A "squared square" is called perfect if no two tiles are the same size, and simple if there is no tiled subrectangle. Same terminology for a squared square torus. (But we require all tiles to be aligned with the sides of the square from which we can get the torus by identifying opposite sides. Otherwise, e.g., a square torus of side 5 is tiled by two square tiles of sides 3 and 4.) The smallest-sided perfect squared square (side = 112) is simple, and uses 21 square tiles of distinct sides. This is known to be the unique example with both the smallest side length and the fewest tiles. I thought I once read that there is a square torus that can be tiled by even fewer square tiles (aligned with the torus), but now I'm unable to locate any reference to such a thing. Does anyone know about perfect squared square tori (that don't come from identifying opposite sides of squared squares)? Of course it would be nicest if such a thing were also simple. Thanks, Dan ________________________________________________________________________________________ It goes without saying that .