Signed Magic arrays with certain property

Abstract

A signed magic array, $SMA(m,n;s,t)$, is an $m\times n$ array with the same number of filled cells $s$ in each row and the same number of filled cells $t$ in each column, filled with a certain set of numbers that is symmetric about the number zero, such that every row and column has a zero sum. We use the notation $SMA(m,n)$ if $m=t$ and $n=s$. In this paper, we prove that for every even number $n\ge 2$ there exists an $SMA(m,n)$ such that the entries $\pm x$ appear in the same row for every $x\in \{1,2,3,\dots ,mn/2\}$ if and only if $m\equiv 0,3(\mathrm{mod}4)$ and $n=2$ or $m\ge 3$ and $n\ge 4$.

arXiv comment10 pages, 9 figures. arXiv admin note: text overlap with arXiv:1701.01649