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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 :
βDoes every plane simple closed curve contain all four vertices of some square?β
or :
βDoes every continuous simple closed curve in the plane Ξ³ : S1 β R2 contains four points that are the vertices of a square?β
or :
βDoes every Jordan curve have an inscribed square?β
It remains unsolved.
For technical details see :A survey on the square peg problem, 
Notices of the American Mathematical Society, 61 (4): 346β352.
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