REVIEW 4 major objections 6 minor 2 cited by
A Fragile Number Sense: Probing the Elemental Limits of Numerical Reasoning in LLMs
T0 review · 4 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash
Pith's one-line read LLMs ace arithmetic but fall off a cliff on the Game of 24
desk verdict The 24-game failure is real and cleanly reproduced, but the paper overreaches when it turns a single-pass, no-search-control protocol into evidence that LLMs lack 'number sense.' read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing instrument is the four-tier 'divide-and-reconstruct' test, culminating in the Game of 24: given four numbers, use each exactly once with additions, subtractions, multiplications, and divisions to obtain 24. The game demands heuristic search because the space of possible operation trees is large and no deterministic algorithm tells the solver when to stop trying; success requires recognizing number relationships such as factor pairs of 24 to prune the search. The earlier sets are designed to isolate the component skills—basic operations, advanced operations such as exponentiation and logarithms, and deterministic trial division for primality—so that the 24-game score can be read as a measure of skill integration under search pressure. The framework's logic is that if the elementary components are intact but integrated search fails, the deficit is in generative problem-solving rather than in arithmetic knowledge.
What would settle it
Run the identical 25 puzzles from the paper's main study with a human baseline, and give the same LLM a version equipped with explicit backtracking search; if the search-augmented model solves near 100% while the baseline solves 73% or less, the bottleneck is search scaffolding rather than number sense.
Extended reading notes
Core claim
The central claim is that state-of-the-art LLM agents do not possess a human-like number sense; their numerical competence is mostly pattern-matched execution of known algorithms. The evidence is the steep performance cliff: on sets requiring straightforward calculation and deterministic factor search, all five main-study models remained strong, while on the Game of 24—which uses only the same four basic operations—the top model, ChatGPT o1, solved 73% and every other model fell below 50%. In a follow-up with newer reasoning models and explicit instructions not to use code, the same pattern appeared: strong on easier puzzles, sharp decline on a harder set, with the best model dropping from 86% to 65%. Error analysis shows the failure is not simply arithmetic: models often declared unsolvable puzzles that were solvable, broke the game rules by omitting or reusing numbers, and miscalculated simple expressions inside longer chains. The authors conclude that reasoning models' advantage over ordinary LLMs comes mostly from added computation, not from an improved number sense.
Load-bearing premise
The Game of 24 is a valid measure of number sense, so the performance gap reflects a missing human-like numerical intuition rather than puzzle difficulty, missing search scaffolding, or unmeasured tool use.
Editorial extensions
If this is right
- High LLM scores on arithmetic benchmarks can coexist with near-total failure on structurally simple but non-algorithmic tasks, so aggregate benchmark scores overstate reasoning ability.
- Reasoning models that 'think longer' buy accuracy with computation but do not escape the heuristic-search bottleneck; the follow-up shows even the newest reasoning models fall on harder 24 puzzles.
- Errors compound inside long reasoning chains: basic operations that models pass in isolation fail in the middle of multi-step search, so chain-of-thought traces are not reliable evidence of underlying competence.
- The same elementary-skill decomposition can be applied to other combinatorial reasoning tasks to locate where model reasoning breaks, not just whether it succeeds.
Reading between the lines
- A direct test the paper does not run: feed the same 24 games to a solver that combines the same LLM with an explicit backtracking search (e.g., Tree-of-Thought style). If solve rates climb toward perfect, the bottleneck is search scaffolding rather than number sense, and the paper's conclusion would need to be narrowed.
- The paper treats 'number sense' as a unified construct, but the data may be read as a difficulty effect: performance falls as search depth and branching factor increase. That predicts a smooth gradient on intermediate puzzles, not a sharp cliff—an observable pattern the authors do not report.
- Because the main study did not verify that models abstained from external tools and did not include a human baseline, a replication could compare humans on the identical 25 puzzles; if humans also miss several, part of the gap is task difficulty rather than missing number sense.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This paper reports two empirical studies of numerical reasoning in commercial LLM agents. In the main study, five agents (ChatGPT o1, o1-mini, Gemini 1.5, Claude Sonnet 3.7, and Microsoft Copilot) are tested on a 100-problem battery: 25 basic arithmetic items, 25 advanced operation items, 25 primality checks, and 25 Game-of-24 items. Accuracy is high on the first three sets and drops sharply on Game of 24, with o1 at 73% and the other agents below 50%. A follow-up study tests three reasoning models (ChatGPT o3, Gemini 2.5 Pro, DeepSeek R1) on an easier and a harder set of 24 games, again finding lower accuracy on the harder set and a qualitative error analysis. The authors interpret the Set 4 collapse as evidence that LLM numerical proficiency is largely confined to algorithmic execution and that LLMs lack a flexible number sense and heuristic search ability; they also propose a divide-and-reconstruct framework for stepwise evaluation.
Significance. The paper's strengths are its transparency and its fine-grained error analysis: the full problem sets are printed in the appendices, raw data are posted on OSF, the scoring rule is explicit (two-decimal tolerance), and the error taxonomy distinguishes false 'no solution' claims, rule violations, and arithmetic miscalculations within long chains. If the central inference is supported, the paper offers a useful behavioral probe of LLM numerical reasoning and reinforces prior evidence that single-pass generation is fragile on combinatorial search tasks. The main quantitative observation, a large and consistent drop on Game of 24 across two studies, is likely real. What is not yet established is the paper's stronger claim that the drop reveals an absent generative problem-solving capacity; the current protocol lacks the controls needed to rule out alternative explanations.
major comments (4)
- [Section 5.2, Fig. 2d, and Abstract] The central inference from the Set 4 collapse to the conclusion that the agents' proficiency is largely confined to recalling and executing known algorithms, rather than performing generative problem-solving, requires the single-pass protocol to be a valid measure of generative problem-solving. The protocol gives each model one free-form response with no opportunity for explicit backtracking, verification, or search, and the paper includes no search-augmented control such as Tree of Thoughts (ToT). This is not a minor omission: the authors cite Yao et al. (2023), whose ToT method raised GPT-4's Game-of-24 accuracy from roughly 4% under chain-of-thought to 74% with explicit tree search, demonstrating that the base model can propose and evaluate partial expressions even when single-pass generation fails. The observed collapse is therefore compatible with the weaker claim that LLMs cannot maintain a search tree in free-form generation; it does not by itself establish that they lack number sense or generative numerical problem-solving. The Limitations section concedes that causal attribution requires tracing paths from input to output; adding a ToT control or explicitly weakening the Abstract's conclusion is necessary before the headline claim can be accepted.
- [Section 4.3 and Fig. 2] The main study did not control for external tool use. Only the follow-up study instructed models not to write code and required disabling coding abilities for the ChatGPT model; no equivalent control or monitoring is reported for Study 1. If any Study 1 agent used a code interpreter or other tool for primality checks or the 24 game, the high Set 3 scores and the 73% o1 score on Set 4 may partially reflect automated search rather than the model's own reasoning. At minimum, the paper should state whether tool use occurred in Study 1, report the relevant platform settings, and analyze the Set 4 result with tool access held constant; otherwise the contrast between deterministic and search-like tasks is confounded by a procedural difference.
- [Section 5.3 and Section 7] The paper calls the 24 games relatively easy for humans and concludes that there is a significant gap between human and LLM performance, but no human baseline is reported on the same 25 items. In addition, the model scores are averaged over three trials without error bars, confidence intervals, or significance tests; with 25 items per set, differences of a few percentage points (e.g., the 58-63% range among Gemini, Claude, and Copilot in Fig. 1) are within sampling noise. A human baseline on the same problems and a report of per-model variability would be needed to support the human-comparison claim and the finer cross-model comparisons.
- [Section 4.2] The main study deliberately excluded three models that scored below 80% on basic arithmetic. As a result, the observation in Section 5.2 that all tested agents possess elementary mathematical skills is true by construction, and the paper does not report how the excluded models performed on Set 4. The headline claim about LLM numerical reasoning would be more robust if the excluded models' Set 4 results were reported as a sensitivity check, or if the scope of the conclusion were explicitly restricted to models above the competence threshold.
minor comments (6)
- [Appendix A, Prompt 3] Items 22 and 23 are identical (both are log_5(1/625)); Appendix B, Prompt 2 contains duplicates (2,3,9,12) at items 4 and 24 and (2,3,8,13) at items 10 and 12. The authors should state whether duplicates are intentional and ensure the scoring accounts for them.
- [Section 5.3] The claim that every Game-of-24 item has at least one solution should be supported by a solution key or a machine-generated verification; no solutions are provided in the appendices.
- [Figure 3] The annotation scheme in Figure 3 is not defined; a legend or caption should explain what problem features are marked and how the annotations were determined.
- [Section 1] The statement that the probability of a correct final answer drops exponentially with the number of reasoning steps is asserted without derivation or citation; as a load-bearing motivational claim it should be supported or explicitly labeled as an assumption.
- [Table 1] Microsoft Copilot is listed without a version identifier; because commercial model behavior changes over time, the exact version and access date are needed for reproducibility.
- [Section 4.3] The procedure says the 24-game rules were verified before Prompt 6, but no verification criterion is described; specify what constituted a correct statement of the rules.
Circularity Check
No circularity: the paper is an empirical benchmark whose central interpretation imports an external cognitive-science construct rather than deriving the conclusion from its own fitted inputs.
full rationale
The paper contains no fitted parameters, no equations that are reused as predictions, and no self-citations. Its central chain is: (i) define a 100-problem battery, (ii) measure accuracy, (iii) observe a drop on the Game of 24, and (iv) attribute the drop to a lack of 'number sense' and heuristic search. Steps (i)-(iii) are empirical and self-contained; step (iv) is an interpretive claim that imports the cognitive-science construct of number sense from Dehaene (2001) rather than defining success as the target conclusion. The paper even concedes in Limitations and Future Directions that causal attribution would require tracing input-to-output paths, which further shows the conclusion is an interpretation, not a construction. The absence of a Tree-of-Thoughts control and the possible use of external tools are threats to external validity, not circularity: the observed performance gap is real and not an artifact of the analysis reducing to its inputs. The comparison with Yao et al. (2023) is an external benchmark, not a self-citation chain, and it strengthens rather than circularizes the finding.
Assumptions & free parameters
assumptions (4)
- domain assumption The Game of 24 is a valid operationalization of 'number sense' and heuristic search.
- domain assumption Success on Sets 1 and 3 implies the elementary numerical skills needed for Set 4 are present, so failure on Set 4 is attributable to search, not missing operations.
- domain assumption Every problem in the 24-game sets is solvable.
- ad hoc to paper Token-by-token generation implies the probability of a correct final answer drops exponentially with the number of reasoning steps.
Cite this review
Pith. "Pith review of A Fragile Number Sense: Probing the Elemental Limits of Numerical Reasoning in LLMs." pith.science (2026). https://pith.science/paper/4ZZE2XKS
@misc{pith2026250906332,
author = {Pith},
title = {Pith review of: A Fragile Number Sense: Probing the Elemental Limits of Numerical Reasoning in LLMs},
year = {2026},
howpublished = {\url{https://pith.science/paper/4ZZE2XKS}},
note = {Machine review of arXiv:2509.06332}
}
read the original abstract
Large Language Models (LLMs) have demonstrated remarkable emergent capabilities, yet the robustness of their numerical reasoning remains an open question. While standard benchmarks evaluate LLM reasoning on complex problem sets using aggregated metrics, they often obscure foundational weaknesses. In this work, we probe LLM mathematical numeracy by evaluating performance on problems of escalating complexity, from constituent operations to combinatorial puzzles. We test several state-of-the-art LLM-based agents on a 100-problem challenge comprising four categories: (1) basic arithmetic, (2) advanced operations, (3) primality checking, and (4) the Game of 24 number puzzle. Our results show that while the agents achieved high accuracy on the first three categories, which require deterministic algorithmic execution, they consistently failed at the number puzzle, underlining its demand for a heuristic search over a large combinatorial space to be a significant bottleneck. These findings reveal that the agents' proficiency is largely confined to recalling and executing known algorithms, rather than performing generative problem-solving. This suggests their apparent numerical reasoning is more akin to sophisticated pattern-matching than flexible, analytical thought, limiting their potential for tasks that require novel or creative numerical insights.
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Forward citations
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Reviewed August 15, 2026 · model on record in the stance chip above.
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