Strategy Executability in Mathematical Reasoning: Leveraging Human-Model Differences for Effective Guidance

Weida Liang; Yiyou Sun; Shuyuan Nan; Chuang Li; Dawn Song; Kenji Kawaguchi

cs.AIFeb 26, 2026

Strategy Executability in Mathematical Reasoning: Leveraging Human-Model Differences for Effective Guidance

Authors: Weida Liang, Yiyou Sun, Shuyuan Nan, Chuang Li, Dawn Song, Kenji Kawaguchi

Organizations: National University of Singapore · University of California, Berkeley

Abstract

Example-based guidance is widely used to improve mathematical reasoning at inference time, yet its effectiveness is highly unstable across problems and models-even when the guidance is correct and problem-relevant. We show that this instability arises from a previously underexplored gap between strategy usage-whether a reasoning strategy appears in successful solutions-and strategy executability-whether the strategy remains effective when instantiated as guidance for a target model. Through a controlled analysis of paired human-written and model-generated solutions, we identify a systematic dissociation between usage and executability: human- and model-derived strategies differ in structured, domain-dependent ways, leading to complementary strengths and consistent source-dependent reversals under guidance. Building on this diagnosis, we propose Selective Strategy Retrieval (SSR), a test-time framework that explicitly models executability by selectively retrieving and combining strategies using empirical, multi-route, source-aware signals. Across multiple mathematical reasoning benchmarks, SSR yields reliable and consistent improvements over direct solving, in-context learning, and single-source guidance, improving accuracy by up to $+13$ points on AIME25 and $+5$ points on Apex for compact reasoning models. Code and benchmark are publicly available at: https://github.com/lwd17/strategy-execute-pipeline.

Figures & tables

Figure 1: An example illustrating that strategies which appear valid in isolation may fail when transferred as guidance. In this AIME-level problem, a human-derived structural strategy and a model-derived procedural strategy are each insufficient on their own, while selectively combining them enables successful execution.

Figure 2: Performance gains from Selective Strategy Retrieval (SSR) on closed-source reasoning models (GPT-4.1 and o3-mini), measured by pass@1 and averaged over five runs.

Figure 3: Normalized strategy usage in human-written and model-generated solutions, aggregated across problems with per-problem normalization. For each problem, strategies contribute equally, ensuring that multi-strategy solutions do not dominate the statistics.

Figure 4: Strategy-level divergence between human-written and model-generated solutions. (a) Normalized differences in strategy usage. (b) Normalized differences in strategy-guided accuracy.

Figure 5: Multi-route strategy retrieval in Selective Strategy Retrieval (SSR). Complementary retrieval routes capture category-level regularities, problem-specific transfer, and semantic coverage, forming the candidate set $\mathcal{S}(x)$ .

	HM-ReasoningBench					AIME25					Apex
Model	DS	ICL	H	M	SSR	DS	ICL	H	M	SSR	DS	ICL	H	M	SSR
Qwen3-8B	63.80	64.20	66.00	65.40	68.60	67.33	68.00	70.67	69.33	74.00	8.16	8.16	8.57	8.98	13.06
Qwen3-14B	67.40	67.60	69.40	68.20	70.20	70.66	70.33	72.67	72.00	74.67	11.02	11.43	13.47	12.65	14.69
R1-Distill-7B	49.20	49.80	51.00	50.80	52.40	42.00	48.00	51.33	52.00	53.13	4.08	4.90	6.53	6.12	7.75

Table 1: Accuracy (%) comparison between Selective Strategy Retrieval (SSR), single-source guidance (H/M), in-context learning (ICL), and direct solving (DS) across three benchmarks. Best results are shown in bold.

	HM-ReasoningBench				AIME25				Apex
Model	SSR	w/o Cat	w/o Tran	w/o Sem	SSR	w/o Cat	w/o Tran	w/o Sem	SSR	w/o Cat	w/o Tran	w/o Sem
Qwen3-8B	68.60	66.00	63.60	67.20	74.00	70.67	67.33	72.67	13.06	12.24	11.02	13.06
Qwen3-14B	70.20	67.60	65.00	69.00	74.67	71.33	68.67	73.33	14.69	13.06	11.84	13.88
R1-Distill-7B	52.40	50.60	48.80	51.20	53.13	52.00	50.67	51.33	7.75	6.94	6.12	7.35

Table 2: Ablation study of retrieval routes in Selective Strategy Retrieval (SSR) across three benchmarks. Best result is shown in bold.

Method	Qwen3-8B	Qwen3-14B	R1-Distill-7B
BM25 retrieval	67.33	69.00	47.33
Dense retrieval	68.00	70.33	48.00
MMR semantic reranking	68.67	71.00	49.33
SSR	74.00	74.67	53.13

Table 3: End-to-end retrieval baselines on AIME25 (accuracy %).

Figure 6: Topic-wise gains on HM-ReasoningBench using Qwen3-14B. Results exhibit domain-dependent behavior.

Figure 7: Average output token consumption per problem under direct solving (DS) and Selective Strategy Retrieval (SSR) using Qwen3-14B, decomposed into reasoning and final-answer tokens.

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

Difficulty Level	Count	Percentage
Level 1	78	1.6%
Level 2	303	6.1%
Level 3	386	7.8%
Level 4	742	15.0%
Level 5	1140	23.1%
Level 6	1197	24.3%

Appendix

Table 4: Difficulty level distribution in HM-ReasoningBench after GPT-5.1 re-annotation.

Subject	Count	Percentage
Combinatorics	1188	24.1%
Number Theory	1167	23.7%
Algebra	1164	23.6%
Geometry	994	20.2%
Mixed Topics	419	8.5%

Appendix

Table 5: Subject distribution of HM-ReasoningBench.

Subject	Template	Description
Algebraic	algebraic_general	General symbolic manipulation not covered by specialized algebraic templates.
	inequality	Inequality-based reasoning via bounding, convexity, or classical inequalities.
	polynomial_analysis	Polynomial structure analysis (factorization, roots–coefficients relations, divisibility).
	algebraic_manipulation	Canonical algebraic transformations (substitution, expansion, identity rewriting).
	functional_equation	Functional equations and recursive functional constraints.
	symmetric_sum	Symmetric polynomial arguments and symmetric-sum identities.

Appendix

Table 6: Strategy taxonomy used throughout the paper. Extracted strategies are mapped to fine-grained templates capturing distinct reasoning operations.

Subject	Template	Representative strategy
Algebraic	algebraic_general	Write down the given quantities and translate all relationships into equations, then solve for the target variable.
	algebraic_manipulation	Introduce intermediate variables to simplify expressions or factor a constraint into a product form, then analyze solutions.
	polynomial_analysis	Use coefficient extraction (with index shifts) to derive constraints on the coefficient sequence.
Combinatorial	counting_principle	Count objects via a direct combination argument (e.g., choose vertices and apply a closed-form expression).
	bijection	Analyze injectivity or surjectivity of a mapping and account for collisions by characterizing exceptional patterns.
	counting_principle	Correct overcounting caused by symmetry or indistinguishable elements by dividing by the number of equivalent permutations.

Appendix

Table 7: Representative realizations of strategy templates. Each row provides a neutral action description illustrating how a template may be instantiated in solutions. These examples are for interpretability only and do not affect the taxonomy or experiments.

Figure 8: Trace-level case studies illustrating strategy executability differences between human-written and model-generated solutions. For each problem, we contrast (i) human-derived strategies, which emphasize structural recognition or theorem-level reasoning, and (ii) model-derived strategies, which rely on procedural or algebraic transformations. Although both solution sources reach the correct final answer, the extracted strategies exhibit different executability properties.

	HM-ReasoningBench				AIME25				Apex
Model	SC@8	L2M	ToT	SSR	SC@8	L2M	ToT	SSR	SC@8	L2M	ToT	SSR
Qwen3-8B	66.40	65.60	67.20	68.60	72.00	69.33	72.33	74.00	11.42	10.61	12.24	13.06
Qwen3-14B	68.80	67.40	69.60	70.20	74.33	73.33	75.00	74.67	13.88	11.83	15.91	14.69
R1-Distill-7B	50.60	48.80	51.20	52.40	48.67	45.33	51.33	53.13	8.97	7.35	9.79	7.75

Appendix

Table 8: Accuracy (%) comparison with inference-time baselines. All methods use the same base prompt and model backbone. Self-Consistency (SC) uses majority voting over

N{=}8

samples. Tree-of-Thoughts (ToT) uses branching factor

B{=}3

and depth

D{=}2

Prior $(\alpha,\beta)$	Spearman vs. $(1,1)$	Top-20 overlap	HM-RB	AIME25	Apex
$(0.5,0.5)$	0.972	93.0%	70.00	74.33	14.69
$(1,1)$	1.000	100.0%	70.20	74.67	14.69
$(2,2)$	0.963	91.0%	69.80	74.33	14.69

Appendix

Table 9: Sensitivity to the Beta prior in Eq. ( 2 ) (Qwen3-14B).

Method	Offline cost (GPU-h)	Online latency (s)	Completion tokens
DS	0.0	7.2	16286
ToT ( $B{=}3,D{=}2$ )	0.0	27.8	35715
SSR	6.4	8.1	14114

Appendix

Table 10: Average cost per query, averaged across HM-RB, AIME25, and Apex.

Setting	HM-RB	AIME25	Apex
w/o contrastive representation learning	68.60	70.33	13.06
Full SSR	70.20	74.67	14.69

Appendix

Table 11: Ablation of contrastive representation learning (Qwen3-14B, accuracy %).

Problem family (preferred style comparison)	Qwen3-8B	Qwen3-14B	R1-Distill-7B
Geometry / invariant-driven (Structural vs. Procedural)	71.67 vs. 67.67	74.67 vs. 71.67	51.33 vs. 48.67
Counting / modular / algebraic decomposition (Procedural vs. Structural)	73.33 vs. 68.67	75.00 vs. 71.67	53.00 vs. 49.67

Appendix

Table 12: Controlled style-pool comparison (accuracy %).

Model	Gap metric	95% CI
R1-Distill-7B	0.238	[0.214, 0.263]
Qwen3-8B	0.221	[0.199, 0.243]
Qwen3-14B	0.205	[0.184, 0.227]
GPT-4.1	0.186	[0.166, 0.206]
GPT-5.1	0.173	[0.154, 0.193]

Appendix

Table 13: Capability trend of the usage–executability gap.

Metric	Same-series pairs	Heterogeneous pairs
Mean Spearman correlation	0.84	0.76
Mean overlap@10	0.76	0.68
Mean Kendall $\tau$	0.66	0.57

Appendix

Table 14: Cross-model consistency of executability profiles.

Comparison	Spearman $\rho$	Kendall $\tau$	Flip rate	$\Delta$ downstream EM
$T=3$ vs. $T=10$	0.91	0.79	5.8%	+0.28
$T=3$ vs. $T=20$	0.89	0.76	7.1%	+0.36

Appendix

Table 15: Robustness of executability estimation under higher trial counts.

Figure 9: Failure-mode analysis on HM-ReasoningBench using Qwen3-14B. Human guidance primarily reduces structural failures, model guidance reduces algebraic errors, while SSR mitigates both error types.

Figure 10: Strategy adherence analysis. Left: average number of strategies correctly executed in the generated solution. Right: percentage of problems for which at least one provided strategy is correctly executed. Results are reported separately for correct and incorrect final answers. Correct solutions consistently exhibit stronger strategy execution, with the gap widening for larger models.

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