ArithMark

A procedurally generated benchmark for evaluating arithmetic reasoning in language models. Each problem presents a numeric expression and asks the model to identify the correct result.

Unlike knowledge-based benchmarks, ArithMark contains no facts a model could have memorised from pretraining. Every problem is generated fresh from random integers and operator sequences, so a model cannot pattern-match to training data — it must actually compute. This makes ArithMark a direct probe of numerical reasoning capability: the arithmetic structure that has been built into the model's weights through training, independent of world knowledge or surface-level heuristics.

Evaluation is log-likelihood multiple-choice, no chain-of-thought, no prompting tricks. Models are scored purely on how well they assign probability to the correct completion.

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Benchmark Results

Evaluated using average log-likelihood over ending tokens, normalised by length. Random chance = 25%.

Company	Model	Params	Stage 1	Stage 2	Stage 3	Stage 4	Stage 5	Avg
Alibaba	Qwen2.5-3B	3.1B	97.30%	75.60%	63.90%	55.40%	53.50%	69.14%
Alibaba	Qwen2.5-Math-1.5B	1.5B	97.70%	77.90%	63.20%	51.00%	44.10%	66.78%
Alibaba	Qwen2.5-1.5B	1.5B	94.70%	69.10%	58.90%	49.90%	46.80%	63.88%
Alibaba	Qwen2.5-Coder-1.5B	1.5B	93.00%	65.20%	57.60%	51.30%	49.20%	63.26%
Alibaba	Qwen2.5-0.5B	494M	85.50%	57.60%	48.20%	40.10%	38.80%	54.04%
HuggingFace	SmolLM2-1.7B	1.7B	82.70%	54.50%	44.90%	35.20%	33.30%	50.12%
EleutherAI	pythia-2.8b	2.8B	52.40%	42.20%	33.30%	28.90%	27.20%	36.80%
OpenAI	gpt2-xl	1.6B	39.90%	38.30%	36.10%	32.80%	33.50%	36.12%
OpenAI	gpt2-medium	345M	37.70%	37.20%	33.60%	30.90%	32.80%	34.44%
Axiomic Labs	GPT-X2-125M (unreleased)	125M	39.10%	39.20%	34.80%	28.20%	29.50%	34.16%
HuggingFace	SmolLM2-135M	135M	41.20%	37.40%	32.80%	28.70%	25.90%	33.20%
HuggingFace	SmolLM-135M	135M	42.70%	36.50%	32.60%	25.30%	23.90%	32.20%
OpenAI	gpt2	124M	35.70%	32.80%	31.90%	28.00%	29.60%	31.60%
Meta	MobileLLM-125M	125M	35.90%	35.00%	32.90%	27.40%	24.60%	31.16%
Axiomic Labs	GPT-X-125M	125M	38.10%	33.20%	29.00%	26.90%	25.40%	30.52%
EleutherAI	pythia-31m	30M	36.40%	31.00%	29.30%	27.80%	26.60%	30.22%
EleutherAI	pythia-160m	162M	35.80%	30.30%	28.00%	28.20%	27.00%	29.86%
EleutherAI	pythia-70m	70M	36.30%	30.10%	28.50%	27.30%	26.90%	29.82%
EleutherAI	pythia-14m	14M	34.50%	29.00%	26.30%	24.40%	24.00%	27.64%

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Task Format

Each problem is presented as a numeric expression followed by four answer choices — a mix of integers and floats so the answer type is never a giveaway:

-13 * 3 + -1 =
A. -40
B. -4.2317
C. 26
D. -33.8851

Evaluation is log-likelihood multiple-choice: the model scores each continuation and the highest wins. No generation, no chain-of-thought.

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Difficulty Stages

Problems are organized into 5 stages of increasing complexity by operand count. All stages draw from {+, -, *, /}.

Stage	Operands	Operators	Example
1	2	1	5 * -3 =
2	3	2	7 - 2 * 4 =
3	4	3	-6 + 10 / 2 - 1 =
4	5	4	3 * -2 + 8 - 1 / 4 =
5	6	5	-4 + 2 * 7 - 3 / 1 + 6 =

1,000 problems per stage, 5,000 total. Division by zero is rejected and regenerated.

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Dataset Format

{
  "id": "math_s3_00000",
  "stage": 3,
  "n_operands": 4,
  "expression": "-13 * 3 + -1",
  "options": [-40, -4.2317, 26, -33.8851],
  "answer_index": 0,
  "answer": -40
}

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Design Decisions

Left-to-right evaluation — expressions are evaluated left-to-right (no operator precedence). Division by zero is rejected and regenerated.

Integer-safe results — float results are rounded to 4 decimal places; exact integer floats are cast to int.

Operand range — integers sampled from [-20, 20] across all stages.

Distractor generation — each wrong answer is generated by adding a random float offset (±0.5–15.0) to the correct answer. Integer answers get 2 integer distractors and 1 float; float answers get 2 float distractors and 1 integer. This ensures the answer type is never uniquely identifiable.

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