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 (Zhao et al., 2025)
• d2: Improved Techniques for Training Reasoning Diffusion Language Models (Wang et al., 2026)
• SPG: Sandwiched Policy Gradient for Masked Diffusion Language Models (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
• 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

@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}
}