Magic rectangles with empty cells

Abstract

A magic rectangle of order $m\times n$ with precisely $r$ filled cells in each row and precisely $s$ filled cells in each column, denoted $MR(m,n;r,s)$, is an arrangement of the numbers from 0 to $mr-1$ in an $m\times n$ array such that each number occurs exactly once in the rectangle and the sum of the entries of each row is the same and the sum of entries of each column is also the same. In this paper we study the existence of $MR(m,n;r,2)$, $MR(m,km;ks,s)$, and $MR(am,bm;bs,as)$. We also prove that there exists a magic square set $MSS(m, s; t)$ if and only if $m=s=t=1$ or $3\leq s\leq m$ and either $s$ is even or $mt$ is odd.