The four-color theorem is one of the most famous problems in mathematics. It says that, given any map on a flat plane, at most four colors are needed to color the different regions of the map in such a way that no two adjacent regions have the same color. The history of this deceptively simple theorem was the subject of a witty lecture by Robin Wilson of the Open University, United Kingdom. The first known mention of the question was in an 1852 letter from Augustus De Morgan to William Rowan Hamilton. In 1879, Alfred Kempe published what he thought was a correct proof; indeed, the error in the paper eluded the notice of many mathematicians. It was only some years later that Percy Heawood published a paper giving an example for which Kempe's map-coloring method failed. Nevertheless, some key ideas in Kempe's paper proved fruitful and were used by Kenneth Appel and Wolfgang Haken in their computer-aided proof, completed in the 1970s. The use of a computer generated controversy: How could one be sure that there is not some unnoticed glitch in the computer calculations? Is it really a proof if no human can carry it out from start to finish? How illuminating can such a proof be? Nevertheless, no serious problems have ever been found in the Appel-Haken proof. In 1994, Neil Robertson, Daniel Sanders, Paul Seymour, and Robin Thomas came up with a new proof that required less computer checking, and it seems that most mathematicians accept this proof without reservation. But that is not the end of the story: As Wilson pointed out, the four-color theorem is really just a special case of more general mathematical questions waiting to be answered.
--- Allyn Jackson, Deputy Editor, Notices of the AMS
[This MAA Invited Address was given January 15, 2003.]