Quandaries and Queries


I am running a golf tournament with twelve golfers. We will be playing four rounds of golf. I don't want to schedule the same two players together more than twice (if possible). How would I make a schedule that each player would play at least once with every other player? I would appreciate any help that you can give me. Thank you.



In short the answer is no, it's not possible.

Twelve golfers determine 66 pairs ( 1211/2=66).

A 4-some determines 6 pairs ( 43/2=6) so three 4-somes determine 18 pairs. However we must have a repetition in each 4-some after the first round giving at most 15 new pairs each round for a total of at most 18+15+15+15 = 63 < 66.

We probably know more. In the 2nd round we need to repeat 3 pairs from the 1st round - 15 new pairs at most; in the 3rd round we must repeat 3 pairs from the 1st and 2nd rounds - 12 new pairs at most; and so on for a total of 18+15+12+9 = 54 pairs at most if we don't have any pairs more than twice.



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