Does every closed loop you could draw contain the four corners of some square?
Draw any simple closed curve, one continuous loop that never crosses itself, however wobbly or irregular. Otto Toeplitz asked in 1911 whether every such curve must pass through all four corners of at least one square somewhere inside it. It sounds like it should be easy to settle one way or the other for a question this simple to state. Is it true for every possible closed curve?
Reveal the answer
Nobody knows for certain: it's proven true for convex curves and for curves that are smooth enough (piecewise smooth), and true in several other special cases mathematicians have worked out since 1911, but the fully general case, for every conceivable wobbly closed curve, remains an open problem to this day.
— Otto Toeplitz, Inscribed square problem — Proposed 1911; also known as the Toeplitz conjecture