学術雑誌論文

学術雑誌論文

text

PLoS ONE. 2015, 10(5), p. e0125062

Numerous but Rare: An Exploration of Magic Squares

PLOS

英語

text

2015-05-14

https://doi.org/10.1371/journal.pone.0125062

Kitajima A., Kikuchi M.. (2015) Numerous but Rare: An Exploration of Magic Squares. PLoS ONE 10(5): e0125062. doi: 10.1371/journal.pone.0125062.

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 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) × 10²⁰⁵⁶ and the corresponding probability is as small as 10^-212. Thus the MMC is effective for counting very rare configurations.

This article is licensed under a Creative Commons Attribution 4.0 International License.

93263

Numerous but Rare: An Exploration of Magic Squares

PLOS

英語

2015-05-14

PLoS ONE

10

5

e0125062

https://doi.org/10.1371/journal.pone.0125062

https://hdl.handle.net/11094/93263