Number of Magic Squares from Parallel Tempering Monte Carlo
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)×10^{20}.
 Publication:

International Journal of Modern Physics C
 Pub Date:
 1998
 DOI:
 10.1142/S0129183198000443
 arXiv:
 arXiv:condmat/9804109
 Bibcode:
 1998IJMPC...9..541P
 Keywords:

 Statistical Mechanics;
 Monte Carlo Methods;
 Simulated Tempering;
 Magic Squares;
 Condensed Matter  Statistical Mechanics
 EPrint:
 8 pages, no figures