Golf Foursomes: Sixteen Players Question: Is it possible to arrange a multi-round tournament involving sixteen players in such a way that each foursome is unique; i.e., each player is partnered with a different collection of players every round with no repeats? Answer: Yes! In fact, with sixteen players it is possible to form foursomes in such a way that each players golfs with everyone else, incurring no repeats! I first encountered this problem in 1999 when serving as "Trail Boss", responsible for coordinating a group of co-workers and friends on a multi-day trip on the Robert Trent Jones Golf Trail. I was curious if there was a tested methodology to define foursomes other than through trial-and-error. Through some research, I ran across a mathematical field called combinatorics that seemed promising. One particular implementation that seemed rather elegant from a mathematical standpoint uses a mathematical construct called a Latin square. Background on Latin squares A Latin square of order n is a permutation of n symbols arranged in n rows and n columns, such that each symbol appears only once in each row, and only once in each column: If all pairs of symbols are different when two Latin squares of the same order are superimposed, the Latin squares are said to be mutually orthogonal. Latin squares are used commonly in the design of experiments that will be subjected to statistical analysis. For instance, if four species of peas A, B, C, D are to be tested with four separate fertilizers a, b, c, d, the plots can be laid out such that each pea species and each fertilizer appears in each row and each column, removing any positional dependencies 1. The methodology for sixteen players If you look carefully at the combination of the pair of orthogonal Latin squares, this seems to be the right track for creating unique golf foursomes. Each row could represent a round (four rounds, above), and the groupings could represent the golf foursomes (four per round). All we would have to do is extend this from two symbols to four. In the case of 4x4 Latin squares, there exists three that are mutually orthogonal: There is a fourth square mutually orthogonal to the three above, even though it doesn't fit the definition of a Latin square: We can use these squares to form our foursomes. If each letter represents one of our 16 players (labeled A-P), we can define each square to represent a pool of players to be mixed with the other pools such that each row represents a different round of golf. Assigning foursomes from the four pools The process for assigning foursomes is straight-forward. Staring with the first row, and matching the first column of each of the four squares yields the first foursome: AEIM. Repeating this process for columns 2-4 of the first row yields the four foursomes for the first round: Repeating for rows 2-4 yields the following table of foursomes: The rows represent the four rounds, and the columns represent the four foursomes for each round. Careful inspection should reveal that no two letters, representing players, are repeated. There is a fifth round that can be formed by virtue of the fact that players in the same square are never paired together in any foursome. Thus, the above can be augmented with a fifth round as follows: Synopsis There is a nice byproduct of this method. If each pool consist of players of a similar skill level (for instance, the ABCD pool consists of 4 scratch players, the EFGH consists of 4 low-handicappers, etc.), each of the foursomes achieve a good balance of players, consisting of one player from each skill level; the exception is the last round which consists of foursomes containing players in the same skill level. In other words, this method avoids matching a scratch player with three high handicappers, and vice-versa. Next: Twenty Players __________________________________ 1 Mathematics: From the Birth of Numbers, by Jan Gullberg, 1997.