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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 number. For this purpose, we formulated the problem as a combinatorial optimization problem and applied the Multicanonical M...

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Detalles Bibliográficos
Autores principales: Kitajima, Akimasa, Kikuchi, Macoto
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Public Library of Science 2015
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4431883/
https://www.ncbi.nlm.nih.gov/pubmed/25973764
http://dx.doi.org/10.1371/journal.pone.0125062
Descripción
Sumario: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.