Euler at the bowling green
Abstract
Suppose that, in competitive play in flat-green bowling, a team has 4n players (n ≥ 4), of whom n play in Lead position, n in position Two, n in position Three, and n in Skip position. Suppose further that the team's players are to be assigned to n rinks within each of n successive games, always with each rink containing one player for each playing position. A natural requirement is that no two players from the team are to be together in a rink more than once, i.e. any 2 of the team's players who have different playing positions are to be together in a rink exactly once. This can be achieved if there are three pairwise orthogonal n × n Latin squares, i.e. if a resolvable transversal design RTD(4, n) exists. Thus the requirement can be met unless n = 6 or possibly n = 10. For n = 6, various assignments of players are available that nearly meet the requirements. These near-solutions are of two types. In a Type I near-solution, players' playing positions are fixed throughout but a few within-rink pairings of players are repeated; in Type II, no pairings are repeated but a few players change playing position for a single game. A further natural requirement of the competition is that each player from the team should appear exactly once in each rink; then a set of four pairwise orthogonal n × n Latin squares is needed, or equivalently a resolvable transversal design RTD(5, n). Near-solutions for n = 6 are given; they do not correspond to sets of four almost pairwise orthogonal 6 × 6 Latin squares. For n = 10, an excellent nearsolution is obtainable from Brouwer's set of four pairwise orthogonal incomplete 10 × 10 Latin squares. An Appendix consists of tables in which assignments of players are printed in full.
