Linear Magic Rectangles

Our purpose is to introduce a linear-algebraic construction for magic rectangles of size pr × ps, where p is prime and 1 ≤ r ≤ s. For fixed p, r, and s, this method allows for the production of many different magic rectangles. It follows that, for many non-coprime dimensions, this construction will significantly augment the sparse collection of “known” magic rectangles. A magic rectangle of size m × n is a rectangular array containing integers 0, 1, . . . ,mn − 1 such that the row sums are constant and the column sums are constant. (The constants are different if m 6= n.) When m = n and both diagonal sums are equal to the row/column sums, one obtains a magic square. The construction of magic squares is a venerable pastime, dating back to the Lo-Shu square of ancient China (ca 650 bce). A variety of magic square constructions may be found in [1]; connections among magic squares, magic rectangles, orthogonal latin squares, and statistical design are described in [6], [9], and [10]. Efforts to construct magic rectangles are much more recent and far less numerous than those for their square counterparts. Sun [11] showed that a magic rectangle of order m×n exists if and only if m and n have the same parity, are both larger than 1, and are not both 2. (We declare such sizes m × n to be admissible.) Similar (partial) results were achieved independently in [2], and were further refined in [7]. Other constructions of magic rectangles may be found in [3], [4], and [5]. Invariably these constructions have a combinatorial flavor and produce a single example (and its obvious permutations) for each admissible size. We take a different approach: instead of a combinatorial method producing one example for each of a large set of sizes, we introduce a linear-algebraic method that will produce a variety of examples for a rather more limited set of sizes, namely pr × ps where p is prime. This, together with the magic rectangle product theorem given in [2], will increase the number of magic rectangle examples for many admissible sizes m × n where gcd(m,n) > 1. The paper is organized as follows: The construction is given in Section 2, conditions we must place upon certain aspects of the construction are developed in Section 3, and in Section 4 we show that these conditions can be satisfied for each