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Enumeration of $4 \times 4$ magic squares

2010; American Mathematical Society; Volume: 80; Issue: 273 Linguagem: Inglês

10.1090/s0025-5718-10-02347-1

1088-6842

Matthias Beck, Andrew van Herick,

Advanced Combinatorial Mathematics

A magic square is an $n \times n$ array of distinct positive integers whose sum along any row, column, or main diagonal is the same number. We compute the number of such squares for $n=4$, as a function of either the magic sum or an upper bound on the entries. The previous record for both functions was the $n=3$ case. Our methods are based on inside-out polytopes, i.e., the combination of hyperplane arrangements and Ehrhart’s theory of lattice-point enumeration.