Existence of centre-complementary magic rectangles

Abstract A magic rectangle of size ( m 1 , m 2 ) is an m 1 × m 2 array consisting of m 1 m 2 consecutive integers in which the sum of each row is a constant and the sum of each column is another (different if m 1 ≠ m 2 ). It is centre-complementary if the sum of any pair of centrally symmetric positions is constant. As a natural generalization of symmetric magic squares, centre-complementary magic rectangles are instrumental in the construction of 3-dimensional rectangles. In this paper, we focus our attention on the existence of centre-complementary magic rectangles and prove that the necessary conditions for the existence of centre-complementary magic rectangles are also sufficient.