Triple youden rectangles - a new class of fully balanced combinatorial arrangements

Triple Youden rectangles are defined and examples are given. These combinatorial arrangements constitute a special class of k x v row-and-column designs, k < v, with superimposed treatments from three sets, namely a single set of v treatments and two sets of k treatments. The structure of each of these row-and-column designs incorporates that of a symmetrical balanced incomplete block design with v treatments in blocks of size k. Indeed, when either of the two sets of k treatments is deleted from a k x v triple Youden rectangle, a k x v double Youden rectangle is obtained; when both are deleted, a k x v Youden square remains. The paper obtains an infinite class of triple Youden rectangles of size k x (k + 1). Then it presents a 4 x 13 triple Youden rectangle which provides a balanced layout for two packs of playing-cards, and a 7 x 15 triple Youden rectangle which incorporates a particularly remarkable 7 x 15 Youden square. Triple Youden rectangles are fully balanced in a statistical as well as a combinatorial sense, and those discovered so far are statistically very efficient.