Author(s)

Qihe Yang

Corresponding Author

Qihe Yang

Abstract

This paper conducts an in - depth study and generalization of the 15 - puzzle using group - theoretic methods. Starting with two - dimensional puzzles, relevant concepts are defined in detail, and the solvability of Lloyd’s Puzzle, n×n, and n×m puzzles is studied by constructing permutations. The research is extended to three - dimensions, where multiple puzzle types are defined, and the solvability conditions of n^3, n^2 m, and nmk puzzles are obtained. Further generalization to d - dimensions is carried out, with the definitions and solvability conclusions of n^d and s -sliced n^d puzzles given. The characteristics of puzzles with multiple empty blocks and generalized initial configurations are also explored. The research results deepen the understanding of the 15 - puzzle and its extended forms, providing a theoretical basis for subsequent related research.

Keywords

15-Puzzle; Group Theory; Solvability; D-Dimensional Puzzles