map-colouring problem

The four-colour map problem is a mathematical puzzle in topology that asks for the minimum number of colors needed to color a map so that no two adjacent regions share the same color. It was first posed in the 1850s and remained unsolved until 1976.

While three colors are insufficient, as a map with four regions, each touching the others, can be constructed, it was proven that five colors are always enough. The problem was finally solved by Kenneth Appel and Wolfgang Haken at the University of Illinois, using a combination of computer calculations and theoretical reasoning. Their proof involved identifying and analyzing a large number of "unavoidable" configurations, demonstrating that any map could indeed be colored with four colors. This solution was groundbreaking as it was one of the first major mathematical proofs to heavily rely on computer assistance.12