No, the answer wasn’t obvious…

Squares Puzzle Stephen Morris has provided a very detailed response to this question, which I have copied to the end of the original post.

It turns out that my intuition was correct (the tiling is possible if and only if r is a rational number lying in the half open interval (0,1]), but the proof is not obvious. The maths involved seems to be rather lovely though, so Stephen’s comments are well worth a look.

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