Foundation models such as GPT and Claude now solve olympiad-level mathematics with remarkable proficiency, so much so that geometry problem solving has become a standard proxy for their mathematical reasoning. Yet solving a geometry problem and drawing the figure it depends on are not the same skill: progress often hinges on a faithful diagram with the right auxiliary constructions and incidences, and it is unclear that a model which reasons its way to the answer can also produce one. A growing collection of benchmarks, including MathVista, and MathVerse, measures whether models reach the correct answer, but to our knowledge, none isolate the distinct ability to construct the diagram itself, leaving this capability unmeasured. We introduce an open-source benchmark that targets this gap: 954 self-contained olympiad geometry problems, with a 297-problem hard subset, each paired with its solution and a human-authored, high-fidelity diagram in renderable Asymptote code, together with a suite of text-, code-, image-, VLM-, and constraint-based metrics for what we term diagrammatic reasoning. Evaluating current foundation models reveals a pronounced gap between solving and drawing: their diagrams are markedly less faithful, with an average compile success rate of only 36.14\%. Strong mathematical reasoning, we find, does not imply the ability to construct accurate geometric diagrams. Our benchmark and dataset can be accessed at https://huggingface.co/datasets/max98765/hard_geometry_problems_with_diagrams.
Understanding how (multimodal) large language models perform on physics problems requires benchmarks that reflect the difficulty and breadth of expert-level physical reasoning. Existing physics benchmarks remain limited in the following two important ways: (1) short of high-difficulty datasets, and (2) lack of comprehensive coverage of visual forms, knowledge points, and step-by-step solution processes. As a result, model performance on current datasets may not be fully representative of their ability to solve complex physics problems. To address these issues, we present PhysElite, a large-scale bilingual multimodal benchmark for Olympiad-level physics reasoning. PhysElite contains 11,586 Olympiad-tier problems. For each problem, we provide corresponding visual diagrams, step-by-step bilingual Chinese-English solution derivations, and the final answer. We benchmark 18 open-source and closed-source MLLMs, and find that even the strongest model reaches only 33.7% answer accuracy. We additionally conduct step-level process evaluation to diagnose where models fail in the reasoning chain. Our datasets are released at https://huggingface.co/datasets/physelite/PhysElite.
Ruoran Xu, Wending Gao, Liyunfeng Chen et al.· 0 citations
We introduce PolyComp, a procedurally generated and verified benchmark that stresses visual recognition and compositional spatial reasoning. In each problem, a model must identify which of four options shows a pair of polycube components that can be combined to form a target solid. The benchmark contains 120 problems across four geometry families, and each problem has three different presentation formats using either a single image or multiple images. The random guessing baseline is 25%. Across the three presentations (360 presented problems per model), GPT-5.6 Sol with max effort attains 50.0% accuracy (95% problem-cluster CI 43.3-56.7%) at a mean cost of \$0.951 per presented problem, Claude Fable 5 with max effort attains 39.4% (33.1-46.1%) at \$0.701, and Gemini 3.1 Pro Preview with thinking level high attains 27.5% (22.8-32.5%), near the 25% random guessing baseline, at \$0.350. The observed accuracy spread across geometry families is larger than across presentation formats. We present a problem development and evaluation protocol, cost and token accounting, and release the 120 problems.
This paper examines how bichromatic separability problems, a classic topic in computational geometry, can be adapted for secondary mathematics education through the use of BichromaticSolver. The software computes simple or convex polygons that separate two finite sets of points under different optimisation criteria, including maximum area, minimum area, maximum perimeter, or minimum perimeter. Unlike traditional approaches, the number of polygon sides k is not fixed in advance but chosen by the user, enabling the exploration of diverse and potentially more effective configurations. Three classroom tasks were designed in which students alternated between paper-and-pencil methods and digital exploration with the software. This two-phase structure encouraged them to verify constructions, compare alternative outcomes, and refine their strategies. Classroom observations from this exploratory study document how these activities created opportunities for students to express and refine geometric reasoning while making computational thinking (CT)-related practices visible, for example decomposition, abstraction, strategic planning, and comparative evaluation of solutions. The findings suggest that integrating computational geometry problems with digital tools can enrich traditional mathematics instruction, highlight the relevance of geometry in authentic contexts, and offer a promising and transferable context for developing CT alongside core geometry content in secondary mathematics education.
R. Molano, MohammadHossein Homaei, M. Ávila et al.· International Electronic Jou...· 0 citations
In this work, we explore an alternative paradigm for spatial reasoning by explicitly disentangling 3D perception from reasoning, rather than jointly acquiring implicit 3D perception and reasoning through large-scale training. Our key observation is that modern perception models excel at estimating continuous 3D geometry, whereas large language models (LLMs) are particularly effective at compositional and symbolic reasoning. Motivated by these complementary strengths, we propose the Disentangled Spatial Reasoner (DiSR), a simple yet effective framework that reconstructs the physical world into structured 3D evidence using off-the-shelf expert perception models and fine-tunes an LLM with LoRA to perform reasoning solely over this explicit geometric evidence. Without large-scale 3D VQA training or complex tool-use policies, DiSR achieves competitive performance on popular spatial reasoning benchmarks. Beyond its strong performance, DiSR offers improved interpretability, modularity, and computational efficiency, demonstrating that explicit separation of perception and reasoning is a scalable and effective alternative paradigm to end-to-end modeling for spatial intelligence.
Haoze Sun, Jiequan Cui, Qingshan Xu et al.· 0 citations
Reasoning or inference-scaling models are the new generation of Large Language Models (LLMs) capable of complex problem solving. To investigate their problem-solving capability in physics, we evaluated model o4-mini by OpenAI on solving traditional, end-of-chapter problems from Halliday and Resnick's"Fundamentals of Physics,"spanning core topics in the undergraduate physics curriculum. Performance was analyzed across modality and problem difficulty. The model solved the problems with overall accuracy of about 90%, but performance depended strongly on representation: accuracy was much higher on text-only problems (96%) than on problems requiring coordinated interpretation of text and images (79%). Accuracy also declined significantly as the problem difficulty increased from low to medium to high. These results show that state-of-the-art LLMs can solve much of the standard introductory physics problems, but that their performance remains uneven and constrained by problem modality and problem difficulty.
