@article{10.1371/journal.pone.0125062,
doi = {10.1371/journal.pone.0125062},
author = {Kitajima, Akimasa AND Kikuchi, Macoto},
journal = {PLOS ONE},
publisher = {Public Library of Science},
title = {Numerous but Rare: An Exploration of Magic Squares},
year = {2015},
month = {05},
volume = {10},
url = {https://doi.org/10.1371/journal.pone.0125062},
pages = {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.},
number = {5},
}