Solution of the 15 puzzle problem

TL;DR

This paper analyzes the mixing time of the generalized 15 puzzle with periodic boundary conditions, establishing an order of at least n^4 and at most n^4 log n for the total variation mixing time, with convergence to a Poisson distribution.

Contribution

It provides the first rigorous bounds on the asymptotic total variation mixing time for the generalized 15 puzzle with periodic boundaries, including convergence results.

Findings

Total variation mixing time is at least order n^4.

Convergence to a Poisson distribution for fixed points after n^4 f(n) moves.

Upper bound of order n^4 log n for mixing time.

Abstract

A generalized `$15$ puzzle' consists of an $n \times n$ numbered grid, with one missing number. A move in the game switches the position of the empty square with the position of one of its neighbors. We solve Diaconis' `15 puzzle problem' by proving that the asymptotic total variation mixing time of the board is at least order $ n^4 $ when the board is given periodic boundary conditions and when random moves are made. We demonstrate that for any $f(n) \to \infty$ with $n$, the number of fixed points after $n^4 f(n)$ moves converges to a Poisson distribution of parameter 1. The order of total variation mixing time for this convergence is $n^4$ without cut-off. We also prove an upper bound of order $n^{4 }\log n$ for the total variation mixing time.