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

A r-magic rectangle set MRS Gamma(a, b; c) is a collection of c arrays of size a x b whose entries are the elements of an abelian group r of order abc, each one appearing once and in a unique array in such a way that the sum of the entries of each row is equal to a constant omega is an element of r and the sum of the entries of each column is equal to a constant delta is an element of r. 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 MRS Gamma(a, b; c). We also generalize this problem describing constructions of r-magic rectangle sets whose elements are partially filled arrays. (c) 2025 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).