The number of "magic" squares and hypercubes

2002/01/31 by Matthias Beck, Moshe Cohen, Jessica Cuomo +1
Mathematics · #math.CO #math.NT #msc:05A15 #msc:52C07 #msc:11H06 #msc:11P21

paper · pdf

published as American Mathematical Monthly 110, no. 8 (2003), 707-717 · 11 pages, 2 figures

arxiv created 2005/01/02 · arxiv updated 2009/11/30

Abstract

We define a magic square to be a square matrix whose entries are nonnegative integers and whose rows, columns, and main diagonals sum up to the same number. We prove structural results for the number of such squares as a function of the size of the matrix and the line sum. We give examples for small sizes and show similar results for symmetric, pandiagonal magic squares, and magic hypercubes.

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