Enumeration of 4×4 magic squares
Matthias Beck, Andrew van Herick · Mathematics of Computation · 2010
A magic square is an n × n 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 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 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.