A quotient-lifting approach to the Hamiltonicity of the cylindrical 5-puzzle graph

TL;DR

This paper presents a novel quotient-lifting method to explicitly construct Hamiltonian cycles in the state graph of the 5-puzzle on a toroidal grid, demonstrating a new approach to solving symmetric combinatorial puzzles.

Contribution

It introduces a general quotient-lifting technique for Hamiltonian cycle construction in symmetric graphs, with explicit cycle encodings for the 5-puzzle's state graph.

Findings

Explicit Hamiltonian cycle with 48 moves over {L,R,V}

A shorter 24-move cycle that forms a 2-cycle cover

The method's potential applicability to broader symmetric configuration spaces

Abstract

We construct an explicit Hamiltonian cycle in the state graph of the 5-puzzle on a toroidal 2x 3 grid, a graph with 720 vertices. The cycle is described by a short symbolic sequence of 48 moves over the alphabet {L,R,V}, repeated $15$ times, which can be verified directly. We also find a shorter 24-move sequence whose repetition yields a 2-cycle cover, which can be spliced into a Hamiltonian path. These constructions arise naturally from a general method: lifting Hamiltonian cycles from a quotient graph under the action of the puzzle's symmetry group. The method produces compact, human-readable cycle encodings and appears effective in broader settings, suggesting a combinatorial grammar underlying Hamiltonian paths in symmetric configuration spaces.