Geometric Problem Solving (GPS) remains at the heart of enhancing mathematical reasoning in large language models because it requires the combination of diagrammatic understanding, symbolic manipulation and logical inference. In existing literature, researchers have chiefly focused on synchronising the diagram descriptions with text literals and solving the problem. In this vein, they have either taken a neural, symbolic or neuro-symbolic approach. But this solves only the first two of the requirements, namely diagrammatic understanding and symbolic manipulation, while leaving logical inference underdeveloped. The logical inference is often limited to one chain-of-thought (CoT). To address this weakness in hitherto existing models, this paper proposes MARS-GPS, that generates multiple parallel reasoning rollouts augmented with Python code execution for numerical verification, ranks them using token-level entropy as a confidence signal, and aggregates answers through a multi-stage voting and self-verification pipeline. Empirical results show that MARS-GPS with 8 parallel rollouts achieves 88.8% on Geometry3K, a nearly +11% improvement over the prior state-of-the-art, with accuracy scaling consistently as the number of rollouts increases from 1 to 16 (+6.0% on ablation subset). We provide our code and data in an anonymous repository: https://anonymous.4open.science/r/MARS-GPS-DE55.
Figures & tables
Figure 1: Overview of the M ulti-path A ggregated R easoning S ystem for G eometry P roblem S olving ( MARS-GPS ) pipeline. Left: the problem parsing stage takes the diagram and problem text as input and produces a unified formal context F∗ via PGDPNet and a rule-based semantic parser. Right: the inference-time ensemble reasoning stage samples k parallel rollouts from fθ , each augmented with a Python sandbox E for numerical computation. The rollout outputs feed into the answer aggregation pipeline, which applies majority voting, entropy-ranked self-verification, and a weighted fallback to produce the final answer a∗ .
Category
Method
Geometry3K
PGPS9K
MLLMs
Qwen-VL ( Bai et al., 2023 )
26.7
23.2
GPT-4o ( OpenAI et al., 2024 )
58.6
51.0
Claude 3.5 Sonnet ( Anthropic, 2024 )
56.4
45.9
Gemini 2 ( Google Gemini Team, 2023 )
60.7
56.8
Proprietary LLMs
GPT-5 ( OpenAI Team, 2025 )
61.5
–
GPT-5.2 ( Zhang and others, 2026 )
73.1
–
Table 1: Comparison on Geometry3K and PGPS9K. Best results in bold . ∗ indicates models trained on the larger PGPS9K dataset.
Figure 2: Ablation studies on a subset of Geometry3K. (a) Entropy-weighted voting outperforms majority voting and entropy sorting by 2.0 percentage points. (b) Removing self-verification causes the largest single-component accuracy drop ( −4.5 pp), followed by code augmentation ( −2.5 pp).
Figure 3: Accuracy vs. number of CoT samples.
Appendix figures & tables6 assets
Supplementary material from the paper’s appendix.
Appendix
Run
Accuracy (%)
CoT 1
82.0
CoT 2
85.0
CoT 4
86.5
CoT 8
87.5
CoT 16
88.0
Appendix
Table 2: Accuracy vs. number of CoT samples on a subset of Geometry3K.
Problem subset
% of problems
Accuracy
All problems
100%
85.0%
Did not attempt Python
58.0%
92.2%
Attempted Python (sandbox blocked)
42.0%
75.0%
Appendix
Table 3: Accuracy breakdown by whether fθ attempted to invoke the Python sandbox E during the ablation run. Problems where fθ attempted code execution but was blocked are substantially harder, with accuracy dropping to 75.0%.
Python calls attempted
% of problems
Accuracy
0 calls
58.0%
92.2%
1–2 calls
17.0%
79.4%
3–5 calls
10.0%
90.0%
6+ calls
15.0%
60.0%
Appendix
Table 4: Accuracy by number of Python calls attempted per problem when the sandbox is disabled. Problems requiring many code calls (6+) drop to 60.0% accuracy, suggesting these are the most computationally intensive cases and benefit most from sandbox access. The recovery in the 3–5 calls bucket (90.0%) suggests these problems have sufficient symbolic structure that fθ can partially compensate without execution.
Outcome
Accuracy
Avg time (s)
All problems
88.8%
47.0
Correct predictions
—
34.6
Wrong predictions
—
146.1
Appendix
Table 5: Average execution time per problem by prediction outcome. Wrong predictions consume 4.2 × more time than correct ones, reflecting the cost of exhausting the verification budget on hard cases.
Time range
% of problems
Accuracy
0–10s
35.3%
98.6%
10–30s
31.2%
94.1%
30–60s
14.8%
89.9%
60–120s
9.0%
72.2%
120–180s
3.0%
61.1%
180–300s
3.5%
47.6%
Appendix
Table 6: Accuracy as a function of execution time range on Geometry3K. Problems resolved within 10 seconds achieve 98.6% accuracy, while problems exceeding 300 seconds drop to 42.1%, confirming that execution time is a reliable proxy for problem difficulty.
Category
Accuracy
Avg time (s)
Correct (s)
Wrong (s)
Similar figures
96.2%
12.9
11.0
60.0
Trigonometry
85.7%
12.8
5.4
57.1
Triangle
100%
31.2
31.2
—
Quadrilateral
96.9%
34.9
28.7
227.7
Length/Other
85.6%
48.3
27.1
173.6
Angle
91.8%
51.7
45.5
121.1
Appendix
Table 7: Per-category accuracy and average execution time on Geometry3K. Area problems have the lowest accuracy (77.4%) and Circle problems have the highest wrong-prediction time (181.8s), indicating the system spends its full verification budget on these categories before falling back.
School of Computer Science and Technology, Xi’an Jiaotong University, Xi’an, China · Shaanxi Province Key Laboratory of Big Data Knowledge Engineering, Xi’an, China