Balanced Magic Rectangles

✍ Scribed by Thomas Bier; Douglas G. Rogers

• Book ID: 102570388
• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 392 KB
• Volume: 14
• Category: Article
• ISSN: 0195-6698
• DOI: 10.1006/eujc.1993.1032

✦ Synopsis

A magic rectangle is an (m \times n) array the entries of which are the first (m n) positive integers, the rows of which have constant sum and the columns of which have constant sum; these two constants are the same just in case (m=n) when we have the famous magic squares (without diagonal conditions) of which magic rectangles are an obvious but apparently neglected generalization. A necessary condition for there to be such a magic rectangle is that (m) and (n) be both even, but not both 2 ; or be both odd. We investigate the sufficiency of this condition. We are also considering related questions concerning the existence of certain orthogonal pairs of quasi-Latin rectangles. We confirm that the condition is sufficient at least when (m) and (n) are both even, and more generally when (m) and (n) are not coprime, and also when (n) is a multiple of 3 and (m) is any odd positive integer greater or equal to three. Our main tool is the notion of a balanced magic rectangle.