Number of Magic Squares from Parallel Tempering Monte Carlo
- 1 June 1998
- journal article
- Published by World Scientific Pub Co Pte Ltd in International Journal of Modern Physics C
- Vol. 9 (4) , 541-546
- https://doi.org/10.1142/s0129183198000443
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
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This publication has 2 references indexed in Scilit:
- Exchange Monte Carlo Method and Application to Spin Glass SimulationsJournal of the Physics Society Japan, 1996
- Mathematical GamesScientific American, 1976