Enumeration of $4 \times 4$ magic squares

Abstract

A magic square is an n × 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.

Six Little Squares and How Their Numbers Grow

“…Thus we begin by getting all the cross-sectional generating functions E u (x) from LattE. Then we either sum them by ( 4) and apply inside-out reciprocity (2), or apply ordinary reciprocity (1) first and then sum by (3). (We did whichever of these seemed more convenient.…”

“…From the geometrical or generating-function denominator we know that the period of S a (t) divides 840 = lcm (3,4,6,7,8). This is long, but it can be simplified.…”

“…Stanley's idea of Möbius inversion over the partition lattice [17,Exercise 4.10] is similar to ours in spirit, but it is less flexible and requires more computation. Beck and van Herick [3] have counted 4 × 4 magic squares using the same basic geometrical setup as ours but with a more direct counting method.…”