Centrally symmetric and magic rectangles Original Research Article

We define a (4 × n)-rectangle R with ground set G(R) = ±[c + 1, c + 2n] to be centrally symmetric with threshold c if all row sums and all column sums of R are equal to zero. A (p × q)-rectangle A with ground set [1, pq] is called magic if A has constant row sums and A has constant column sums, the two constants not necessarily equal. In this paper we solve the problem of the existence of centrally symmetric and magic rectangles by determining all pairs of integers (n, c) resp. (p, q), for which there exists a centrally symmetric (4 × n)-rectangle with threshold c resp. a magic (p × q)-rectangle.