Odd Magic Powers

Abstract

(which is obtained from the famous example in Diirer's Melancholia by interchanging the first two rows and the first two columns) has the additional property that all of its 8 diagonals (e.g. 5,2, 12, 15; 11, 13,6,4; 10, 13,7,4) also have the same sum. A magic square with this extra property is called a pandiagonal magic square. Conventionally, additional restrictions are placed on the entries: (a) that they be (non-negative) integers; and often, (b) that they be the consecutive integers from 1 to n2 (when the array is n x n). We shall not require either of these restrictions and the numbers may be rational, real or complex. The purpose of this note is to consider the multiplicative properties of sets of magic squares (using the normal matrix multiplication). As van den Essen [31 has shown, some of these turn out to be quite surprising. We extend his results by showing that the product of any odd number of 3 x 3 magic squares is magic, that if such a square is invertible, then the inverse is magic and that these results extend to 4 x 4 and 5 x 5 pandiagonal magic squares. Because our method of proof is somewhat different from that of [3] and for completeness, we present the whole argument. As C. Small [2] points out, results of this type provide good examples for a Linear Algebra class. The first question of a multiplicative type is to ask whether a magic square is necessarily invertible. Clearly the answer is no-take the trivial case with all entries equal (even 0!) but a less trivial one is the example above (the vector (3, 1, 3, 1)T is in the null space). Our notation and terminology is standard except that we shall use first and second diagonal to mean the main diagonal and the 'other' one respectively. Also, we shall call a matrix with constant row sums an affine matrix and an affine matrix with constant column sums will be called doubly affine. Since we often need various permutation matrices, these will be defined by listing the columns as elements of the usual basis {el, e2,... , ej} of R'. For example: O 0 1 p = [e3 ,e2, el] = 1 0. -1 0 0-