title: Numerous but rare: an exploration of magic squares.
abstract: 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 number. 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, …, n(2) 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) × 10(2056) and the corresponding probability is as small as 10(-212). Thus the MMC is effective for counting very rare configurations.
authors.affiliation: Research and Legislative Reference Bureau, National Diet Library, Chiyoda-ku, Tokyo, Japan; Department of Physics, Graduate School of Science, Osaka university, Toyonaka, Osaka, Japan.
authors.forename: Akimasa
authors.initials: A
authors.lastname: Kitajima
authors.affiliation: Large-Scale Computational Science Division, Cybermedia center, Osaka University, Toyonaka, Osaka, Japan; Department of Physics, Graduate School of Science, Osaka university, Toyonaka, Osaka, Japan.
authors.forename: Macoto
authors.initials: M
authors.lastname: Kikuchi
mesh_terms: D000465:Algorithms; D003198:Computer Simulation; D006801:Humans; D015233:Models, Statistical; D009010:Monte Carlo Method