Any square torus of side C can certainly be tiled with can certainly be tiled with two squares, one of side A and one of side B, as long as A^2 + B^2 = C^2. There are probably similar examples for any higher number of squares as well. On Fri, Jul 3, 2020 at 11:48 AM Dan Asimov <dasimov@earthlink.net> wrote:
A squared square <https://en.wikipedia.org/wiki/Squaring_the_square> is an NxN square that's tiled with smaller KxK squares, where no K occurs twice.
I've long wondered about squared *tori*, (using the NxN square torus), which may be easier to find than squared squares.
(If you don't require the K to be different, there is an amusing example of squared tori — where the tiles are all 1x1 squares — for any Pythagorean hypotenuse, i.e., any N such that
N^2 = A^2 + B^2
for integers A, B.) Amusing because the sides of the tiles are not parallel to those of the NxN square.
Can there be a smaller NxN squared torus than the smallest squared square?
But how about the famous expression
1^2 + 2^2 + 3^2 + ... + L^2 = N^2
for L = 24 and N = 70, the unique nontrivial such equation.
QUESTION: --------- Can a 70x70 square torus be tiled with one KxK square tile for each K in the range 1 ≤ K ≤ 24 ???
Possibly in a non-parallel fashion?
—Dan
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