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Tarski's circle-squaring problem
Tarski's circle-squaring problem is the challenge, posed by Alfred Tarski in 1925, to take a circle (including its interior) in the plane, cut it into finitely many pieces, and reassemble the pieces so as to get a square of equal area. This was proven to be possible by MiklĂłs Laczkovich in 1990; the decomposition makes heavy use of the axiom of choice and is therefore non-constructive.
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