Existence of Z-cyclic triplewhist tournaments for a prime number of players

A Z-cyclic triplewhist tournament for 4n + 1 players, or briefly a TWh(4n 1), is equivalent to a n-set {(a(i), b(i), c(i), d(i)) \i = 1,..., n} of q uadruples partitioning Z(4n + 1) -{0} with the property that U-i = 1(n) {+/ -(a(i) - c(i)) +/-(b(i)-d(i))} = U-i=1(n) {+/-(a(i)-b(i)), +/-(c(i) - d(i)) } = U-i=1(n) {+/-(a(i) - d(i)), +/- (b(i) - c(i))} = Z(4n + 1) - {O} The ex istence problem for Z-cyclic TWh(p)'s with p a prime has been solved for p not equivalent to 1 (mod 16). I. Anderson el al. (1995, Discrete Math. 138, 31-41) treated the case of p = 5 (mod 8) while Y. S. Liaw (1996, J. Combin . Drs. ii, 219-233) and G. McNay (1996, Utilitas Math. 49, 191-201) treated the case of p = 9 (mod 16). In this paper, besides giving easier proofs of these authors' results, we solve the problem also for primes p = 1 (mod 16 ). The final result is the existence of a Z-cyclic TWh(upsilon) for any ups ilon whose prime factors are all = 1 (mod 4) and distinct from 5, 13, and 1 7. (C) 2000 Academic Press.