NUMBER OF MAGIC SQUARES FROM PARALLEL TEMPERING MONTE-CARLO

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 n umber of higher order magic squares. We propose a method to estimate t hese numbers by Monte Carlo simulating magic squares at finite tempera ture. One is led to perform low temperature simulations of a system wi th many ground states that are separated by energy barriers. The Paral lel Tempering Monte Carlo method turns out to be of great help here. O ur estimate for the number of 6 by 6 magic squares is (0.17745 +/- 0.0 0016) x 10(20).