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, …, 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) × 102056 and the corresponding probability is as small as 10−212. Thus the MMC is effective for counting very rare configurations.
Highlights
Data Availability Statement: All relevant data are within the paper
We focus on classic magic squares
They, provide some special classes of magic squares, which gives rise to the question: Among all the possible arrangements of numbers in a square of a given size, how many of them form magic squares? Putting the question in another form: Is there any chance of making a magic square by putting numbers randomly in a square? It may be surprising to know that the exact number of possible magic squares is so far only known up to order 5
Summary
Data Availability Statement: All relevant data are within the paper. It may be surprising to know that the exact number of possible magic squares is so far only known up to order 5. We apply a Monte Carlo method to this problem, and estimate the number of the magic squares of each size up to order 30. To state the problem explicitly, we consider classic magic squares order n.
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