---
task_categories:
- image-text-to-text
language:
- en
tags:
- visual-reasoning
- symbolic-reasoning
- math
- matchstick-puzzles
- benchmark
---

# MathSticks: A Benchmark for Visual Symbolic Compositional Reasoning with Matchstick Puzzles

**Paper:** [MathSticks: A Benchmark for Visual Symbolic Compositional Reasoning with Matchstick Puzzles](https://huggingface.co/papers/2510.00483)
**Code:** [https://github.com/Yuheng2000/MathSticks](https://github.com/Yuheng2000/MathSticks)

## Overview
MathSticks is a benchmark for Visual Symbolic Compositional Reasoning (VSCR) that unifies visual perception, symbolic manipulation, and arithmetic consistency. Each task presents an incorrect matchstick equation in a seven-segment style. The goal is to move exactly one or two sticks—under strict stick-conservation and digit-legibility constraints—to make the equation mathematically correct.

- Two evaluation regimes:
  - Text-guided: the equation string is provided to the model.
  - Pure-visual: only the rendered puzzle image is provided.
- Systematic coverage: digit scale (Levels 1–4), move complexity (1/2 sticks), solution multiplicity, and operator variation.
- Scale: 1.4M generated instances; a curated test set of 400 items is released.

<p align="center">
  <img src="https://raw.githubusercontent.com/Yuheng2000/MathSticks/main/example.png" alt="MathSticks example predictions" width="300"/>
  <br/>
  <em>Example: input puzzle, reasoning trace, and move-format predictions.</em>
  <br/>
</p>

Evaluations across 14 VLMs reveal substantial limitations: closed-source models succeed only on simple cases, open-source models fail in the pure-visual regime, while humans exceed 90% accuracy. These results establish MathSticks as a rigorous, diagnostic testbed for advancing compositional reasoning across vision and symbols.

## Task Definition
- Input: a rendered image of an incorrect equation composed of matchsticks (seven-segment digits). Optionally, a text equation string (text-guided regime).
- Constraints: move one or two sticks only; no addition/removal; preserve digit legibility.
- Objective: reach a valid arithmetic equation (addition or subtraction). Operator flips may be required by the minimal-move constraint in some cases.
- Output format: a boxed sequence of Move operations, e.g. `\boxed{Move(A0, C3)}` or `\boxed{Move(A0, C3), Move(E1, F4)}`.

## Data Format (Benchmark JSONL)
Each line is one sample with the following fields:
- `id` (string): unique sample identifier, e.g., `"00075585"`.
- `level` (int): difficulty level (1–4) indicating digit scale.
- `image` (string): image path relative to repo root, e.g., `level1/00075585_8-9=3.png`.
- `problem` (string): the displayed (incorrect) equation string, e.g., `8-9=3`.
- `solution_num` (list[int, int]): counts of solvable solutions by move budget `[one_move_count, two_move_count]`.
- `mode_1_solution` (list): list of one-move solutions. Empty when `solution_num[0] == 0`.
- `mode_2_solution` (list): list of two-move solutions. Each item has:
  - `solution` (string): corrected equation (e.g., `"8 - 6 = 2"`).
  - `moves` (list[string]): standardized move format strings, e.g., `["Move(B2, B5)", "Move(C3, C5)"]`.
- `option_answer` (object): order-invariant representation of moves, for robust parsing:
  - `mode_1` (list): each one-move answer as `{ "pick": [from_label], "place": [to_label] }`.
  - `mode_2` (list): each two-move answer as `{ "pick": [from_label_1, from_label_2], "place": [to_label_1, to_label_2] }`.

Example:
```json
{
  "id": "00075585",
  "level": 1,
  "problem": "8-9=3",
  "image": "level1/00075585_8-9=3.png",
  "solution_num": [0, 4],
  "mode_1_solution": [],
  "mode_2_solution": [
    {"solution": "8 - 6 = 2", "moves": ["Move(B2, B5)", "Move(C3, C5)"]},
    {"solution": "9 - 9 = 0", "moves": ["Move(A5, C5)", "Move(C0, C6)"]},
    {"solution": "6 + 3 = 9", "moves": ["Move(A2, G0)", "Move(B6, C6)"]},
    {"solution": "9 - 0 = 9", "moves": ["Move(A5, B5)", "Move(B0, C6)"]}
  ],
  "option_answer": {
    "mode_1": [],
    "mode_2": [
      {"pick": ["B2", "C3"], "place": ["B5", "C5"]},
      {"pick": ["A5", "C0"], "place": ["C5", "C6"]},
      {"pick": ["A2", "B6"], "place": ["G0", "C6"]},
      {"pick": ["A5", "B0"], "place": ["B5", "C6"]}
    ]
  }
}
```

Notes:
- Pure-visual regime uses only `image` for model input; text-guided may also use `problem`.
- The string move format is strict for parsing; `option_answer` provides an order-invariant equivalent when needed.

## Sample Usage

### 1) Run evaluation
```bash
python eval.py \
  --input MathSticks_bench_400.jsonl \
  --image-dir ./image \
  --output predictions.jsonl
```

### 2) Score predictions
```bash
python cal_score.py \
  --pred predictions.jsonl \
  --label MathSticks_bench_400.jsonl \
  --output score.json
```

### 3) Generate data (research use only)
```bash
python match_gen_flt.py
```
This enumerates positional encodings and writes the discovered solvable cases to `match_gen.jsonl`. It is computationally intensive and may take a long time.

## Citation
If you find MathSticks useful, please cite the paper:
```bibtex
@article{mathsticks2025,
  title   = {MathSticks: A Benchmark for Visual Symbolic Compositional Reasoning with Matchstick Puzzles},
  author  = {Anonymous},
  journal = {arXiv preprint arXiv:TODO},
  year    = {2025},
  url     = {https://huggingface.co/papers/2510.00483}
}
```