Magic rectangles with empty cells

TL;DR

This paper investigates the existence conditions for specific types of magic rectangles with empty cells, focusing on cases with fixed row and column fill counts and their relation to magic square sets.

Contribution

It establishes new existence criteria for magic rectangles with particular parameters and characterizes when magic square sets exist based on these parameters.

Findings

Existence of $MR(m,n;r,2)$ established.

Conditions for $MR(m,km;ks,s)$ proven.

Characterization of $MSS(m, s; t)$ existence provided.

Abstract

A magic rectangle of order $m\times n$ with precisely $r$ filled cells in each row and precisely $s$ filled cells in each column, denoted $MR(m,n;r,s)$, is an arrangement of the numbers from 0 to $mr-1$ in an $m\times n$ array such that each number occurs exactly once in the rectangle and the sum of the entries of each row is the same and the sum of entries of each column is also the same. In this paper we study the existence of $MR(m,n;r,2)$, $MR(m,km;ks,s)$, and $MR(am,bm;bs,as)$. We also prove that there exists a magic square set $MSS(m, s; t)$ if and only if $m=s=t=1$ or $3\leq s\leq m$ and either $s$ is even or $mt$ is odd.