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 $n\geq 2$ there exists an $SMA(m,n)$ such that the entries $\pm x$ appear in the same row for every $x\in\{1, 2, 3,\ldots, mn/2\}$ if and only if $m\equiv 0, 3(\mod4)$ and $n=2$ or $m\geq 3$ and $n\geq 4$.