Cut Any Angle Into Three Exactly Equal Parts — Compass Only
Ancient Greek geometers could bisect any angle perfectly using just a compass and an unmarked straightedge, and assumed trisecting one into three equal parts would be just as easy. Given any angle, try to construct exactly one-third of it using only those two classical tools — no measuring, no marked ruler, no shortcuts. Some special angles fall quickly; the challenge is doing it for every angle, every time.
Reveal the answer
It can't be done in general — with only an unmarked straightedge and compass, most angles (including a plain 60° angle) can never be trisected exactly, a fact only proved rigorously in 1837 by French mathematician Pierre Wantzel using the algebra of constructible numbers. It's one of the three famous construction problems handed down from antiquity, alongside squaring the circle and doubling the cube, and settling all three took mathematics over two thousand years.
— Heinrich Dörrie, 100 Great Problems of Elementary Mathematics: Their History and Solution — Dover Publications, English translation 1965; trisection discussed as one of antiquity's three classical construction problems
Go deeper: get the book →