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We construct new families of whist tournaments that are at the same time both triplewhist tournaments and directedwhist tournaments. In particular, we construct such a design on v elements whenever v is a produce of primes pi pi ≥ 29, pi ≡ 5 (mod 8). It follows that, for such v, two SOLSSOMs exist sharing the same mate. © John Wiley & Sons, Inc. J Combin Designs 5: 397–406, 1997

A pitch tournament is a resolvable or near resolvablevY 8Y 7 BIBD that satis®es certain criteria in addition to theusual condition that v 0 or 1 mod 8. Here we establish that for the case v 8n the necessary condition forpitch tournaments is suf®cient for all n b 1615, with at most 187 smaller exceptions. This complements our earlier study of the v 8n 1 case, where we established suf®ciency for all n b 224, with at most 28 smaller exceptions. The four missing cases for vY 8Y 7 BIBDs are provided, namely v Pf48Y 56Y 96Y 448g, thereby establishing that the necessary existence conditions are suf®cient without exception. Some constructions for resolvable designs are also provided, reducing the existence question for vY 8Y 7 RBIBDs to 21 possible exceptions.

Let E denote the group of units (i.e., the reduce set of residues) in the ring Z3p,,n. Here we consider q,p to be primes, q = 3 (mod 4), q 2 7, p = 1 (mod 4). Let W denote a common primitive root of 3, q, and p 2 . If H denotes the (normal) subgroup of E that is generated by {-1, W } , we show that the factor group E / H is cyclic by demonstrating the existence of an element x in E such that the coset X H has order equal to IE/HI. This order is given by gcd(p"-'(p -1 ) , q -1). This representation of E / H is exploited via an appropriate construction to produce Z-cyclic whist tournaments for 3qp" players. Consequently these results extend those of an early study of Wh(3qp") that was restricted to gcd(p"-'(p -I), q -1) = 2. 0 1995 John Wiley & Sons, he.

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