Color any map so neighbors never match — how few colors do you need?
Take any flat map divided into regions — countries, counties, anything — where neighboring regions share a real border, not just a single point. Color every region so no two neighbors share a color. What's the smallest number of colors that's guaranteed to work, no matter how bizarrely the map is drawn?
Reveal the answer
Four — always. This is the Four Color Theorem. In 1852, law student Francis Guthrie noticed while coloring a map of England's counties that four colors sufficed, and passed the question via his brother to mathematician Augustus De Morgan. Alfred Kempe published a proof in 1879 that stood for 11 years until Percy Heawood found a flaw in it in 1890. The theorem wasn't actually proved until 1976, when Kenneth Appel and Wolfgang Haken cracked it by having a computer check thousands of special map configurations by brute force — the first major theorem ever proved with essential computer assistance, which was controversial at the time since no human could verify the case-check by hand.