{"id":"84a1ed8f-8256-4933-8443-9719fe149564","arxiv_id":"2509.06332","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"LLMs solve routine arithmetic and primality checks well but consistently fail at the Game of 24, which requires trial-and-error search.","lead":"This paper tests five leading AI chatbots on arithmetic, prime checking, and a number puzzle called the 24 Game. The models handle straightforward math well but often fail at the puzzle, which the authors say shows AI lacks the flexible number intuition that humans use.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 24-game performance gap is real, but the paper has not shown it is caused by a missing 'number sense' rather than by the lack of a tree-search scaffold: the protocol needs a Tree-of-Thoughts control before the Abstract's conclusion about generative problem-solving is supported.","rationale":"I agree with the reader's identification of measurement validity as the weak point, and I single out the omitted search-augmented control because it is the most direct test of the Abstract's generative-problem-solving claim. The paper has real strengths: a clear divide-and-reconstruct design, published prompts and data, and a useful decomposition of error types. But those strengths do not close the gap between a performance drop and a capacity conclusion. The absence of a ToT arm is especially telling because the authors themselves cite Tree of Thoughts' success on Game of 24 but do not include it. The reader's verdict of CONDITIONAL is appropriate; the condition should specifically be the ToT control (or an equivalent search baseline). Adding a human baseline and variance or error reporting would strengthen the paper further, but the ToT control is the load-bearing requirement.","tokens_in":12190,"tokens_out":11988,"duration_ms":108856,"concrete_test":"Run a controlled comparison on the exact 25 puzzles in Appendix A, Prompt 6. For each of o1, o1-mini, and Gemini 2.5 Pro, record accuracy with the paper's direct-answer protocol and with a Tree-of-Thoughts controller that uses the same model to propose candidate expressions and to evaluate or branch on partial results, with no external calculator or solver. If ToT accuracy is markedly higher than the reported Set 4 scores (e.g., o1 from 73% to over 90%), the low unaided scores should be attributed to missing search scaffolding rather than absent number sense. If ToT fails to improve accuracy, the authors' interpretation survives this test.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 5.2's Figure 2d is the keystone: o1 scores 73% on Set 4 while the other main-study models are below 50%, and the text interprets this as evidence that the agents' proficiency is largely confined to recalling and executing known algorithms, rather than performing generative problem-solving (Abstract). For that interpretation to hold, the direct-answer protocol must be a fair test of generative problem-solving. It is not, because the paper provides no search-augmented control. The paper itself cites Yao et al. (2023), whose Tree of Thoughts on the Game of 24 raised GPT-4 from roughly 4% under chain-of-thought to 74% with explicit tree search. That prior result demonstrates that the base model can propose and evaluate partial expressions correctly; what the single-pass protocol suppresses is the search loop. The observed Set 4 collapse is therefore compatible with the weaker claim that LLMs cannot maintain a search tree in free-form generation, and does not by itself establish that they lack number sense or are incapable of generative numerical problem-solving. The Limitations section even concedes that causal attribution would require tracing the paths from input to output.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":12284,"tokens_out":8861,"duration_ms":77033,"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":[{"comment":"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":"Section 5.2, Fig. 2d, and Abstract"},{"comment":"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":"Section 4.3 and Fig. 2"},{"comment":"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":"Section 5.3 and Section 7"},{"comment":"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.","section":"Section 4.2"}],"minor_comments":[{"comment":"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":"Appendix A, Prompt 3"},{"comment":"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.","section":"Section 5.3"},{"comment":"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":"Figure 3"},{"comment":"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.","section":"Section 1"},{"comment":"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":"Table 1"},{"comment":"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.","section":"Section 4.3"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the empirical observation is solid enough to merit a revision, but the manuscript currently overclaims. The key request should be a Tree-of-Thoughts control for the 24 task and a human baseline; without these, the Abstract's conclusion is not supported. The OSF data and appendices are a genuine strength and make the requested analyses feasible."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline result is credible: across two studies and eight model runs, LLMs do fine on arithmetic and primality checks and fall off a cliff on the 24 game. That is a real replication of a known phenomenon, and the paper has real virtues. The full problem sets are in the appendix, the prompts are exact, the raw data is on OSF, and the follow-up explicitly forbade code use, which rules out the easiest 'they used a solver' objection. The difficulty manipulation (easy vs. harder sets) is also a sensible design choice.\n\nWhat is genuinely new is modest. The 24-game failure was already documented in Yao et al. and Ding et al., and the authors cite both. The contribution is a compact, transparent probe plus an update on three current reasoning models. That is useful, but it is an extension, not a discovery.\n\nThe main soft spot is the interpretive leap in the Abstract and Section 6. The Abstract says the agents' proficiency is 'largely confined to recalling and executing known algorithms, rather than performing generative problem-solving.' The protocol, however, is direct-answer only. There is no Tree-of-Thoughts control and no search-augmented comparator. This matters because the paper itself cites Yao et al., whose tree search raised GPT-4 on the 24 game from roughly 4% under chain-of-thought to 74%. That cited result shows the base model can propose and evaluate partial expressions; what single-pass generation suppresses is the search loop. So the observed collapse supports 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 are incapable of generative numerical problem-solving. The stress-test note is right about this.\n\nTwo smaller issues: there is no human baseline, so 'relatively easy for humans' is asserted rather than measured, and the main study gives no error bars or variance on the three trial averages. The 80% inclusion threshold for models also weakens generality. These are not fatal, but they support a conditional verdict rather than a strong one.\n\nTo the authors' credit, the Limitations section concedes that causal attribution would require tracing input-to-output paths, and it calls for replication on larger task sets. The abstract and conclusions just do not honor that caution.\n\nWho this is for: people working on LLM evaluation or reasoning who want a short, reproducible probe and an updated check on current reasoning models. I would send it to a serious referee. The right outcome is likely major revision, with tempered claims or added controls—preferably both.","headline":"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.'","tokens_in":12928,"tokens_out":2947,"would_cite":true,"duration_ms":27434,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"LLMs ace arithmetic but fall off a cliff on the Game of 24","keywords":["LLMs","numerical reasoning","number sense","Game of 24","heuristic search","large reasoning models","benchmark design","mathematical problem solving"],"falsifier":"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.","tokens_in":11858,"feed_emoji":"🎲","tokens_out":7388,"duration_ms":64712,"temperature":0.7,"pith_summary":"The paper asks whether LLM-based agents have a genuine number sense—an intuitive grasp of numbers and their relationships—or merely the ability to execute known routines. To separate the two, the authors built a 100-problem test in four escalating tiers: basic arithmetic, advanced operations, primality checking, and the Game of 24, where four numbers must be combined once each with addition, subtraction, multiplication, and division to make 24. The agents scored between 74% and 95% on the first three tiers but dropped to between 11% and 73% on the 24 game, with only the strongest model crossing 50%. The authors interpret the drop as evidence that LLM proficiency is confined to recalling and executing deterministic algorithms, while heuristic, trial-and-error search over a combinatorial space remains a bottleneck.","feed_headline":"LLMs ace arithmetic, fall off a cliff on the Game of 24","feed_subtitle":"A 100-problem probe shows strong arithmetic but weak trial-and-error search, pointing to brittle number sense.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Provides the closest precedent: controllable puzzle environments show reasoning collapses as complexity grows, which the paper extends to numerical search.","marker":"Shojaee et al. (2025)"},{"why":"Identifies heuristic tree search over reasoning paths as a known LLM weakness; the 24-game condition is designed to test exactly this bottleneck.","marker":"Yao et al. (2023)"},{"why":"Establishes chain-of-thought prompting as the mechanism for multi-step reasoning; the paper's failure analysis targets breakdowns inside such chains.","marker":"Wei et al. (2022)"},{"why":"Supplies the cognitive analysis of the number game that frames the 24 puzzle as a test of number relationships, not raw arithmetic.","marker":"Van der Maas and Nyamsuren (2017)"},{"why":"Prior work using the 24 game to evaluate LLMs; the paper builds on it while adding a difficulty progression.","marker":"Ding et al. (2023)"},{"why":"Defines number sense as an intuitive grasp of numbers and relationships, the construct the paper claims LLMs lack.","marker":"Dehaene (2001)"}],"fun_headline_variants":["LLMs crack arithmetic, crumble on number puzzle Game of 24","Number sense gap: LLMs ace algorithms but fail Game of 24","Brittle number sense: LLMs strong on math rules, weak on novel puzzles","Math algorithms yes, heuristic search no: LLMs' number sense is shallow","Game of 24 exposes LLM numerical limits beyond arithmetic"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["LLMs crack arithmetic, crumble on number puzzle Game of 24","Number sense gap: LLMs ace algorithms but fail Game of 24","Brittle number sense: LLMs strong on math rules, weak on novel puzzles","Math algorithms yes, heuristic search no: LLMs' number sense is shallow","Game of 24 exposes LLM numerical limits beyond arithmetic"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000512,"raw_usage":{"total_tokens":2498,"prompt_tokens":965,"completion_tokens":1533,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":581,"completion_tokens_details":{"reasoning_tokens":1436}},"tokens_in":581,"tokens_out":1533,"duration_ms":9866,"temperature":1.0,"reasoning_tokens":1436,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:17:27.233177+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Identifies heuristic tree search over reasoning paths as a known LLM weakness; the 24-game condition is designed to test exactly this bottleneck."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the cognitive analysis of the number game that frames the 24 puzzle as a test of number relationships, not raw arithmetic."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines number sense as an intuitive grasp of numbers and relationships, the construct the paper claims LLMs lack."}],"review_version":2}