Constant and nearly constant block-sum partially balanced incomplete block designs and magic rectangles (Q6100196)

In this paper, the author establishes a connection between magic rectangles and constant block-sum partially balanced incomplete block designs. A magic rectangle of order \(a_1\times a_2\) is an array in elements \(1, 2,\dots, a_1a_2\), each appearing only once so that each of the \(a_1\) rows adds to a constant \(A_1\) and each of the \(a_2\) columns adds to a constant \(A_2\). A magic rectangle of order \(a_1\times a_2\) exists if and only if \(a_1\) and \(a_2\) have the same parity. A pseudo-magic rectangle is an array as above with constant row-sum and column-sum, without the requirement that the elements are in \(\{1,2,\dots,a_1a_2\}\). A magic rectangle set using integers \(\{1,2,\dots,a_1a_2c\}\) is a set of \(c\) pseudo-magic rectangles, each of order \(a_1\times a_2\), \(2\le a_1\le a_2\), so that all integers from \(1\) through \(a_1a_2\) appear once and only once within the set and all rows in any pseudo-magic rectangle have a constant row-sum \(A_1^{\star}\) and, similarly, all columns in any pseudo-magic rectangle have a constant column-sum, \(A_2^{\star}\). If \(a_1\) or \(a_2\) is odd and \(a_1a_2c\) is even, the magic rectangle set does not exist. \textit{D. Froncek} [AKCE Int. J. Graphs Comb. 10, No. 2, 119--127 (2013; Zbl 1301.05289); Australas. J. Comb. 67, Part 2, 345--351 (2017; Zbl 1375.05234)] has shown that if \(a_1\equiv a_2\mod 2\) and \(a_2\ge 4\) a magic rectangle set exists for every \(c\) and that if \(a_1\), \(a_2\), \(c\) are all positive odd integers so that \(1\le a_1\le a_2\), a magic rectangle exists. The author shows that magic rectangles and magic rectangles sets are essentially constant block-sum partially balanced incomplete block designs. A nearly magic rectangle of order \(p\times q\) is an array of order \(p\times q\) that contains integers from the set \(\{1,2,\dots,pq\}\) exactly once, that has constant column-sums and that hat row-sums that differ from each other by no more than \(1\). Such rectangles are then associated with nearly constant block-sum PBIB designs, which minimize the corrected sum of squares. In the appendix, the author provides some examples of such designs.