Modeling the 15 Tile Puzzle Through the Lens of Group Theory

TL;DR

This paper models the 15 Tile Puzzle using group theory to analyze its solvability and reachability of configurations, providing a mathematical framework for understanding its underlying structure.

Contribution

It introduces a group-theoretic model of the 15 Tile Puzzle, demonstrating how to determine the solvability of configurations through permutation analysis.

Findings

Proves the puzzle's configurations form a specific permutation group.

Provides criteria to determine if a configuration is reachable.

Shows the puzzle's solvability depends on permutation parity.

Abstract

In this piece, we examine one variant of the infamous 15 Tile Puzzle and develop a mathematical backing behind why it is unsolvable. Using concepts of permutations, bijectivity, and cycle transpositions, we not only prove how to model this puzzle as a group but also how to determine if a certain configuration of the puzzle is reachable.