Signed Magic Arrays with Certain Property

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 chapter, 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 (a) \(n=2\) and \(m\equiv 0, 3\pmod 4\) , or (b) \(n\geq 4\) and \(m\geq 3\) .