Draw a map on a piece of paper. It is a famous theorem that you can color the regions with only four colors so that no two regions that share a border have the same color.
(Equivalently, you can consider the "dual graph": draw a vertex for each region, and an edge between regions that share a border. This dual graph is "planar"; its edges do not cross. Then you can color the vertices of the dual graph with four colors so that no two adjacent vertices have the same color. This is the formulation that we will be using for our visualization.)
This theorem is very hard to prove; in fact, the only known proofs require computer assistance to check a huge number of cases, and there is no easy-to-understand algorithm for four-coloring a planar map.
However, if you relax the statement of the theorem to five colors, then simple proofs are known, as are simple algorithms. This page presents a visualization of an algorithm based on Thomassen's proof of the stronger "5-choosability" theorem.
Go to visualization