Build a cube with exactly double the volume of another, using only a compass
Legend says the plague-stricken citizens of Delos consulted an oracle, who told them to double the size of their cubic altar to Apollo. Simply doubling each edge would make the volume eight times bigger, not two — so what edge length actually doubles the volume, and can it be constructed with just a compass and straightedge?
Reveal the answer
It can't. The required edge is the cube root of 2 times the original edge, and that kind of length can never be built with compass-and-straightedge constructions alone — only square roots can. Pierre Wantzel proved this impossibility in 1837, using the same algebraic reasoning that also ruled out trisecting an arbitrary angle. Ancient mathematicians had already found workarounds using other tools; they just could never do it within the two allowed instruments.
— Pierre Wantzel, Doubling the cube — Proof of impossibility, 1837