Try to draw a map that needs a fifth color. Mapmakers couldn't either
Take any map divided into regions, like countries or counties, and color each region so that no two regions sharing a border have the same color. Mapmakers had long noticed four colors always seemed to be enough, no matter how tangled the borders got. Can you sketch a map that genuinely needs a fifth?
Reveal the answer
You can't — it was proven impossible in 1976 by Kenneth Appel and Wolfgang Haken, who showed every possible map reduces to one of 1,936 unavoidable configurations, each checkable by computer. It was the first major theorem ever proved with essential help from a computer, and the reliance on a machine-checked case analysis — rather than a proof a human could fully verify by hand — was controversial among mathematicians for years afterward.