---
dataset_info:
  features:
    - name: puzzle
      dtype: string
    - name: solution
      dtype: string
  splits:
    - name: train
      num_examples: 1000000
    - name: test
      num_examples: 500
license: mit
task_categories:
  - text-generation
tags:
  - sudoku
  - constraint-satisfaction
  - planning
  - reasoning
  - diffusion-language-model
  - reinforcement-learning
size_categories:
  - 1M<n<10M
---

# 4x4 Sudoku Dataset

Standard benchmark dataset for evaluating reasoning capabilities of diffusion language models (dLLMs). This is the dataset used in the following papers:

- [**d1: Scaling Reasoning in Diffusion Large Language Models via Reinforcement Learning**](https://arxiv.org/abs/2504.12216) (Zhao et al., 2025)
- [**d2: Improved Techniques for Training Reasoning Diffusion Language Models**](https://arxiv.org/abs/2509.21474) (Wang et al., 2026)
- [**SPG: Sandwiched Policy Gradient for Masked Diffusion Language Models**](https://arxiv.org/abs/2510.09541) (Facebook Research, 2025)

## Dataset Description

4x4 Sudoku puzzles represented as 16-character strings, where `0` denotes an empty cell and digits `1-4` denote filled cells.

| Split | Examples |
|-------|----------|
| Train | 1,000,000 |
| Test  | 500 |

### Data Format

Each example contains two fields:

- **`puzzle`**: 16-character string representing the puzzle (`0` = empty cell)
- **`solution`**: 16-character string representing the completed grid

```
puzzle:   0010000402400421
solution: 4312213412433421
```

Every 4 characters form one row of the 4x4 grid:

```
Puzzle:             Solution:
0 0 | 1 0          4 3 | 1 2
0 0 | 0 4          2 1 | 3 4
----+----          ----+----
0 2 | 4 0          1 2 | 4 3
0 4 | 2 1          3 4 | 2 1
```

### Rules

- Fill empty cells (0s) with digits 1-4
- Each row must contain digits 1-4 exactly once
- Each column must contain digits 1-4 exactly once
- Each 2x2 box must contain digits 1-4 exactly once

## Typical Experimental Setup

The standard setup used in d1/d2/SPG:

- **Base model**: [LLaDA-8B-Instruct](https://huggingface.co/GSAI-ML/LLaDA-8B-Instruct)
- **Training**: RL fine-tuning (diffu-GRPO, d2-StepMerge, SPG, etc.)
- **Prompt format**: System prompt with CoT reasoning (`<reasoning>...</reasoning><answer>...</answer>`)
- **Evaluation metric**: Cell-level accuracy (fraction of correctly filled empty cells)

### Model Input Prompt Example

The following is the actual prompt format used in d1/d2/SPG for training and evaluation:

```
[System]
Please solve the following 4x4 Sudoku puzzle. The puzzle is provided as a 16-character
string reading left-to-right, top-to-bottom, where '0' represents empty cells.

Rules:
- Fill empty cells with digits 1-4
- Each row must contain digits 1-4 exactly once
- Each column must contain digits 1-4 exactly once
- Each 2x2 box must contain digits 1-4 exactly once

Important: Your solution must be a COMPLETE 16-character string with only the digits 1-4,
representing your final solved grid.

Respond in this exact format:
<reasoning>
Your step-by-step solving process
</reasoning>
<answer>
[16-character solution string with no spaces or separators]
</answer>

[User]
Solve the following Sudoku puzzle: 0321003004002100
```

**Expected model output (example - actual CoT content varies by model):**

```
<reasoning>
The puzzle grid is:
0 3 | 2 1
0 0 | 3 0
----+----
0 4 | 0 0
2 1 | 0 0

Row 1: has 3,2,1 → missing 4 → position 0 = 4
Row 2: has 3 → need 1,2,4
  Col 0: has 2 → need 1 or 4; Box0: has 3 → R1C0=4, so need 1 → position 4 = 1
  Col 1: has 3,4,1 → need 2 → position 5 = 2
  Col 3: has 1 → need 2,3,4; Row2 needs 4 → position 7 = 4
Row 3: has 4 → need 1,2,3
  Col 2: has 2,3 → need 1 or 4; Row3 needs 1,2,3 → position 10 = 1
  Col 3: has 1,4 → need 2 or 3; Box3: has 1 → position 11 = 2
Row 4: has 2,1 → need 3,4
  Col 2: has 2,3,1 → need 4 → position 14 = 4
  Col 3: has 1,4,2 → need 3 → position 15 = 3
</reasoning>
<answer>
4321123434122143
</answer>
```


## Citation

```bibtex
@article{zhao2025d1,
    title={d1: Scaling Reasoning in Diffusion Large Language Models via Reinforcement Learning},
    author={Zhao, Yanchen and Diao, Shitian and Bao, Hanze and Nie, Shuai and Wang, Juntao and Zhang, Min},
    journal={arXiv preprint arXiv:2504.12216},
    year={2025}
}

@article{wang2026d2,
    title={d2: Improved Techniques for Training Reasoning Diffusion Language Models},
    author={Wang, Guanghan and Turok, Gilad and Schiff, Yair and Arriola, Marianne and Kuleshov, Volodymyr},
    journal={arXiv preprint arXiv:2509.21474},
    year={2026}
}

@article{spg2025,
    title={SPG: Sandwiched Policy Gradient for Masked Diffusion Language Models},
    author={Facebook Research},
    journal={arXiv preprint arXiv:2510.09541},
    year={2025}
}
```