Numerous but rare: An exploration of magic squares

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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 x 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.

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APA

Kitajima, A., & Kikuchi, M. (2015). Numerous but rare: An exploration of magic squares. PLoS ONE, 10(5). https://doi.org/10.1371/journal.pone.0125062

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