Symmetric Sudoku-Type Games from Perfect Codes

TL;DR

This paper introduces a new method for creating symmetric Sudoku-like puzzles using perfect codes, providing detailed enumeration and a solver to evaluate game difficulty, thus expanding Sudoku variants.

Contribution

The paper develops a novel construction approach for symmetric Sudoku-type games based on Lee distance perfect codes, including enumeration of solutions and a solver for difficulty assessment.

Findings

17 inequivalent solutions for 5x5 Sudoku

232,735 and 304,014 solutions for two 8x8 Sudoku variants

Balanced difficulty levels from Easy to Hard for 5x5 Sudoku

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.