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That's a really cool solution. One potential problem is that it doesn't find the loop as soon as possible (e.g. if it's going around the perimeter of a square, the cycle is detectable after 4N steps, but the algorithm will take N^2 steps to realize that)

That's nice (and can be easily fixed to handle negative coordinates), but extremely slow (imagine a simple square cycle with side length N. Her solution takes N^2 steps.

Perfection! A beautiful hack around not knowing the size of the board, and even faster detecting cycles to boot.

I think it's the best answer so far!