An even side analogue of Room squares (Q797589)

Let F be a one-factor of the complete graph, K(2n), of order 2n. A house of order n is equivalent to a pair of orthogonal one-factorizations of K(2n)\(\cup F\), such that F is used twice in the first one-factorization. The authors show houses of order n exist for all positive integers n except \(n=2\). They use these arrays to obtain the following theorem: For all \(n\geq 4\), and all odd \(r\geq 7\) (except \(r=11)\) there is a Room square of side \(nr+n-1\) which contains subsquares of side r and 2n-1. This paper is distinguished by its clear exposition and the elegance of its techniques.