The Four Color Problem
The four-color problem states that any map in a plane can be colored using four-colors in such a way that regions sharing a common boundary (other than a single point) do not share the same color. This problem was first stated in 1852 in England.This problem was solved by Appel and Haken in 1977, who constructed a computer-assisted proof that four colors were sufficient. However, because part of the proof consisted of a long run by a computer, some mathematicians do not accept it.
The magazine Scientific American ran the following as an April Fools joke in 1976. They said that the figure on the left needed 5 colors. They did not publish the four color solution on the right until the next month (May).
Although it's been proven impossible, many amateur mathematicians have tried to come up with a map that needs 5 different colors to color. If someone could have come up with such a map, they would be famous.
See just how hard it is by trying to color the map on the left with 4 colors without looking a the solution on the right.