347333", + "40744972888206223162387414347329", + "40744972888206223162387414347331", + "40744972888206223162387414347330" + ], + "answer": "40744972888206223162387414347330", + "correct_value": 40744972888206223162387414347330, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6862483568417665 × 8565433400238616", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 6862483568417665 × 8565433400238616?", + "options": [ + "58780145965513351320172549551639", + "58780145965513351320172549551640", + "58780145965513351320172549551650", + "58780145965513351320172549551641" + ], + "answer": "58780145965513351320172549551640", + "correct_value": 58780145965513351320172549551640, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4940459504155873 × 9166926327531731", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4940459504155873 × 9166926327531731?", + "options": [ + "40398223522224633587972777506163", + "45288828298750833587972777506163", + "53887091955096243587972777506163", + "53877558375831943587972777506163" + ], + "answer": "45288828298750833587972777506163", + "correct_value": 45288828298750833587972777506163, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001576", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7213731379752666 × 3087612137773096", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 7213731379752666 × 3087612137773096?", + "options": [ + "4454640913351798894893701814784", + "2227320456675899447446850907390", + "22273204566758994474468509073936", + "222732045667589944744685090739402" + ], + "answer": "22273204566758994474468509073936", + "correct_value": 22273204566758994474468509073936, + "explanation": "Product has 32 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8239672695843940 × 5633478364354926", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 8239672695843940 × 5633478364354926?", + "options": [ + "46418017861402862781759266248439", + "46418017861402862781759266248441", + "46418017861402862781759266248420", + "46418017861402862781759266248440" + ], + "answer": "46418017861402862781759266248440", + "correct_value": 46418017861402862781759266248440, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9153190170892871 × 8012294981653827", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 9153190170892871 × 8012294981653827?", + "options": [ + "82214400917405455471844824167317", + "72759159757192865471844824167317", + "73338059672368085471844824167317", + "81720260669730595471844824167317" + ], + "answer": "73338059672368085471844824167317", + "correct_value": 73338059672368085471844824167317, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001577", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7542925189441836 × 6889952272681644", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 7542925189441836 × 6889952272681644?", + "options": [ + "51970394551662398057883282858385", + "51970394551662398057883282858386", + "51970394551662398057883282858384", + "51970394551662398057883282858383" + ], + "answer": "51970394551662398057883282858384", + "correct_value": 51970394551662398057883282858384, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9210232229508215 × 5365136469613626", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 9210232229508215 × 5365136469613626?", + "options": [ + "49414152828145340193657042937590", + "49414152828145340193657042937589", + "49414152828145340193657042937591", + "49414152828145340193657042937600" + ], + "answer": "49414152828145340193657042937590", + "correct_value": 49414152828145340193657042937590, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3172711683938625 × 3534762873314635", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 3172711683938625 × 3534762873314635?", + "options": [ + "10077434871648968201345654276875", + "13440725944866368201345654276875", + "11539368232928098201345654276875", + "11214783468117808201345654276875" + ], + "answer": "11214783468117808201345654276875", + "correct_value": 11214783468117808201345654276875, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_001578", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6014195717486709 × 7956601730014575", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 6014195717486709 × 7956601730014575?", + "options": [ + "47852560050400996983988938783674", + "47852560050400996983988938783675", + "47852560050400996983988938783676", + "47852560050400996983988938783677" + ], + "answer": "47852560050400996983988938783675", + "correct_value": 47852560050400996983988938783675, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3050834015351900 × 2874530445678702", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 3050834015351900 × 2874530445678702?", + "options": [ + "8769715261841241086482865233800", + "8769715261841241086482865233799", + "8769715261841241086482865233810", + "8769715261841241086482865233801" + ], + "answer": "8769715261841241086482865233800", + "correct_value": 8769715261841241086482865233800, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5314545849152773 × 9870476216872057", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5314545849152773 × 9870476216872057?", + "options": [ + "43767791207246466556521587764061", + "62644484343349236556521587764061", + "46356535312419606556521587764061", + "52457098407538556556521587764061" + ], + "answer": "52457098407538556556521587764061", + "correct_value": 52457098407538556556521587764061, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_001579", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3818572167003824 × 9061807225483345", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 3818572167003824 × 9061807225483345?", + "options": [ + "3460316485398484668988886331128", + "346031648539848466898888633112850", + "346031648539848466898888633112819", + "34603164853984846689888863311280" + ], + "answer": "34603164853984846689888863311280", + "correct_value": 34603164853984846689888863311280, + "explanation": "Product has 32 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2673641747012725 × 4152329224454090", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 2673641747012725 × 4152329224454090?", + "options": [ + "11101840761841426698276408295250", + "11101840761841426698276408295249", + "11101840761841426698276408295270", + "11101840761841426698276408295251" + ], + "answer": "11101840761841426698276408295250", + "correct_value": 11101840761841426698276408295250, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4706119946682963 × 8530440955304553", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4706119946682963 × 8530440955304553?", + "options": [ + "40145278333760026924071801430539", + "37541484802808856924071801430539", + "33487148554433976924071801430539", + "47938211279771816924071801430539" + ], + "answer": "40145278333760026924071801430539", + "correct_value": 40145278333760026924071801430539, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001580", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4046203707448456 × 3085079634328149", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 4046203707448456 × 3085079634328149?", + "options": [ + "2496572130838456682148121477584", + "1248286065419228341074060738794491", + "12482860654192283410740607387944", + "1248286065419228341074060738794406" + ], + "answer": "12482860654192283410740607387944", + "correct_value": 12482860654192283410740607387944, + "explanation": "Product has 32 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4332547192037445 × 8680380667498968", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 4332547192037445 × 8680380667498968?", + "options": [ + "37608158886788776325392354856761", + "37608158886788776325392354856740", + "37608158886788776325392354856759", + "37608158886788776325392354856760" + ], + "answer": "37608158886788776325392354856760", + "correct_value": 37608158886788776325392354856760, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6370520203591549 × 4500299432176029", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6370520203591549 × 4500299432176029?", + "options": [ + "28495948632628028625618184778921", + "28669248454888968625618184778921", + "31215493599097218625618184778921", + "32552224609684938625618184778921" + ], + "answer": "28669248454888968625618184778921", + "correct_value": 28669248454888968625618184778921, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001581", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3513412892349729 × 6833798894328363", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 3513412892349729 × 6833798894328363?", + "options": [ + "24009957139058593898970360063626", + "24009957139058593898970360063628", + "24009957139058593898970360063627", + "24009957139058593898970360063629" + ], + "answer": "24009957139058593898970360063627", + "correct_value": 24009957139058593898970360063627, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4427988671788410 × 1107538359129044", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 4427988671788410 × 1107538359129044?", + "options": [ + "4904167307794530576781453580039", + "4904167307794530576781453580030", + "4904167307794530576781453580041", + "4904167307794530576781453580040" + ], + "answer": "4904167307794530576781453580040", + "correct_value": 4904167307794530576781453580040, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5647493434129129 × 9322004850093027", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5647493434129129 × 9322004850093027?", + "options": [ + "56440578684648225435972380483483", + "52645961183820265435972380483483", + "52656314680372005435972380483483", + "45308846106272905435972380483483" + ], + "answer": "52645961183820265435972380483483", + "correct_value": 52645961183820265435972380483483, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001582", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9592419845424676 × 8870522514249578", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 9592419845424676 × 8870522514249578?", + "options": [ + "170179552409948090618673727573448", + "178688530030445495149607413952120", + "153161597168953281556806354816104", + "85089776204974045309336863786728" + ], + "answer": "85089776204974045309336863786728", + "correct_value": 85089776204974045309336863786728, + "explanation": "Product has 32 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4486065545000635 × 4422630625996590", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 4486065545000635 × 4422630625996590?", + "options": [ + "19840210869547892056939057834650", + "19840210869547892056939057834660", + "19840210869547892056939057834649", + "19840210869547892056939057834651" + ], + "answer": "19840210869547892056939057834650", + "correct_value": 19840210869547892056939057834650, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1424388053159637 × 4010367837730863", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 1424388053159637 × 4010367837730863?", + "options": [ + "5712320036839486977091580776731", + "6180263020519376977091580776731", + "6176532293478200977091580776731", + "5480936340458304977091580776731" + ], + "answer": "5712320036839486977091580776731", + "correct_value": 5712320036839486977091580776731, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001583", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3158730931756257 × 7196143162153134", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 3158730931756257 × 7196143162153134?", + "options": [ + "22730679995639386563768196659438", + "22730679995639386563768196659440", + "22730679995639386563768196659437", + "22730679995639386563768196659439" + ], + "answer": "22730679995639386563768196659438", + "correct_value": 22730679995639386563768196659438, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3208924632123675 × 5539427187449428", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 3208924632123675 × 5539427187449428?", + "options": [ + "17775604349662039420918304007910", + "17775604349662039420918304007900", + "17775604349662039420918304007901", + "17775604349662039420918304007899" + ], + "answer": "17775604349662039420918304007900", + "correct_value": 17775604349662039420918304007900, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8063666274288853 × 7268169544787743", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 8063666274288853 × 7268169544787743?", + "options": [ + "60665662560300588295200155928779", + "65851364106591698295200155928779", + "69350865676344998295200155928779", + "58608093634118288295200155928779" + ], + "answer": "58608093634118288295200155928779", + "correct_value": 58608093634118288295200155928779, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_001584", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5654104903419940 × 8700877718348512", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 5654104903419940 × 8700877718348512?", + "options": [ + "49195675371371621350997610129283", + "49195675371371621350997610129281", + "49195675371371621350997610129280", + "49195675371371621350997610129279" + ], + "answer": "49195675371371621350997610129280", + "correct_value": 49195675371371621350997610129280, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4905119398787375 × 5265820313663736", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 4905119398787375 × 5265820313663736?", + "options": [ + "25829477371080611172221912133000", + "25829477371080611172221912132980", + "25829477371080611172221912132999", + "25829477371080611172221912133001" + ], + "answer": "25829477371080611172221912133000", + "correct_value": 25829477371080611172221912133000, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2692400653145773 × 2472176115730175", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 2692400653145773 × 2472176115730175?", + "options": [ + "6494516266841127270722609800275", + "5504667816994923270722609800275", + "7895237308481934270722609800275", + "6656088588683303270722609800275" + ], + "answer": "6656088588683303270722609800275", + "correct_value": 6656088588683303270722609800275, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_001585", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3081219179671170 × 6765911169612164", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 3081219179671170 × 6765911169612164?", + "options": [ + "20847255263760398308201952111880", + "20847255263760398308201952111883", + "20847255263760398308201952111879", + "20847255263760398308201952111881" + ], + "answer": "20847255263760398308201952111880", + "correct_value": 20847255263760398308201952111880, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6599370524333905 × 6041163997091612", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 6599370524333905 × 6041163997091612?", + "options": [ + "39867879615073580824893562704861", + "39867879615073580824893562704860", + "39867879615073580824893562704840", + "39867879615073580824893562704859" + ], + "answer": "39867879615073580824893562704860", + "correct_value": 39867879615073580824893562704860, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6857477010799669 × 8809265397860525", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6857477010799669 × 8809265397860525?", + "options": [ + "60409334947861549825471978166225", + "51112069759274839825471978166225", + "69054792109891869825471978166225", + "49460947518466639825471978166225" + ], + "answer": "60409334947861549825471978166225", + "correct_value": 60409334947861549825471978166225, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001586", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2093060111840670 × 8462091475368140", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 2093060111840670 × 8462091475368140?", + "options": [ + "17711666129840019314861274253800", + "17711666129840019314861274253803", + "17711666129840019314861274253801", + "17711666129840019314861274253799" + ], + "answer": "17711666129840019314861274253800", + "correct_value": 17711666129840019314861274253800, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5950628423508205 × 5575117212768094", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 5950628423508205 × 5575117212768094?", + "options": [ + "33175450950687661107050571211269", + "33175450950687661107050571211260", + "33175450950687661107050571211271", + "33175450950687661107050571211270" + ], + "answer": "33175450950687661107050571211270", + "correct_value": 33175450950687661107050571211270, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5351806069188143 × 2214994378330827", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5351806069188143 × 2214994378330827?", + "options": [ + "11854220357168537715711359784261", + "12965801481961057715711359784261", + "13758693394640697715711359784261", + "10919298566312827715711359784261" + ], + "answer": "11854220357168537715711359784261", + "correct_value": 11854220357168537715711359784261, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001587", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3472488829255530 × 1565961625968832", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 3472488829255530 × 1565961625968832?", + "options": [ + "5437784253219595596461543640960", + "5437784253219595596461543640963", + "5437784253219595596461543640959", + "5437784253219595596461543640961" + ], + "answer": "5437784253219595596461543640960", + "correct_value": 5437784253219595596461543640960, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8038867654527440 × 7368580173693902", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 8038867654527440 × 7368580173693902?", + "options": [ + "59235040818100094411659019670860", + "59235040818100094411659019670881", + "59235040818100094411659019670879", + "59235040818100094411659019670880" + ], + "answer": "59235040818100094411659019670880", + "correct_value": 59235040818100094411659019670880, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4491516580243833 × 9541993687427473", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4491516580243833 × 9541993687427473?", + "options": [ + "42858022855662485473788843024009", + "45965550121476975473788843024009", + "46259919548416575473788843024009", + "44729011486175795473788843024009" + ], + "answer": "42858022855662485473788843024009", + "correct_value": 42858022855662485473788843024009, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001588", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4940674088413356 × 2487942053436903", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 4940674088413356 × 2487942053436903?", + "options": [ + "12292110836889623770509928476467", + "12292110836889623770509928476468", + "12292110836889623770509928476469", + "12292110836889623770509928476470" + ], + "answer": "12292110836889623770509928476468", + "correct_value": 12292110836889623770509928476468, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2522750164650235 × 2987719761061568", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 2522750164650235 × 2987719761061568?", + "options": [ + "7537270519146831444931020668480", + "7537270519146831444931020668481", + "7537270519146831444931020668479", + "7537270519146831444931020668460" + ], + "answer": "7537270519146831444931020668480", + "correct_value": 7537270519146831444931020668480, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1010350976013247 × 9780388433150063", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 1010350976013247 × 9780388433150063?", + "options": [ + "9881624999221837712085426884561", + "8319912307394945712085426884561", + "9453748496225351712085426884561", + "8306060953259991712085426884561" + ], + "answer": "9881624999221837712085426884561", + "correct_value": 9881624999221837712085426884561, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001589", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6318317909183010 × 6478399989537806", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 6318317909183010 × 6478399989537806?", + "options": [ + "8186518135349548852890593575212", + "409325906767477442644529678760643", + "4093259067674774426445296787606", + "40932590676747744264452967876060" + ], + "answer": "40932590676747744264452967876060", + "correct_value": 40932590676747744264452967876060, + "explanation": "Product has 32 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6050979803273290 × 3260366717753688", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 6050979803273290 × 3260366717753688?", + "options": [ + "19728413160391993237058369393540", + "19728413160391993237058369393521", + "19728413160391993237058369393520", + "19728413160391993237058369393519" + ], + "answer": "19728413160391993237058369393520", + "correct_value": 19728413160391993237058369393520, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1577805754726375 × 1952636988418475", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 1577805754726375 × 1952636988418475?", + "options": [ + "3541789761293581907367619778125", + "3501883157567717907367619778125", + "3201716167643728907367619778125", + "3080881877218247907367619778125" + ], + "answer": "3080881877218247907367619778125", + "correct_value": 3080881877218247907367619778125, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_001590", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4867467700520370 × 8481282108163580", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 4867467700520370 × 8481282108163580?", + "options": [ + "41282366720487536736991082124599", + "41282366720487536736991082124601", + "41282366720487536736991082124603", + "41282366720487536736991082124600" + ], + "answer": "41282366720487536736991082124600", + "correct_value": 41282366720487536736991082124600, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2291961403253545 × 6027333872420382", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 2291961403253545 × 6027333872420382?", + "options": [ + "13814416600110242101198771754180", + "13814416600110242101198771754190", + "13814416600110242101198771754189", + "13814416600110242101198771754191" + ], + "answer": "13814416600110242101198771754190", + "correct_value": 13814416600110242101198771754190, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7230482141277227 × 5832457397683871", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 7230482141277227 × 5832457397683871?", + "options": [ + "45071839796679328696235517505717", + "44854116159560578696235517505717", + "42171479053713478696235517505717", + "48710698922281768696235517505717" + ], + "answer": "42171479053713478696235517505717", + "correct_value": 42171479053713478696235517505717, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001591", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4163832105432513 × 5881874901286401", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 4163832105432513 × 5881874901286401?", + "options": [ + "24491139554114009642884190155714", + "24491139554114009642884190155713", + "24491139554114009642884190155712", + "24491139554114009642884190155716" + ], + "answer": "24491139554114009642884190155713", + "correct_value": 24491139554114009642884190155713, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5786743253304555 × 2385886041704744", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 5786743253304555 × 2385886041704744?", + "options": [ + "13806509954988437483523620308940", + "13806509954988437483523620308921", + "13806509954988437483523620308920", + "13806509954988437483523620308919" + ], + "answer": "13806509954988437483523620308920", + "correct_value": 13806509954988437483523620308920, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1752368282729443 × 2377934162326559", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 1752368282729443 × 2377934162326559?", + "options": [ + "4390889937035171746971610176637", + "4558470843516006746971610176637", + "4167016404479868746971610176637", + "4434979362846124746971610176637" + ], + "answer": "4167016404479868746971610176637", + "correct_value": 4167016404479868746971610176637, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001592", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7288523537004620 × 9450332716510620", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 7288523537004620 × 9450332716510620?", + "options": [ + "6887897243681246291761321906440", + "137757944873624925835226438128800", + "144645842117306172126987760035240", + "68878972436812462917613219064400" + ], + "answer": "68878972436812462917613219064400", + "correct_value": 68878972436812462917613219064400, + "explanation": "Product has 32 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3569997895960310 × 1497389614887062", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 3569997895960310 × 1497389614887062?", + "options": [ + "5345677774579630223806684509221", + "5345677774579630223806684509220", + "5345677774579630223806684509210", + "5345677774579630223806684509219" + ], + "answer": "5345677774579630223806684509220", + "correct_value": 5345677774579630223806684509220, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8651630699585929 × 9862165857536867", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 8651630699585929 × 9862165857536867?", + "options": [ + "94840481676710358040220751944443", + "85323816897474148040220751944443", + "71171274968004378040220751944443", + "77600304805015808040220751944443" + ], + "answer": "85323816897474148040220751944443", + "correct_value": 85323816897474148040220751944443, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001593", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9819596545092505 × 8066982446959455", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 9819596545092505 × 8066982446959455?", + "options": [ + "79214512965484946284338959384778", + "79214512965484946284338959384774", + "79214512965484946284338959384776", + "79214512965484946284338959384775" + ], + "answer": "79214512965484946284338959384775", + "correct_value": 79214512965484946284338959384775, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1693037065815600 × 3585193583416618", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 1693037065815600 × 3585193583416618?", + "options": [ + "6069865624848587497580763640800", + "6069865624848587497580763640780", + "6069865624848587497580763640801", + "6069865624848587497580763640799" + ], + "answer": "6069865624848587497580763640800", + "correct_value": 6069865624848587497580763640800, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3828154302928653 × 2028008895077469", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 3828154302928653 × 2028008895077469?", + "options": [ + "7763530978068396120062014819257", + "7972882821006243120062014819257", + "7648408341006217120062014819257", + "8320836777979039120062014819257" + ], + "answer": "7763530978068396120062014819257", + "correct_value": 7763530978068396120062014819257, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001594", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2010477542044524 × 4542838048666181", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 2010477542044524 × 4542838048666181?", + "options": [ + "10137934000127485057121365697552", + "18266547747977450553371830085684", + "19179875135376323081040421589968", + "9133273873988725276685915042844" + ], + "answer": "9133273873988725276685915042844", + "correct_value": 9133273873988725276685915042844, + "explanation": "Product has 31 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5787539708367700 × 1635481249144116", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 5787539708367700 × 1635481249144116?", + "options": [ + "9465412671712378819868419453199", + "9465412671712378819868419453201", + "9465412671712378819868419453200", + "9465412671712378819868419453210" + ], + "answer": "9465412671712378819868419453200", + "correct_value": 9465412671712378819868419453200, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3104034420574535 × 6003579712097427", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 3104034420574535 × 6003579712097427?", + "options": [ + "16922672258666820471327235221445", + "18635318073013370471327235221445", + "16358528682367520471327235221445", + "15226349316690090471327235221445" + ], + "answer": "18635318073013370471327235221445", + "correct_value": 18635318073013370471327235221445, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001595", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8786150048903330 × 7943979796163156", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 8786150048903330 × 7943979796163156?", + "options": [ + "69796998474545978574499551709479", + "69796998474545978574499551709480", + "69796998474545978574499551709481", + "69796998474545978574499551709483" + ], + "answer": "69796998474545978574499551709480", + "correct_value": 69796998474545978574499551709480, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3983807888961460 × 1115986206578610", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 3983807888961460 × 1115986206578610?", + "options": [ + "4445874653740040108252750370610", + "4445874653740040108252750370601", + "4445874653740040108252750370599", + "4445874653740040108252750370600" + ], + "answer": "4445874653740040108252750370600", + "correct_value": 4445874653740040108252750370600, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3283180606240433 × 3181297353322457", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 3283180606240433 × 3181297353322457?", + "options": [ + "11247361891508039353220320303881", + "10444773773112309353220320303881", + "10315012892916029353220320303881", + "11651367799884419353220320303881" + ], + "answer": "10444773773112309353220320303881", + "correct_value": 10444773773112309353220320303881, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001596", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5190766652348470 × 7440968699690902", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 5190766652348470 × 7440968699690902?", + "options": [ + "38624332187524291172181392619940", + "38624332187524291172181392619943", + "38624332187524291172181392619941", + "38624332187524291172181392619939" + ], + "answer": "38624332187524291172181392619940", + "correct_value": 38624332187524291172181392619940, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5062586882880255 × 1439710909431754", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 5062586882880255 × 1439710909431754?", + "options": [ + "7288661565228800601232876617269", + "7288661565228800601232876617271", + "7288661565228800601232876617280", + "7288661565228800601232876617270" + ], + "answer": "7288661565228800601232876617270", + "correct_value": 7288661565228800601232876617270, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2125794470535237 × 6763442068799137", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 2125794470535237 × 6763442068799137?", + "options": [ + "13505177930659719417950233690469", + "11910741032003219417950233690469", + "14377687751638609417950233690469", + "16745019515681829417950233690469" + ], + "answer": "14377687751638609417950233690469", + "correct_value": 14377687751638609417950233690469, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001597", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5709251040734865 × 1567481107898032", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 5709251040734865 × 1567481107898032?", + "options": [ + "17898286293198156828617534571360", + "894914314659907841430876728568", + "8949143146599078414308767285680", + "19688114922517972511479288028496" + ], + "answer": "8949143146599078414308767285680", + "correct_value": 8949143146599078414308767285680, + "explanation": "Product has 31 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5641156343740710 × 4646356529142848", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 5641156343740710 × 4646356529142848?", + "options": [ + "26210823609655244092986262942100", + "26210823609655244092986262942081", + "26210823609655244092986262942080", + "26210823609655244092986262942079" + ], + "answer": "26210823609655244092986262942080", + "correct_value": 26210823609655244092986262942080, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4676073520580933 × 6726916458916965", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4676073520580933 × 6726916458916965?", + "options": [ + "36400464051454975674494809228345", + "31455555928701675674494809228345", + "27238457052799345674494809228345", + "31085427057357575674494809228345" + ], + "answer": "31455555928701675674494809228345", + "correct_value": 31455555928701675674494809228345, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001598", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8143624831666060 × 9930551964356277", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 8143624831666060 × 9930551964356277?", + "options": [ + "177915517051980285048445323488964", + "80870689569081947749293328858620", + "145567241224347505948727991945516", + "8087068956908194774929332885862" + ], + "answer": "80870689569081947749293328858620", + "correct_value": 80870689569081947749293328858620, + "explanation": "Product has 32 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8868286539897005 × 2371388150621778", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 8868286539897005 × 2371388150621778?", + "options": [ + "21030149617030365345694829974889", + "21030149617030365345694829974891", + "21030149617030365345694829974870", + "21030149617030365345694829974890" + ], + "answer": "21030149617030365345694829974890", + "correct_value": 21030149617030365345694829974890, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6804183951133473 × 3655748730770145", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6804183951133473 × 3655748730770145?", + "options": [ + "27242142628002574229284978563585", + "23020721595821604229284978563585", + "24874386843282784229284978563585", + "26246684950080544229284978563585" + ], + "answer": "24874386843282784229284978563585", + "correct_value": 24874386843282784229284978563585, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001599", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3469029693142585 × 6143139281443207", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 3469029693142585 × 6143139281443207?", + "options": [ + "21310732576437088490590030670098", + "21310732576437088490590030670094", + "21310732576437088490590030670096", + "21310732576437088490590030670095" + ], + "answer": "21310732576437088490590030670095", + "correct_value": 21310732576437088490590030670095, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8230061953763220 × 9646648567167258", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 8230061953763220 × 9646648567167258?", + "options": [ + "79392515353967730172568268650759", + "79392515353967730172568268650750", + "79392515353967730172568268650760", + "79392515353967730172568268650761" + ], + "answer": "79392515353967730172568268650760", + "correct_value": 79392515353967730172568268650760, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4999642993506255 × 4334160961894347", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4999642993506255 × 4334160961894347?", + "options": [ + "21467379488442782642624393640485", + "21669257485863402642624393640485", + "22333236401719532642624393640485", + "22781047019545072642624393640485" + ], + "answer": "21669257485863402642624393640485", + "correct_value": 21669257485863402642624393640485, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001600", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2319400042090665 × 1754247088243002", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 2319400042090665 × 1754247088243002?", + "options": [ + "4068800770308245357261635776330", + "4068800770308245357261635776329", + "4068800770308245357261635776331", + "4068800770308245357261635776332" + ], + "answer": "4068800770308245357261635776330", + "correct_value": 4068800770308245357261635776330, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6774265252377280 × 8616372167914848", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 6774265252377280 × 8616372167914848?", + "options": [ + "58369590578656248992772609853450", + "58369590578656248992772609853439", + "58369590578656248992772609853440", + "58369590578656248992772609853441" + ], + "answer": "58369590578656248992772609853440", + "correct_value": 58369590578656248992772609853440, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3046691762033043 × 6916284114945829", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 3046691762033043 × 6916284114945829?", + "options": [ + "23671041938071683066570853027647", + "22977119835209293066570853027647", + "21071785836885453066570853027647", + "19789342930086383066570853027647" + ], + "answer": "21071785836885453066570853027647", + "correct_value": 21071785836885453066570853027647, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_001601", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2843786192974500 × 6848318830608338", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 2843786192974500 × 6848318830608338?", + "options": [ + "194751545355712652648967213810018", + "19475154535571265264896721381000", + "1947515453557126526489672138100", + "1947515453557126526489672138100012" + ], + "answer": "19475154535571265264896721381000", + "correct_value": 19475154535571265264896721381000, + "explanation": "Product has 32 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5167599468238550 × 6802723170459728", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 5167599468238550 × 6802723170459728?", + "options": [ + "35153748638241753340537872114400", + "35153748638241753340537872114401", + "35153748638241753340537872114420", + "35153748638241753340537872114399" + ], + "answer": "35153748638241753340537872114400", + "correct_value": 35153748638241753340537872114400, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5317971905409333 × 8283401077171239", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5317971905409333 × 8283401077171239?", + "options": [ + "49083108893223245289162029773587", + "39473099415937845289162029773587", + "44050894209634055289162029773587", + "44854781254400605289162029773587" + ], + "answer": "44050894209634055289162029773587", + "correct_value": 44050894209634055289162029773587, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + } +] \ No newline at end of file