Balanced Magic Rectangles

Abstract A magic rectangle is an m × n array the entries of which are the first mn positive integers, the rows of which have constant sum and the columns of which have constant sum; these two constants are the same just in case m = n when we have the famous magic squares (without diagonal conditions) of which magic rectangles are an obvious but apparently neglected generalization. A necessary condition for there to be such a magic rectangle is that m and n be both even, but not both 2; or be both odd. We investigate the sufficiency of this condition. We are also considering related questions concerning the existence of certain orthogonal pairs of quasi-Latin rectangles. We confirm that the condition is sufficient at least when m and n are both even, and more generally when m and n are not coprime, and also when n is a multiple of 3 and m is any odd positive integer greater or equal to three. Our main tool is the notion of a balanced magic rectangle.