A Construction of Addition-Multiplication Magic Squares Using Orthogonal Diagonal Latin Squares

Abstract

An addition-multiplication magic square of order $n$ is an $n\times n$ matrix whose entries are $n^2$ distinct positive integers such that not only the sum but also the product of the entries in each row, column, main diagonal, and back diagonal is a constant. It is shown in this paper that such a square exists for any order $mn$, where $m$ and $n$ are positive integers and $m,n\notin \{1,2,3,6\}$.