Signed magic rectangles with two filled cells in each column

Abstract

A signed magic rectangle $SMR(m,n;r,s)$ is an $m\times n$ array with entries from $X$, where $X=\{0,\pm 1,\pm 2,\dots ,$ $\pm (ms-1)/2\}$ if $mr$ is odd and $X=\{\pm 1,\pm 2,\dots ,\pm mr/2\}$ if $mr$ is even, such that precisely $r$ cells in every row and $s$ cells in every column are filled, every integer from set $X$ appears exactly once in the array and the sum of each row and of each column is zero. In this paper we prove that a signed magic rectangle $SMR(m,n;r,2)$ exists if and only if either $m=2$ and $n=r\equiv 0,3(\mathrm{mod}4)$ or $m,r\ge 3$ and $mr=2n$.

18 pages, 14 figures