Higher-dimensional cubical sliding puzzles

TL;DR

This paper introduces higher-dimensional cubical sliding puzzles inspired by the 15 Puzzle, analyzing their solvability across different dimensions and providing algorithms and specific puzzles with known minimal solutions.

Contribution

It extends classical sliding puzzles into higher dimensions, characterizes solvability regimes, and develops algorithms for analyzing these complex puzzles.

Findings

Solvability depends on the number of tokens and empty vertices.

Identified regimes from unsolvable to fully solvable puzzles.

Algorithms implemented for 3D, 4D, and 5D cubes to analyze puzzles.

Abstract

We introduce higher-dimensional cubical sliding puzzles that are inspired by the classical 15 Puzzle from the 1880s. In our puzzles, on a $d$-dimensional cube, a labeled token can be slid from one vertex to another if it is topologically free to move on lower-dimensional faces. We analyze the solvability of these puzzles by studying how the puzzle graph changes with the number of labeled tokens vs empty vertices. We give characterizations of the different regimes ranging from being completely stuck (and thus all puzzles unsolvable) to having only one giant component where almost all puzzles can be solved. For the Cube, the Tesseract, and the Penteract ($5$-dimensional cube) we have implemented an algorithm to completely analyze their solvability and we provide specific puzzles for which we know the minimum number of moves needed to solve them.