On balanced room squares and complete balanced howell rotations

Abstract

The first part of this paper deals with established results concerning complete balanced Howell rotations for an even number of teams. The point of view used in this paper gives rise to independent proofs of known properties of these rotations. Further, the terminology of Room squares readily lends itself to this study and the existing theory of Room squares enables us to give a short proof of the important Berlekamp and Hwang [1] result. Two new constructions are given in section 5. A Room square (RS) or a Room design of order 2n, where n is a positive integer, is an arrangement of 2n elements (symbols) in a square array of side 2 n 1 such that (a) each of the ( 2 n 1) 2 cells of the array is either empty or contains an unordered pair of distinct elements, (b) each of the 2n objects appears precisely once in each row and in each column, and (c) each unordered pair of distinct objects occurs in exactly one cell of the array. For a description of results about RS's and an extensive bibliography, the reader is referred to the paper by Mullin and Stanton [9]. References for some of the more recent results are available in Horton's paper [5]. Now let us consider duplicate bridge tournaments and Howell rotations (progressions, movements). The ideal situation in a duplicate bridge tournament for 2n teams (partnerships) is to arrange that