[math-fun] Torus tiling puzzle

It's known that a certain torus T^2 (namely, the hexagonal torus obtained by identifying opposite edges of a regular hexagon) can be tiled by regular hexagons so that each hexagon shares an edge with each other one. Puzzle: For which N ≥ 1 does there exist a tiling of some torus (either the square or hexagonal one) by N regular polygons, any two of which share an edge? —Dan