I asked this question about a month ago. I found this interesting because the connection graph of such a tiling (since any two tiles share an edge) is the complete graph K(N). I'll post the answer in a couple of days. —Dan ----- 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? -----