1. Skedulering van gade-vermydende gemengdedubbels rondomtalie-tennistoernooie.
- Author
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BURGER, A. P. and VAN VUUREN, J. H.
- Subjects
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SCHEDULING , *TENNIS tournaments , *DOUBLES tennis , *MARRIED people , *TENNIS players - Abstract
The problem of scheduling a spouse-avoiding mixed doubles round-robin tennis tournament (SMDRTT) of order n involves finding a playing schedule for n married couples in such a way that no player teams up with his/her spouse, no player opposes his/her spouse, each player opposes every other player of the same sex exactly once, each player teams up with every player of the opposite sex (except his/her spouse) exactly once, and each player opposes every player of the opposite sex (except his/her spouse) exactly once. These mixed doubles tennis matches have to be partitioned into the smallest number of rounds so that no player plays more than once per round and so that each round comprises the maximum number of matches. If n is even, then each player may be scheduled to compete in every round and hence an SMDRTT of even order n comprises n - 1 rounds, each containing n/2 matches. However, if n is odd, then one man and one woman must necessarily receive a bye during each round and hence an SMDRTT of odd order n comprises n rounds, each containing (n - 1)/2 matches. The notion of an SMDRTT may be attributed to the director of the Briarcliff Racquet Club in New York, who sought such a schedule for his club in 1972. His motivation was that spouses know each other too well and hence may have an unfair advantage with respect to anticipating elements in each other's play. Although it is known that results from the mathematical subdiscipline of design theory may be used to construct SMDRTTs of virtually any order, neither these techniques nor the application thereof is easily accessible to administrators of tennis clubs, who are typically not mathematicians. The aim in this paper is therefore two-fold: (I) to investigate which techniques from design theory are applicable in the construction of playing schedules for SMDRTTs, and (II) to apply these techniques in the construction of playing schedules for SMDRTTs of order n ≤ 20, and to document the resulting schedules in a way that is easily accessible to non-mathematicians. It transpires that the notion of a Latin square is central in the construction of SMDRTT playing schedules. A Latin square of order n is an n x n array of n symbols arranged so that each row and each column of the array contains every symbol (exactly once). Two Latin squares of order n are said to be orthogonal if, when super-imposed on one another, all n² ordered pairs of symbols appear (exactly once). A Latin square is called self-orthogonal if it is orthogonal to its transpose. It is well known that self-orthogonal Latin squares of all orders n > 3 exist, except for n = 6. In fact, if p is a prime number, q is a natural number and λ is an element of the Galois field of order pq (with λ ≠ 0, 1, 2-1), then a well-known 1971-result states that the array whose entry in row y and column z is the linear combination λy +(1-λ)z, is a self-orthogonal Latin square of order pq, where arithmetic is performed over the field. This result may be combined with the well-known result that the Kronecker product of two self-orthogonal Latin squares of orders n1 and n2 is a self-orthogonal Latin square of order n1n2, in order to construct a self-orthogonal Latin square of virtually any order. The following characterisation has been known since 1973: The matches of an SMDRTT of order n can be listed if and only if there exists a self-orthogonal Latin square L = [Lij] of order n satisfying Lii = i for all i ∈ {0, ... , n - 1}. (1) If such a Latin square exists, then the matches of the SMDRTT may be read off from the uppertriangular part of the square. More specifically, in the match in which man i opposes man j, woman Lij should be the team mate of man i, while woman Lji should be the team mate of man j.… [ABSTRACT FROM AUTHOR]
- Published
- 2009