Social Golfer Problem

The social golfer problem asks whether golfers can be scheduled for rounds in groups of golfers so that every golfer plays in one group per round and no two golfers meet in the same group more than once. Since a golfer meets others in each round, the elementary upper bound is

where is the floor function.

For example, 20 golfers can play in five groups of four for five rounds without a repeated pair, as shown below.

The 32-golfer instance asks for eight groups of four over as many rounds as possible. The upper bound of 10 rounds is attained by a schedule derived from a resolvable group-divisible design. Shen (1996) established the existence of the required design, Colbourn (1999) independently gave a construction from which it can be obtained, and Aguado (2004) explicitly presented the resulting 10-round schedule.

Finite affine planes provide one special family of exact schedules: order gives rounds for golfers in groups of size , with every pair meeting exactly once.

The general optimization problem remains an unsolved problem. Miller et al. (2026) give best-known schedules for all numbers of players up to 150 with equal group size at least 3 when the group size divides the number of players, and prove many of the schedules maximal. Pegg (2008) gives many explicit schedules for groups of sizes 2 through 5 in a Wolfram Demonstration.