Let T^2 = R^2/Z^2 denote the square torus, i.e., a unit square with its opposite edges identified. Let X = {x_1, ..., x_n} be any set of n distinct points of T^2 where n = 2k is even. Puzzle: ------- Does there necessarily exist n/2 disjoint line segments in T^2 whose n endpoints form the entire set X ??? Proof or counterexample. (Note that on T^2 there are infinitely many line segments between points p and q.) Question: --------- When X is such that there does exist such a collection of n/2 disjoint line segments, can they always be chosen so as to be *shortest* line segments??? (If points p and q of T^2 are less than the maximum possible distance of √(1/2) apart, then there is a unique shortest segment connecting them. If their distance is √(1/2) then there are four such segments.) —Dan