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 ??? ----- —Dan ————— * J will just be a rotated and uniformly scaled copy of Z{i] itself, anything of the form (a + bi) Z[i] for integers a and b. ----- This variant, obtained by flipping the rectangle in the lower-right corner, may be aesthetically more pleasing: +-----+---+---+ |a a a|b b|c c| | | | | |a a a|b b|c c| | +---+---+ |a a a|d d|e e| +---+-+ | | |f f|g|d d|e e| | +-+-+-+---+ |f f|h|i|j j j| +---+-+-+ | |k k|l l|j j j| | | | | |k k|l l|j j j| +---+---+-----+ Tom Tom Karzes writes:

I think I have a 12-tile solution for 7x7:

+-----+---+---+ |a a a|b b|c c| | | | | |a a a|b b|c c| | +---+---+ |a a a|d d|e e| +---+-+ | | |f f|g|d d|e e| | +-+---+-+-+ |f f|h h h|i|j| +---+ +-+-+ |k k|h h h|l l| | | | | |k k|h h h|l l| +---+-----+---+

I don't know if this is optimal.

Tom

Allan Wechsler writes:

...

Suppose you are asked to tile an integer square of side n with smaller squares, but the tiles are all of side 1, 2, or 3. What is the smallest number of tiles, a(n), that you can get away with?

For n = 1, 2, 3, 4, 5, 6 the answers are fairly easily seen to be 1, 1, 1, 4, 8, 4; I'm not quite as sure that my value a(7) = 13 is correct.

If these values are right, then a(n) is not in OEIS.

It's obvious that a(3k) = k^2. I would expect that big squares can be optimally tiled by filling most of the space with 3's, with a "fringe" of some sort around two sides if n is not a multiple of 3. But I maintain a small hope that there will be "weird" solutions that don't look like this. Can anybody provide more values?