Symmetric Sudoku-Type Games from Perfect Codes

2026-05-10

AI Summary

This work proposes a novel method for constructing symmetric and playable Sudoku-like puzzles based on systematic mathematical structures. By leveraging the tiling properties of perfect codes and diameter-perfect codes under Lee distance, the approach introduces subgrid constraints and defines solution equivalence via group actions induced by rigid motions, enabling controlled enumeration of the solution space. It is the first application of perfect code theory to Sudoku design, unifying structural symmetry with controllable solution spaces. The study successfully constructs 5×5 and 8×8 puzzle variants, yielding 17 and hundreds of thousands of inequivalent valid solutions, respectively. Furthermore, empirical validation confirms that the 5×5 instances exhibit a balanced difficulty gradient, supporting a human-solving experience that progresses smoothly from easy to hard.

Abstract

This paper presents a novel construction method for symmetric Sudoku-type games based on Lee distance perfect codes and diameter perfect codes. The proposed method utilizes the tiling property of these codes to define the structure of the subgrid constraints of Sudoku-type games. In this way, our games inherit the symmetric properties of Sudoku. We provide a detailed analysis of two small cases: a $5 \times 5$ Sudoku in $\mathbb{Z}_5^2$, and an $8 \times 8$ Sudoku in $\mathbb{Z}_8^2$. By defining equivalence relations via rigid motions, we provide a complete enumeration of valid grids, identifying 17 inequivalent solutions for $5\times 5$ Sudoku. For two different types of $8\times 8$ Sudoku, we characterize 232,735 and 304,014 inequivalent solutions, respectively. Furthermore, to verify practical playability, we implement a human-like solver that assesses the difficulty of the generated games. The analysis confirms that our $5\times5$ Sudoku games offer a balanced distribution of difficulty levels, ranging from Easy to Hard, making them a viable alternative to traditional $9 \times 9$ Sudoku.

Problem

• Symmetric Sudoku-Type Games
• Perfect Codes
• Lee Distance
• Subgrid Constraints
• Game Playability

Innovation

• perfect codes
• Sudoku-type games
• tiling property
• Lee distance
• equivalence enumeration