Summary: From Hans and Stuart, I've seen ian Gambini's thesis, in which he describes doing an exhaustive computer search for simple perfect squared square tori, of side lengths between 6 and 450. Exactly one such was found! It uses 24 squares, to make a squared torus of side length 181. He also found 107 simple perfect squared cylinders, which if made into a torus would have a "circular fault line". Finally, I made yet another mistake in the original query: Although the simple perfect squared square using the smallest number of tiles is indeed 112x112 with 21 tiles, 112 is not the smallest possible side length. There are 3 distinct simple perfect squared squared of side length 110 (2 using 22 tiles and the other using 23). 110 has been shown to be the smallest side length possible. --Dan ________________________________________________________________________________________ It goes without saying that .