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This is the question on Futility Closet image

Is there really no answer?

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See here. – Ragib Zaman Mar 23 '12 at 9:07
@RagibZaman thanks – yiyi Mar 23 '12 at 9:12
The question is essentially settled for $C^2$ curves, locally graphical curves if we restrict ourselves to one codimension. There is still more to be done (rectifiable curves would be interesting). – Glen Wheeler Mar 23 '12 at 9:15
Another place to look would be Jason Cantarella's page on the topic – Louis Mar 23 '12 at 10:46
up vote 9 down vote accepted

According to Wikipedia (inscribed square problem), this is an open problem - that is, we don't currently know the answer. It's known to be true for "nice enough" curves (Stromquist's Theorem).

So it is true for e.g. all piecewise smooth curves, which are the kind we tend to imagine. Stromquist more generally implies an affirmative answer for curves that are piecewise graphs (as in $y=f(x)$) of continuous functions. Also see this page for further discussion.

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