The four colour theorem states that any map drawn on a flat plane can be coloured using at most four colours so that no two regions sharing a border have the same colour. Two regions that only touch at a single point may share a colour. The theorem applies to any map, no matter how complex.
Regions 1, 2, 3, 4 each border multiple others. The left (4) and right (4) regions share no border, so they can share a colour. Exactly 4 colours needed here.
Francis Guthrie conjectured the theorem in 1852 while colouring a map of English counties. He noticed four colours always seemed sufficient but could not prove it. The problem stumped mathematicians for 124 years. Many false proofs were published and refuted. Five colours always suffice and can be proved by hand using Euler's formula for planar graphs.
The 1976 proof by Kenneth Appel and Wolfgang Haken was the first major theorem proved by computer. It reduced all possible maps to 1,936 configurations and had a computer verify each one over 1,200 hours of CPU time. Many mathematicians were uncomfortable with a proof that could not be checked by hand. A human-readable proof, if one exists, remains undiscovered.
Five outer regions (an odd number) force the ring to use 3 colours: no 2-colouring of a 5-cycle exists. The centre region is adjacent to all five, touching all three ring colours, so it must be a fourth colour. This shows four is genuinely sometimes necessary.
Every map drawn on a flat plane can be coloured using at most four colours so that no two regions sharing a border have the same colour. Conjectured by Francis Guthrie in 1852. Proved by Appel and Haken in 1976 using a computer to verify 1,936 configurations, making it the first major theorem proved with computer assistance. A shorter verification by Robertson, Sanders, Seymour and Thomas in 1997 reduced this to 633 configurations. The theorem does not hold on a torus, where seven colours may be required.