Number of Magic Squares From Parallel Tempering Monte Carlo

TL;DR

This paper introduces a Monte Carlo simulation approach using Parallel Tempering to estimate the number of magic squares of larger sizes, providing the first such estimate for 6x6 squares.

Contribution

It presents a novel application of Parallel Tempering Monte Carlo to estimate the count of higher order magic squares, overcoming computational challenges.

Findings

Estimated 6x6 magic squares count as approximately 1.7745×10^19.

Demonstrated effectiveness of Parallel Tempering in exploring complex combinatorial spaces.

Provided new quantitative insights into the enumeration of magic squares.

Abstract

There are 880 magic squares of size 4 by 4, and 275,305,224 of size 5 by 5. It seems very difficult if not impossible to count exactly the number of higher order magic squares. We propose a method to estimate these numbers by Monte Carlo simulating magic squares at finite temperature. One is led to perform low temperature simulations of a system with many ground states that are separated by energy barriers. The Parallel Tempering Monte Carlo method turns out to be of great help here. Our estimate for the number of 6 by 6 magic squares is 0.17745(16) times 10**20.