Four Colors Suffice: How the Map Problem Was Solved [Paperback]

As a mathematics student - and I have studied quite a lot of mathematics - it seems to me that proofs came in three kinds. There are the mind opening 'obvious' ones that are so stand-alone that once you read them there is nothing to learn. The blinkers have been lifted from the eyes and the world is a different place. Then there are the proofs that take such a lot of work to assimilate and for a long time you just don't see it. Perhaps you never really do, but you do come to accept it because the mathematics community is convinced. Then there are the proofs that even the mathematics community struggle with. The four-colour problem's proof is one of these. Consequently there is left a nagging doubt, which I gather is quite widespread amongst people far wiser and knowledgeable than me - than Mr Wilson also I suspect.

The curious thing is that a conjecture like the four-colour mapping, or Fermat's last theorem, or the conjecture that all even numbers can be made up of the sum of two prime numbers, is so powerful AND there are no counter examples available to challenge the conjecture. So why can they not be proved by some elegant insight such as Fermat claimed for his last theorem but never showed the world before his immanent death in a duel? Why can the four-colour problem only be proved by such inelegant computer-assisted means as this book describes? Perhaps Mr Wilson's greatest achievement is in exposing the doubts and dissatisfactions of the current proof of the four-colour problem despite the appearance that it may well be adequate (this goes for the proof of Fermat's last theorem too).

"What is the least possible number of colors needed to fill in any map (real or invented) so that neighboring countries are always colored differently?"

This apparently simple question was not immediately resolved, and over the years there were many attempts to answer it. In 1879, Alfred Bray Kempe published what was thought to be a proof that four colors were enough, but it turned out that the proof was flawed. It was only when the problem was reduced to a set of special cases that could be examined by a computer that a conclusive "proof" was finally derived by a computer in 1976. Kenneth Appel and Wolfgang Haken, who have been given credit for resolving the problem, programmed the computer. Many mathematicians found this type of proof very unnerving, and it forced the mathematical community to reexamine what the definition of a mathematical proof really is.
This story, told very well by Robin Wilson, has student input, progress that proceeds in fits and starts with one major false proof, reduction of the problem to simpler terms based on new ideas and different approaches to the problem and the unprecedented proof of a major result by computer analysis. It also demonstrates how persistent the mathematical community can be when confronted with an unsolved problem. As befits the topics, Wilson relies heavily on diagrams to get his points across, which is a necessity. Like the statement of the problem, the diagrams are easy to understand and they alter it so that it is more in the nature of a puzzle than a mathematical problem.
I enjoyed this book immensely, both as a historical account of what is right about mathematics and as a description of a persistent process that leads to truth. This book would make an excellent text for history of mathematics courses.

Published in the recreational mathematics e-mail newsletter, reprinted with permission.