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| Math Central | Quandaries & Queries |
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Question from Ian: "We are a group of 8 golfers, one of whom is unpopular. How can we construct a schedule, of two foursomes, so that each person is scheduled to play with him the same number of times. What does the week by week schedule look like?" |
We have two responses for you
Ian,
You're asking for a resolvable design with n=8 and block size 4 the number of weeks you'd need is a multiple of 7. There might not be one with a small number of classes, that is, a schedule that repeats every few weeks. However, you could take all 8 choose 4 (70) four subsets and pair each with its complement to get a balanced schedule that runs over a period of 35 weeks. Every pair would play together 15 times over that period, assuming everyone shows up in every week.
If one more player is added, there is a nice balanced schedule for three threesomes that repeats every fifth week. Each pair of players plays together once in a four week period. You can find that schedule in the Archives. Search the archives for golf 9.
Victoria
Well, if you promise to be nicer to your friend in future, I will give you a schedule that not only lets everybody play that person the same number of times but does the same for every golfer of the eight, so nobody can deduce whom you had in mind.
The following schedule involves seven games. Each player thus has 21 "play-withs", three with each other player.
ABCD EFGH
ABGH EFCD
ABEF CDGH
AFCH EBGD
AECG BDFH
AFGD EBCH
AEHD BFGC
You can repeat this as often as you like, or reassign labels after each seven games.
-RD
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