Four Colors Suffice: How the Map Problem Was Solved

Picture-5-11-aRobin Wilson, Princeton University Press, Princeton and Oxford, 2002
hardcover, ISBN 0-691-11533-8, list $24.95, Amazon $20.48

How many colors does it take to color a map? Since few cartographers or map publishers cared, hand colored maps could have as many colors as a paint box held. But mathematicians think differently. For them the problem was the LEAST number of colors one could use to keep adjacent territories clearly separate. Oxford mathematician Robin Wilson writes that the question goes back to an 1852 letter from Professor Augustus De Morgan written to a colleague, and it took over 100 years to come up with the solution of only four. Now I’ve given the book away…right? Well, not exactly. The author starts us on a journey with simple problems of colouring a flat maps (remember he’s British, so the extra “u” applies to the inside text, while as a compromise, the word “color” appears on the cover of the book.) We then run through examples of Mobius of the twisted strip and the problem of the five princes, to Euler and his colored polyhedrons. From there we romp through a whole list of famous math thinkers with maps on donuts and horseshoes which fascinate some people. We end with German mathematician Wolfgang Haken and Kenneth Appel, an American, who came onto the problem at a crucial time – the explosion of the computer age. In 1976 computers were slow enough to take 2000 hours to check for the goal of a reducible configuration in complex arrangements. In March 1976 Haken and Appel announced “Modulo careful checking, it appears that four colors suffice.” The Appel-Haken proof of the four color theorem was greeted with great enthusiasm within the mathematical community. And so mathematicians had proven what every map colorist had known for 300 years – yeah, four is plenty. A nicely written book with some dense math but on the whole a readable insight into the world of higher mathematics.

Reviewed by Bill Warren
From the Society’s May 2011 Newsletter