科研之友
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摘要
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 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) x 10(2056) and the corresponding probability is as small as 10(-212). Thus the MMC is effective for counting very rare configurations.
- 出版日期2015-5-14
