32 of 32 people found the following review helpful:
5.0 out of 5 stars
Mathematical Teamwork, And The Philosophy of Proof, April 29, 2003
This review is from: Four Colors Suffice: How the Map Problem Was Solved (Hardcover)
One of the most famous theorems in mathematics is the Four Color Map Theorem. It is wonderfully simple to understand, and interesting to spend time doodling on. Mapmakers like to take a map like, say, the states of the U.S. and color in the states with different colors so they are easily told apart; the theorem states that any such map (or any imaginary map of contiguous regions), no matter how complex, only requires four colors so that no state touches a state of the same color. This is not obvious, but if you try to draw blobs on a sheet of paper that need more than four colors (in other words, five blobs each of which touches all the others along a boundary), you will quickly see that the theorem seems to be true. In fact, ever since the question was mentioned, first in 1852, people have tried to draw maps that needed five colors, many of them very complicated, but no one succeeded. But that isn't good enough for mathematics; it's interesting that no one could do it, but can it be proved that it cannot be done? For over a century, there was no counter-example and yet no proof, but in 1976 there was a proof that has held up, but is controversial because it used a computer. The amazing story of the years of competition and cooperation that finally proved the theorem is told in _Four Colors Suffice: How the Map Problem Was Solved_ (Princeton) by Robin Wilson. This is as clear an explanation of the problem, and the attempts to solve it, as non-mathematicians are going to get, and best of all, it is an account, exciting at times, of the triumphs and frustrations along the way, not just with the final proof, but in all the years leading up to it.
Surprisingly, mapmakers aren't very interested in the problem. It was first mentioned in writing in 1852, and in 1879, Alfred Kempe published one of the most famous proofs in mathematics, famous because it proved the theorem and famous because, although it was accepted for about a decade, it was wrong. Kempe's work was useful, as it was an attack on the problem that others eventually used in different ways, but it did not stand. Percy Heawood published a paper in which he included a diagram that Kempe's method could be used on and for which Kempe's method failed. (Not that more than four colors were needed for the map; it simply showed Kempe's method didn't cover all possibilities.) Heawood built on Kempe's work to prove a five color map theorem, but the four color version proved elusive. There was so much data developed in proofs in the 1960s that computers became essential to handle them. Wolfgang Haken worked on the theorem, and was told by computer experts that his ideas could not be programmed, but programmer Kenneth Appel disagreed. In 1972, Haken and Appel teamed up to work on a computer-aided solution, and in 1976, they announced it. They were rushing, as other map-colorers were coming close to a solution themselves. The proof required a thousand hours of computer time, a hundred pages of summary, a hundred pages of detail, and seven hundred pages of back-up work. The computer printouts for it stacked to four feet high. The long hunt was over, but it was not satisfactory to everyone. The problem is that the computer did so much work on the proof that humans cannot check everything the computer did; some mathematicians, especially older ones, have not accepted this proof, although no significant error has been found.
_Four Colors Suffice_ not only explains the theorem and historic attempts at proofs in a clear fashion, it is an inspiring look at something that is really rather lovable in our species, the pursuit of mathematical knowledge for its own sake. To be sure, the theorem does have practical interest, if not to actual mapmakers, then to road, rail, and communications networks, but it has mainly inspired other aspects of pure mathematics like graph theory and algorithms. There are many stories of cooperation between mathematicians here that make the final conquest of the problem seem like a team effort that has been conducted for over a century. One example: when Haken and Appel needed referees to check their paper, one of them was a mathematician who was bitterly disappointed that his own proof had not scooped them. His work as a referee proved to be conscientious and constructive. This may be a tale of a proof that only a computer could crack, but it is a handsome human success story.
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33 of 34 people found the following review helpful:
5.0 out of 5 stars
Solved, April 14, 2008
This review is from: Four Colors Suffice: How the Map Problem Was Solved (Hardcover)
Review of: "Four Colors Suffice - How the Map Problem Was Solved"
By: Robin Wilson
The four color map theorem is easy to understand and hard to prove.
The four color map theorem states that on a plane, which is divided into non-overlapping contiguous regions, the regions can be colored with four colors in such a way that all regions are colored and no two adjacent regions have the same color. In other words you can color any ordinary map with just four colors.
The proof of the four color theorem is very difficult. It is so difficult that the proof took over a century. The search for a proof was so long and became so complex that some mathematicians speculated that it was impossible. The four color served as one of the first real mathematical challenges posed to mathematics undergraduate students.
The statement of the challenge was deceptively simple. Prove that four colors are sufficient. The statement of the problem is so simple that it seems the solution should be equally simple. It is not simple. In 1976 the four-color theorem was finally demonstrated. The authors of the proof are Kenneth Appel and Wolfgang Haken of the University of Illinois.
The book "Four Colors Suffice" is the story of the century long search for the proof. The effort culminated in a computer program. Appel and Haken restated the problem as a collection of 1,936 types of maps. They had a computer program prove each of these 1,936 forms.
The author succeeds in conveying the excitement of the competition in those final months. This book shows the drama of one of the most exciting episodes of modern mathematics.
See also:
Graphs, Colourings and the Four-Colour Theorem (Oxford Science Publications)
The Four-Color Theorem: History, Topological Foundations, and Idea of Proof
Introduction to Graph Theory (4th Edition)
I thoroughly enjoyed this thoughtful and exciting book.
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21 of 23 people found the following review helpful:
5.0 out of 5 stars
Very nicely done, October 21, 2005
This review is from: Four Colors Suffice: How the Map Problem Was Solved (Hardcover)
I am a mathematician extensively familiar with the Four-Color Theorem and I was impressed by Wilson's book. He knows just what to put in and what to leave out; the narrative has just the right mixture of storytelling and math. If I have one complaint it is that the discharging procedure (part of the proof) is rather glanced over, but I can see how it would be daunting to expose "real" discharging procedures to a non-mathematical audience.
Overall, an entertaining and elegant book.
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