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

Robin Wilson
4.4 out of 5 stars  See all reviews (5 customer reviews)

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Book Description

Nov. 7 2004

On October 23, 1852, Professor Augustus De Morgan wrote a letter to a colleague, unaware that he was launching one of the most famous mathematical conundrums in history--one that would confound thousands of puzzlers for more than a century. This is the amazing story of how the "map problem" was solved.

The problem posed in the letter came from a former student: What is the least possible number of colors needed to fill in any map (real or invented) so that neighboring counties are always colored differently? This deceptively simple question was of minimal interest to cartographers, who saw little need to limit how many colors they used. But the problem set off a frenzy among professional mathematicians and amateur problem solvers, among them Lewis Carroll, an astronomer, a botanist, an obsessive golfer, the Bishop of London, a man who set his watch only once a year, a California traffic cop, and a bridegroom who spent his honeymoon coloring maps. In their pursuit of the solution, mathematicians painted maps on doughnuts and horseshoes and played with patterned soccer balls and the great rhombicuboctahedron.

It would be more than one hundred years (and countless colored maps) later before the result was finally established. Even then, difficult questions remained, and the intricate solution--which involved no fewer than 1,200 hours of computer time--was greeted with as much dismay as enthusiasm.

Providing a clear and elegant explanation of the problem and the proof, Robin Wilson tells how a seemingly innocuous question baffled great minds and stimulated exciting mathematics with far-flung applications. This is the entertaining story of those who failed to prove, and those who ultimately did prove, that four colors do indeed suffice to color any map.

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The four-color conjecture, formulated in 1852, was among the most popular unsolved problems in mathematics. Amateurs and professionals alike succumbed to its allure. It is, simply stated: four colors are all that is needed to fill in any map so that neighboring countries are always colored differently. That the proof, which was completed in 1976, consumed a thousand pages and gobs of computer time hints at the hidden complexity encountered by those attempting to solve it. Recreational mathematicians will find Wilson's history of the conjecture an approachable mix of its technical and human aspects, in part because the math involved is understandable even to able middle-schoolers. The conjecture seemed a snap to its originator, one Francis Guthrie, but his claimed proof has never surfaced; those proofs that did surface, prior to the final breakthrough by Kenneth Appel and Wolfgang Haken, contained fatal errors. Wilson explains all with exemplary clarity and an accent on the eccentricities of the characters, Lewis Carroll among them. Gilbert Taylor
Copyright © American Library Association. All rights reserved --This text refers to the Hardcover edition.


"Wilson's lucid history weaves together lively anecdotes, biographical sketches, and a non-technical account of the mathematics."--Science

"An attractive and well-written account of the solution of the Four Color Problem. . . . It tells in simple terms an exciting story. It . . . give[s] the reader a view into the world of mathematicians, their ideas and methods, discussions, competitions, and ways of collaboration. As such it is warmly recommended."--Bjarne Toft, Notices of the American Mathematical Society

"A thoroughly accessible history of attempts to prove the four-color theorem. Wilson defines the problem and explains some of the methods used by those trying to solve it. His descriptions of the contributions made by dozens of dedicated, and often eccentric, mathematicians give a fascinating insight into how mathematics moves forward, and how approaches have changed over the past 50 years. . . . It's comforting to know that however indispensable computers become, there will always be a place for the delightfully eccentric mathematical mind. Let's hope that Robin Wilson continues to write about them."--Elizabeth Sourbut, New Scientist

"Recreational mathematicians will find Wilson's history of the conjecture an approachable mix of its technical and human aspects. . . . Wilson explains all with exemplary clarity and an accent on the eccentricities of the characters."--Booklist

"Robin Wilson appeals to the mathematical novice with an unassuming lucidity. It's thrilling to see great mathematicians fall for seductively simple proofs, then stumble on equally simple counter-examples. Or swallow their pride."--Jascha Hoffman, The Boston Globe

"Wilson gives a clear account of the proof . . . enlivened by historical tales."--Alastair Rae, Physics World

"Earlier books . . . relate some of the relevant history in their introductions, but they are primarily technical. In contrast, Four Colors Suffice is a blend of history anecdotes and mathematics. Mathematical arguments are presented in a clear, colloquial style, which flows gracefully."--Daniel S. Silver, American Scientist

"Wilson provides a lively narrative and good, easy-to-read arguments showing not only some of the victories but the defeats as well. . . . Even those with only a mild interest in coloring problems or graphs or topology will have fun reading this book. . . . [It is] entertaining, erudite and loaded with anecdotes."--G.L. Alexanderson, MAA Online

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First Sentence
Solving any type of puzzle, such as a jigsaw or crossword puzzle, can be enjoyed purely for relaxation and recreation, and certainly the four-colour problem has provided many hours of enjoyment - and frustration - for many people. Read the first page
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Customer Reviews

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Most helpful customer reviews
5.0 out of 5 stars If you like mathematics you'll like this! Jan. 1 2004
By K. D.
This book deserves every star it gets from me! The quality of the writing startled me since afterall it was written by a mathematician. The four color problem was presented in a fascinating manner. Brief histories on the people who worked on the problem were very interesting and added flavor. Also, the book was not dry. It had nice anecdotes and a sense of humor ("humour"-see below). Diagrams and formulas were presented in a very clear concise manner to anyone who has a good geometrical foundation or higher.
My nitpicky thoughts that would probably never bother anyone else:
The title is deceptive. "Colors" is spelled "colour" in the actual text.
Also, the example of the shape of football was used in the text. What he meant was a soccerball. Completely different shapes come to mind.
My last nitpicky thing is on the same British/American culture line of reasoning. Apparently the Brit's use a term called "overleaf" I finally realized that he meant "on the next page" about half way through. Other than the regional differences in language, the work was presented beautifully. I plan on looking for anything else Mr. Wilson has written. I've always loved math but never really liked reading about it. This book has definitely sparked an interest in reading more like this!
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4.0 out of 5 stars When is a proof a proof? Nov. 17 2003
I enjoyed this book very much - it is fresh in expression and introducing complex ideas - even humourous at times! And yet for all that there is a sense of some lack of achievement also, although this may not be a failing of Mr Wilson.
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).
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4.0 out of 5 stars A tale of many great features of mathematics Sept. 10 2003
By Charles Ashbacher TOP 1000 REVIEWER
The history of the four color problem is one that illuminates much of what makes mathematics such a great topic to explore and was the first instance of a whole new movement in mathematics. It started with a letter from Augustus De Morgan in 1852, where he asks a question that was first asked of him by a student.
"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.
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5.0 out of 5 stars Mathematical Teamwork, And The Philosophy of Proof April 29 2003
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.
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