Number of Magic Squares from Parallel Tempering Monte Carlo

K. Pinn and C. Wieczerkowski
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K. Pinn: Institut für Theoretische Physik I, Universität Münster, Wilhelm–Klemm–Str. 9, D–48149 Münster, Germany
C. Wieczerkowski: Institut für Theoretische Physik I, Universität Münster, Wilhelm–Klemm–Str. 9, D–48149 Münster, Germany

International Journal of Modern Physics C (IJMPC), 1998, vol. 09, issue 04, 541-546

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± 0.00016)×1020.

Keywords: Statistical Mechanics; Monte Carlo Methods; Simulated Tempering; Magic Squares (search for similar items in EconPapers)
Date: 1998
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DOI: 10.1142/S0129183198000443

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