In 1911, Otto Toeplitz asked the following question.

**Inscribed Square Problem**: *Does every plane simple closed curve contain all four vertices of some square?*

This question, also known as the **square peg problem** or the **Toeplitz’ conjecture, **is still unsolved in general. (It is known in lots of special cases.)

Thanks to Mark Meyerson (“Equilateral triangles and continuous curves”,*Fundamenta Mathematicae*, 1980) and others, the analog for triangles is true. For any triangle *T* and Jordan curve *C*, there is a triangle similar to *T* and inscribed in *C*. (In particular, the triangle can be equilateral.) The survey page by Mark J. Nielsen has more information on this problem.

Added 2016-11-23: See also this recent post by T. Tao.

Added 2020-07-01: This has apparently been solved by Joshua Greene and Andrew Lobb! See their ArXiV paper (https://arxiv.org/abs/2005.09193).