Balanced Whist Tournament

Summary: Whist tournaments for v players, Wh(v) are known to exist for all v 0, 1 (mod 4). In this paper a new specialization of whist tournament design, namely a balanced whist tournament, is introduced. We establish that balanced whist tournaments on v players, BWh(u), exist for several infinite classes of v. An adaptation of a classic construction due to R. C. Bose and J. M. Cameron enables us to establish that BWh(4n + 1) exist whenever 4n+1 is a prime or a prime power. It is also established that BWh(4n) exist for 4n = 2kA, with k 0 (mod 2, 3 or 5). We demonstrate that a BWh(4n + 1) is equivalent to a conference matrix of order 4n + 2. Consequently, a necessary condition for the existence of a BWh(4n + 1) is that 4n + 1 is a product of primes each of which is 1 (mod 4). Thus, in particular, BWh(21) and BWh(33) do not exist. Specific examples of BWh(v) are given for v = 4, 8, 9, 20, 24, 32. It is also shown that a BWh(12) does not exist.