Why are four colors always enough to color any map?

Nobody has a short, satisfying reason. We know four colors are always enough because a computer checked it: in 1976, Kenneth Appel and Wolfgang Haken proved that any map can be colored with at most four colors so that no two neighboring regions match, as long as "neighboring" means sharing a border, not just touching at a corner.

The puzzle began on October 23, 1852, when Francis Guthrie was coloring a map of England's counties and noticed four colors were all he needed. Proving it for every possible map turned out to be brutal. Alfred Kempe published a celebrated proof in 1879; it was 11 years before anyone spotted the flaw.

Appel and Haken's strategy was to show that a smallest map needing five colors can't exist. They boiled the infinity of possible maps down to 1,834 configurations and had a computer check each one, which took over a thousand hours. It was the first major theorem proved by computer, and some mathematicians refused to accept a proof no human could check by hand.

Funny twist: mapmakers don't care. Maps that use only four colors are rare, and most need just three.