Latin squares

A Latin square is an n-by-n array in which each row and column is a permutation of the same n-symbol set. For example, the composition table of a group is a Latin square. Latin squares are notoriously hard to count, and several false enumerations have appeared in print. Currently the best (practically computable) upper and lower bounds for general n are separated by a superexponential factor. I will discuss these bounds, as well as the problem of near-uniform sampling of Latin squares. Depending on time and audience interest, I will also address some subset of these related topics: Euler's 36-officer problem and mutually orthogonal Latin squares, applications to coding theory, applications to experimental design, and connections to finite geometry. I will also teach you a game.