Signed magic rectangles with three filled cells in each column

Abstract

A {\em signed magic rectangle} $SMR(m,n;k,s)$ is an $m\times n$ array with entries from $X$, where $X=\{0,\pm 1,\pm 2,\dots ,$ $\pm (mk-1)/2\}$ if $mk$ is odd and $X=\{\pm 1,\pm 2,\dots ,\pm mk/2\}$ if $mk$ is even, such that precisely $k$ 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 $SMR(m,n;k,3)$ exists if and only if $3\le m,k\le n$ and $mk=3n$.

23 pages, 5 figures