The square peg problem

The Square Peg Problem (a.k.a. the Inscribed Square Problem and the Toeplitz' Conjecture) is a geometry problem proposed by German mathematician Otto Toeplitz in 1911.

It can be stated as :

"Does every plane simple closed curve contain all four vertices of some square?"

or :

"Does every continuous simple closed curve in the plane γ : S¹ → R² contains four points that are the vertices of a square?"

or :

"Does every Jordan curve have an inscribed square?"

It remains unsolved.

For technical details see :A survey on the square peg problem, Notices of the American Mathematical Society, 61 (4): 346–352.