Four Colors Suffice is a history of the Four Colour Map Problem, from its origins in 1852 down to its proof in 1976 and the response to that proof. Along the way Wilson develops enough of the mathematics involved to give the reader a broad understanding of how the proof works.

After a brief explanation of the four colour map problem, Wilson goes back to its origins, in a letter in 1852 from Augustus De Morgan to William Hamilton. Arthur Cayley revived interest in the problem in 1878 and a year later came Alfred Kempe's "proof", the flaw in which wasn't discovered for eleven years.

"Kempe's proof of the four-colour theorem was a very good one. It was incorrect, but it was a very good incorrect proof. Not only did it convince Victorian mathematicians for eleven years, but most of the ideas it contained were sound and would form the basis for later assaults on the problem, including the one that was ultimately to prove successful."

Wilson introduces some basic graph theory and some key concepts: a minimal criminal is a map that needs five colours, where no smaller map does; an unavoidable set is a group of configurations at least one of which has to be present in any planar graph; and a reducible configuration is an arrangement of countries that cannot occur in a minimal criminal.

"The rest of this book is concerned with the attempt to find an unavoidable set of reducible configurations. Finding such a set proves the four-color theorem: since the set is unavoidable, every map must contain at least one of the configurations, but each configuration is reducible and so cannot be contained in a minimal criminal."

The eventual proof by Wolfgang Haken and Kenneth Appel, in 1976, used a computer program to check the reducibility of configurations; an updated proof, in 1994, used computers to check the unavoidability of sets of configurations as well. Wilson sticks to the history here, and doesn't venture into philosophical debates, or a broader exploration of computer-assisted proof.

Like a lot of graph theory, Four Colors Suffice is understandable without much if any background in mathematics, though it assumes a certain comfort with mathematical thinking. There is no reason why a motivated high school student couldn't follow it, but it will appeal to mathematicians too: this is a treat for anyone curious about the history of mathematics. Highly recommended.

March 2026

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