Algebraic Structure of the Varikon Box

TL;DR

This paper analyzes the algebraic group structure of the Varikon Box, a 3D variant of the 15-Puzzle, to develop a heuristic for solving it efficiently by understanding its configuration space.

Contribution

It introduces the group-theoretic structure of the Varikon Box and proposes a heuristic based on shortest word representations in the symmetry group.

Findings

Identifies reachable configurations using parity arguments.

Defines a generating set for the puzzle's symmetry group.

Provides a method to compute minimal move sequences for configurations.

Abstract

The 15-Puzzle is a well studied permutation puzzle. This paper explores the group structure of a three-dimensional variant of the 15-Puzzle known as the Varikon Box, with the goal of providing a heuristic that would help a human solve it while minimizing the number of moves. First, we show by a parity argument which configurations of the puzzle are reachable. We define a generating set based on the three dimensions of movement, which generates a group that acts on the puzzle configurations, and we explore the structure of this group. Finally, we show a heuristic for solving the puzzle by writing an element of the symmetry group as a word in terms of a generating set, and we compute the shortest possible word for each puzzle configuration.