Numerous but Rare: An Exploration of Magic Squares

Akimasa Kitajima and Macoto Kikuchi

PLOS ONE, 2015, vol. 10, issue 5, 1-7

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, …, 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 corresponding probability is as small as 10−212. Thus the MMC is effective for counting very rare configurations.

Date: 2015
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Persistent link: https://EconPapers.repec.org/RePEc:plo:pone00:0125062

DOI: 10.1371/journal.pone.0125062

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