Inductive extensions of some Z-cyclic whist tournaments Original Research Article

A whist tournament Wh(v) for v players is a special type of resolvable (v, 4, 3)-BIBD; the blocks represent four players, each block having two ‘partner pairs’ and four ‘opponent pairs,’ with every one of the (2v) possible pairs occurring once among the former and twice among the latter. For v = 4n (and v = 4n + 1), n a positive integer, a Wh(v) is known to exist. We consider Wh(v) that are Z-cyclic, that is, having players represented by residues modulo N (depending on v, with an additional symbol ∞ when v = 4n) and all rounds obtained by successively adding 1 (mod N) to symbols from the initial round. We give constructions for Z-cyclic Wh(v) for v = q2k, k > 1, and for v = q2m+1 + 1, m > 0, where q  3 (mod 4), q ⩾ 7, is a prime. We show inductively that whenever a special Z-cyclic Wh(q2) exists (and, in the second case, a Z-cyclic Wh(q + 1) exists), tournaments for all such v also exist. Additional infinite families arise when these constructions are combined with others in the literature.