On the number of transversals in Cayley tables of cyclic groups

It is well known that if nn is even, the addition table for the integers modulo nn (which we denote by BnBn) possesses no transversals. We show that if nn is odd, then the number of transversals in BnBn is at least exponential in nn. Equivalently, for odd nn, the number of diagonally cyclic latin squares of order nn, the number of complete mappings or orthomorphisms of the cyclic group of order nn, the number of magic juggling sequences of period nn and the number of placements of nn non-attacking semi-queens on an n×nn×n toroidal chessboard are at least exponential in nn. For all large nn we show that there is a latin square of order nn with at least (3.246)n(3.246)n transversals.We diagnose all possible sizes for the intersection of two transversals in BnBn and use this result to complete the spectrum of possible sizes of homogeneous latin bitrades.We also briefly explore potential applications of our results in constructing random mutually orthogonal latin squares.