Magic rectangles and modular magic rectangles

Abstract Magic rectangles are m × n matrices with entries 1, …, mn, all row sums being equal and all column sums being equal. Sun established necessary and sufficient conditions for the existence of magic (m, n) rectangles. We introduce modular magic rectangles, variants of magic rectangles, and study two classes of modular magic rectangles: Pseudomagic and complete magic rectangles. We construct classes of pseudomagic, modular magic rectangles that are not magic rectangles, and classes of complete, modular magic rectangles. This suggests the problem of determining the spectra of pseudomagic, modular magic rectangles that are not magic rectangles; complete, modular magic rectangles; and complete, magic rectangles.