On a conjecture by Sylwia Cichacz and Tomasz Hinc, and a related problem

TL;DR

This paper investigates the existence of $ ext{MRS}_ ext{Gamma}(a, b; c)$, a special array set with group-valued entries, providing new evidence for a conjecture and generalizing the problem to partially filled arrays.

Contribution

The paper offers new evidence supporting a conjecture on $ ext{MRS}_ ext{Gamma}$ existence and introduces constructions for partially filled array sets.

Findings

Confirmed cases supporting the conjecture.

Constructed $ ext{MRS}_ ext{Gamma}$ with partially filled arrays.

Extended the problem to broader array configurations.

Abstract

A $\Gamma$-magic rectangle set $\mathrm{MRS}_\Gamma (a, b; c)$ is a collection of $c$ arrays of size $a\times b$ whose entries are the elements of an abelian group $\Gamma$ of order $abc$, each one appearing once and in a unique array in such a way that the sum of the elements of each row is equal to a constant $\omega \in \Gamma$ and the sum of the elements of each column is equal to a constant $\delta \in \Gamma$. In this paper we provide new evidences for the validity of a conjecture proposed by Sylwia Cichacz and Tomasz Hinc on the existence of an $\mathrm{MRS}_\Gamma(a,b;c)$. We also generalize this problem, describing constructions of $\Gamma$-magic rectangle sets, whose elements are partially filled arrays.