Z-cyclic Whist Tournaments

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In [1], Leonard gives a process for constructing of Z-cyclic whist tournaments for q2 players, where q is a prime congruent to 3 modulo 4. The construction seems to work for all such q > 7, but this is not proven. Here we provide data from computations for 7 < q < 5000 (range extended 11/06):

Data file of Z-cyclic whist tournaments

The file provides data for quickly constructing a single Z-cyclic whist tournament on q2 players for each prime considered. It contains a list of the form

[data for prime 1, data for prime 2, ...],

where each term of the list has the form

[q, g, [[A1,B1], [A2,B2], ..., [Am, Bm]].


  • q is a prime congruent to 3 modulo 4;
  • g is an element of the multiplicative group (Z/(q2))× of order (q-1)/2;
  • the pairs [Ai, Bi] provide the needed data to write down the tables of the tournament. The number of these is m = (q+1)/2.

Given this information, one constructs the initial round tables of the form

[ gj Ak, gj Bk; -gj Ak, -gj Bk]

where all computations are done in Z/(q2). Here, 1 ≤ k ≤ (q+1)/2 and 0 ≤ j < (q-1)/2. Subsequent rounds of the whist tournament come successively adding 1 to each element of the previous round, computations again done in Z/(q2).

[1] Leonard, Philip A., Some new Z-cyclic whist tournaments, Utilitas Math. 49 (1996), 223-232.

Last Update: November 14, 2006

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Last Modified: November 14, 2006