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This is the question on Futility Closet image

Is there really no answer?

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    Another place to look would be Jason Cantarella's page on the topic http://www.jasoncantarella.com/webpage/index.php?title=Square_Peg_problem2012-03-23

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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.