For our purposes, a “regular map” on an metric orientable surface M (assumed compact and without boundary) is a tiling of the surface by congruent regular n-gons, such that for any two n-gons P and Q and any orientation-preserving isometry P -> Q (there are n of these), there is an automorphism f: M -> M of the tiling such that f restricted to P is this isometry P -> Q. (There must be an easier way to say this. But a regular map defined as just a tiling of M by congruent polygons need not have the isometry property.) PUZZLE: If M = the torus T, what is the least positive integer L such that there exists no regular map on T having exactly L polygons? (Note: We allow the boundary of any tile to overlap any part of itself. So even one lone square can be a regular map of T.) —Dan