Actually, that's easy: Just tile the plane with squares of sides a and b in a repeating pattern like this |-------- | | | |-------- | | | ----------| | | | | | | |----------- | | | | -------| |--- | | etc. --------- and by connecting e.g. the centers of the 4 big squares shown we get a fundamental domain for the torus having side sqrt(a^2 + b^2), tiled by one a x a and one b x b square. SO . . . suppose some N in Z+ is equal to an arbitrary sum of not necessarily distinct integer squares: N = a_1^2 + ... + a_k^2 . Question: --------- When does that correspond to a tiling of the square torus of side sqrt(N) by one square each of sides a_1, ..., a_n, respectively. (with no condition on the orientation of the tiles) ??? —Dan ----- Usually when tiling a square is mentioned I think also of the corresponding question on a square torus. For the case of a square torus, it doesn't need to be integer-sided. E.g., the square torus of side sqrt5 can be tiled (edge-to-edge) by one 2 x 2 square and one 1 x 1 square. The sides of the tiles will *not* be parallel to the two perpendicular preferred directions on the torus (the sides of its underlying square). In general if J is an ideal* of the Gaussian integers Z[i], we can look at the square torus C / J , which has area = a^2 + b^2 and will contain a^2 + b^2 images of points in Z{i] by the quotient map. Question: --------- Will such a torus — WLOG, coming from the square with vertices at (0,0), (a,b), (-b,a), (a-b,a+b) — always be tilable by one square of side a and one square of side b ??? ----- -----