Signed magic rectangles with two filled cells in each column

A signed magic rectangle $SMR(m,n;r, s)$ is an $m \times n$ array with entries from $X$, where $X=\{0,\pm1,\pm2,\ldots,$ $\pm (ms-1)/2\}$ if $mr$ is odd and $X = \{\pm1,\pm2,\ldots,\pm mr/2\}$ 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 $n=r\equiv 0,3 \pmod 4$ or $m,r\geq 3$ and $mr=2n$.

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