Enumeration of $4 \times 4$ Magic Squares

Details Enumeration of $4 \times 4$ Magic Squares Enumeration of $4 \times 4$ Magic Squares

2009-07-18

arxiv.org/abs/0907.3188v2

Mathematics of Computation 80 (2011), 617-621

Mathematics

Combinatorics

4 pages, accompanying software IOP at http://iop.sourceforge.net/, to appear in Math. Comp

Scientific paper

A \emph{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.

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