# Discrete configuration spaces of squares and hexagons

arXiv: https://arxiv.org/abs/1704.06221 · HTML: https://mlsanity.com/paper/1704.06221/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Apr 2017.

## Built on by (most cited first, 1 of 1)

- [Approximately Optimal Search on a Higher-dimensional Sliding Puzzle](https://mlsanity.com/paper/2412.01937/) (2024)

## Abstract (arXiv)

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.