Four color theorem
Many simpler maps can be colored using three colors. The fourth color is needed for some maps, such as one in which one region is surrounded by an odd number of others, which touch each other in a cycle. One such example is given in the image. The five color theorem states that five colors are enough to color a map. It has a short, elementary proof and was proven in the late 19th century.  Proving that four colors are all that is needed turned out to be much more difficult. Many false proofs and false counterexamples have appeared since the first statement of the four color theorem in 1852.