The Four Color Theorem says that it is possible, given a two dimensional map of nations, to color the map so that no two adjacent nations have the same color, with only four colors.
The proof of this theorem, sometime in the 1970s, took hundreds of pages and the help of a supercomputer.
Extra Credit. Prove Karl's Two Color Theorem: It is possible, given a map made from circles, to color the map so that no two adjacent nations have the same color, with only two colors.
Is the theorem true for maps made of an infinite number of circles?