# Discrete configuration spaces of squares and hexagons

## TL;DR

This paper explores the combinatorial and topological properties of generalized sliding puzzles on square and hexagonal grids, analyzing their configuration spaces and potential implications in statistical mechanics and robotics.

## Contribution

It introduces new combinatorial theorems for the configuration spaces of these puzzles, linking discrete puzzles to topological and statistical mechanics concepts.

## Key findings

- Asymptotic bounds on puzzle solvability speed
- Connections between puzzle configurations and topological properties
- Potential applications in statistical mechanics and robotics

## Abstract

We consider generalizations of the familiar fifteen-piece sliding puzzle on the 4 by 4 square grid. On larger grids with more pieces and more holes, asymptotically how fast can we move the puzzle into the solved state? We also give a variation with sliding hexagons. The square puzzles and the hexagon puzzles are both discrete versions of configuration spaces of disks, which are of interest in statistical mechanics and topological robotics. The combinatorial theorems and proofs in this paper suggest followup questions in both combinatorics and topology, and may turn out to be useful for proving topological statements about configuration spaces.