TY - JOUR
T1 - Numerous but Rare: An Exploration of Magic Squares
A1 - Kitajima, Akimasa
A1 - Kikuchi, Macoto
Y1 - 2015/05/14
N2 - 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.
JF - PLOS ONE
JA - PLOS ONE
VL - 10
IS - 5
UR - https://doi.org/10.1371/journal.pone.0125062
SP - e0125062
EP -
PB - Public Library of Science
M3 - doi:10.1371/journal.pone.0125062
ER -