Abstract Can the sphere be tiled by an odd number of congruent polygons in an edge-to-edge arrangement? The answer is negative under a natural setting where polygons have at least 3 edges and at least 3 polygons meet at each vertex. The parity phenomenon has come to light after a voluminous classification of spherical tilings. We offer a short proof of it and draw a parallel to a result in polyhedra by Branko Grünbaum.
Hoi Ping Luk (Fri,) studied this question.