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Golf Foursomes: Eight Players Question: Is it possible to arrange a multi-round tournament involving eight players in such a way that each player gets matched in a foursome with the other 7 players at least once? Answer: Yes, using three rounds it is possible, but we must be willing to allow pairs of players to be matched and golf together in every round to make it work. This problem is highly constrained in that there are seven other players with whom a given player would like to be matched. That means that to optimize the foursomes from a single player's perspective, we must require three rounds (7 divided by 3 = 2 mod 1). The methodology for eight players This approach will make use of paired sets. Notice that if we pair a given player with another player, we have four such sets of pairs: AB, CD, EF, and GH. If we explore various combinations of these sets to form foursomes in a given round, we find that there are three unique ways to arrange the four sets. This leads us to the following foursome assignments:  Note that besides a given player's pair, with whom he golfs three times, each player is in a foursome with all other players once, exactly. Other Options? Not really. All other solutions in three rounds are either permutations of the above, or result in foursome assignments that result in certain players not being partnered with some of the other players. For instance, if we attempt to optimize foursome pairings for player A, he would be partnered with BCD in round 1, EFG in round 2, and H plus two repeats in round 3. One such arrangement, where I have selected BC at random for player A's repeat in round 3, yields the following foursome assignments:  Inspection of the above reveals that certain players are never paired in a foursome. For instance, player A is matched with DEFG and H only once, and with B and C twice; however, player B is matched with ADH twice, C three times, and EFG four times. Any other permutation based upon different choices for player A's repeats simply shifts the patter to the chosen two for the repeat. Therefore, there appears to be no better choice for the foursome arrangements in three rounds for eight players than the first one offered. At least the paired players don't have to ride in the same cart every round....  On the other hand, this would make for an excellent arrangement for a better ball tournament! Next: Twelve Players12.htmlshapeimage_2_link_0