In regard to the second question: Start with the two-square solution. Replace the smaller square with an appropriately-scaled "squared-square" solution. As long as the squared-square has no internal 4-corners points, then the tiling as a whole will not either. This will tile a square torus with many different sized squares. The process may be repeated by doing the same thing to the new smallest square. Tom James Propp writes:
Oops; I just realized that a “squared square” is an answer to the question as asked!
I meant to specify (in both the puzzle and the question) that the tiling must not have points where the corners of four squares meet.
Jim Propp
On Saturday, July 7, 2018, James Propp <jamespropp@gmail.com> wrote:
Inspired by Scott Kim’s work on motley dissections ( http://www.gathering4gardner.org/g4g13-videos/), I offer a puzzle that I know the answer to and ask a question I don’t know the answer to:
Puzzle: Find a tiling of the square torus by two squares of unequal sizes.
Question: Is there a tiling of some (not necessarily square) torus by more than two squares, all of unequal sizes?
Jim Propp