RESEARCH ARTICLE Numerous but Rare: An Exploration of Magic Squares

How rare are magic squares? So far, the exact number of magic squares of order n is only known for n 5. For larger squares, we need statistical approaches for estimating the num-ber. For this purpose, we formulated the problem as a combinatorial optimization problem and applied the Multicanonical Monte Carlo method (MMC), which has been developed in the field of computational statistical physics. Among all the possible arrangements of the numbers 1; 2,..., n2 in an n × n square, the probability of finding a magic square decreases faster than the exponential of n. We estimated the number of magic squares for n 30. The number of magic squares for n = 30 was estimated to be 6.56(29) × 102056 and the corre-sponding probability is as small as 10−212. Thus the MMC is effective for counting very rare configurations