A ug 2 00 9 ENUMERATION OF 4 × 4 MAGIC SQUARES

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. 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, the magic sum. The history of magic squares is well documented, see, e.g., [8, 9, 21]. The contents of a magic square have varied with time and writer; usually they have been the first n 2 consecutive positive integers, but often any arithmetic sequence and sometimes fairly arbitrary numbers. The fixed ideas are that they are integers, positive, and distinct. In the last century mathematicians took an interest in results about the number of squares with a fixed magic sum, but with simplifications: diagonal sums were often omitted and the fundamental requirement of distinctness was almost invariably neglected [1, 3, 11, 16]. For example, classical formulas of MacMahon [15] include 3 t + 3 4 + t + 2 2 , the number of 3 × 3 squares with (not necessarily distinct) nonnegative integer entries that sum to t along any row and column, and 2 9 t 2 + 2 3 t + 1 if 3|t, the number of such squares in which the two main diagonals also sum to t. The papers [6, 20] form, to the best of our knowledge, the beginning of a theory that tackles counting problems related to magic squares with the distinctness of the entries enforced. Our goal is to show that the ideas in [6] can be used to compute the number of 4×4 magic squares (with distinct entries), as a function of a parameter that is either the magic sum or an upper bound on the entries of the square. To be precise, we define the affine magic counting function a n (t) to be the number of all n × n matrices consisting of distinct positive integers whose sum along any row, column, or main diagonal is the same number t. The cubical …