Puzzle—Tennis Mixed Doubles

Abstract

Introduction The following problem was introduced to the writer by Dick Hess at the eighth “Gathering for Gardner,” a convention for recreational mathematics and magic held every two years in Atlanta in honor of Martin Gardner. Twenty tennis players (ten men and ten women) wish to play several rounds of mixed doubles. Each round will consist of five matches where each match has a man and a woman on each side. No player may oppose the same player more than once and no player may partner with the same player more than once. However, a player may partner someone in one round and oppose that person in another round. What is the maximum number of rounds that may be played without violating these conditions? Each player has only nine possible opponents of the same gender. Thus, nine is an upper bound on the number of rounds. Hess (2008) has identified a sevenround solution using an exhaustive search routine and we identify an alternative schedule using integer programming (IP). We also, unsuccessfully, attempt to identify an eight-round solution. Hess (2008) provides further discussion of this and related problems.