# Parity Property of Hexagonal Sliding Puzzles

## TL;DR

This paper explores the solvability of hexagonal sliding puzzles and how their shape and number of holes affect their complexity and solvability.

## Contribution

The paper introduces new parity properties for hexagonal sliding puzzles and provides solvability criteria based on board shape and hole count.

## Key findings

- Hexagonal puzzles with three or more holes on large boards are always solvable.
- Solvability for puzzles with two or more holes depends on parity and tile placement in corners.
- Puzzle graphs model configuration spaces of hard tiles on tessellated domains.

## Abstract

We study the puzzle graphs of hexagonal sliding puzzles of various shapes, and with various numbers of holes. The puzzle graph is a combinatorial model which captures the solvability and the complexity of sequential mechanical puzzles. Questions relating to the puzzle graph have been previously studied and resolved for the 15 Puzzle, which is the most famous—and unsolvable—square sliding puzzle of all time. It is known that for square puzzles such as the 15 Puzzle, solvability depends on a parity property that splits the puzzle graph into two components. In the case of hexagonal sliding puzzles, we get more interesting parity properties that depend on the shape of the boards and on the missing tiles or holes on the board. We show that for large-enough hexagonal, triangular, or parallelogram-shaped boards with hexagonal tiles, all puzzles with three or more holes are solvable. For puzzles with two or more holes, we give a solvability criterion involving both a parity property, and the placement of tiles in tight corners of the board. The puzzle graph is a discrete model for the configuration space of hard tiles (hexagons or squares) moving on different tessellation-based domains. Understanding the combinatorics of the puzzle graph could lead to understanding some aspects of the topology of these configuration spaces.