Some new Z-cyclic whist tournaments
A whist tournament \(\text{Wh}(v)\), \(v \equiv 0,1 \pmod{4}\), is a schedule of games each involving two players opposing two others such that each player plays in each round (or misses exactly one round when \( v \equiv 1 \pmod{4} \)) and each player is a partner of every other player exactly once and an opponent of every other player exactly twice. Denoting each game by a \( 4 \)-tuple \( (a,b,c,d) \) in which the pairs \( \{ a,c \} \), \( \{ b,d \} \) are partner pairs, a whist tournament \( \text{Wh}(n) \) can be associated with a family of ordered \( 4 \)-tuples divided into rounds of non-overlapping games. A Z-cyclic whist tournament \( \text{Wh}(n) \) is a tournament where each round is obtained from the previous one by applying a cyclic permutation of the players (a cyclic permutation of all but one player when \( v \equiv 0 \pmod{4} \)). Although a whist tournament \( \text{Wh}(v) \) is known to exist for all \( v \equiv 0,1 \pmod{4} \), the existence of Z-cyclic whist tournaments for certain \( v \)'s is still an open question. The present paper fills one such gap by presenting a construction method for two special cases of \( v \): the case \( v = 3^3 p_1 + 1 \), and the case \( v = 3^3 p_1 p_2 + 1 \), where \( p_1, p_2 \) are primes congruent to \( 1 \) modulo \( 36 \). The applicability of the introduced constructions is tied to the existence of solutions for certain sets of linear equations. The authors show the existence of such solutions for all primes \( p_1 \equiv 1 \pmod{36} \) between \( 37 \) and \( 5000 \).
scientific article; zbMATH DE number 1428619