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Numerous but Rare: An Exploration of Magic Squares

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  • Akimasa Kitajima
  • Macoto Kikuchi

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.

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  • Akimasa Kitajima & Macoto Kikuchi, 2015. "Numerous but Rare: An Exploration of Magic Squares," PLOS ONE, Public Library of Science, vol. 10(5), pages 1-7, May.
    • Handle: RePEc:plo:pone00:0125062
    DOI: 10.1371/journal.pone.0125062
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    Cited by:

    1. Shintaro Nagata & Macoto Kikuchi, 2020. "Emergence of cooperative bistability and robustness of gene regulatory networks," PLOS Computational Biology, Public Library of Science, vol. 16(6), pages 1-24, June.

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