Can't you just start with a tiling using identical squares, then replace one or more (but not all) of the squares with four (or nine, or sixteen, etc.) smaller identical squares? Continuing this process, any number of unequal squares can be obtained. You could constrain the sizes so that no square can be tiled by any of the other squares. The same approach would work, but would require that all squares be decomposed, using different component sizes, e.g. a 6x6 square can be decomposed into four 3x3 squares or nine 2x2 squares. Tom James Propp writes:
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