# Existence of Some Signed Magic Arrays

## TL;DR

This paper investigates the existence conditions of signed magic arrays, which are structured arrays with symmetric entries and zero-sum rows and columns, providing complete characterizations for specific parameter cases.

## Contribution

It offers a comprehensive characterization of signed magic arrays for cases where n=s, n=m, and most cases where n=2m, advancing understanding of their existence.

## Key findings

- Complete characterization for n=s and n=m cases.
- Most cases characterized for n=2m.
- Identification of parameter conditions for array existence.

## Abstract

We consider the notion of a signed magic array, which is an $m \times n$ rectangular 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 attempt to make progress toward a characterization of for which $(m, n, s, t)$ there exists such an array. This characterization is complete in the case where $n = s$ and in the case where $n = m$; we also characterize three-fourths of the cases where $n = 2m$.