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Four Colors Suffice: How the Map Problem Was Solved Paperback – 7 Nov 2004
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At first glance Four Colours Suffice seems like such an easy thing to prove. However big and complicated the map, four colours are enough to distinguish each country from its neighbours. How do we prove that only four colours are needed? Once we realise that, if four countries all share borders with each other, then one country must be enclosed by the other three (try it), we seem to be most of the way there. But things turned out to be not quite so simple. Robin Wilson might balk at the idea that his sardonic and lively account of the problem and its solution is in any way farcical--as, indeed, might the dedicated mathematicians and keen amateurs whose 150 years of work he describes. But if the way an apparently simple problem throws out poisoned blossoms of complication, confusion and embarrassment is your definition of farce, then this story surely fits the bill. Proving the four-colour conjecture turned out to be heinously difficult, and has at last been achieved--and that in the ugliest way imaginable--only with the aid of a computer.
This, we can see now, was a landmark moment in mathematics: the moment we realised that there are proofs out there so complicated, that publishing them in full is impractical, working through them by hand is impossible, and explaining them to the public requires writers of a very special stamp indeed. (Robin Wilson, I should add, is most definitely one of them.) The publishers, in deciding to make a black-and-white book out of a colour problem, have not only done justice to Wilson's illustrations, but have also created one of the most visually arresting science books around. --Simon Ings --This text refers to an out of print or unavailable edition of this title.
"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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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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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.
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.
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.
Overall, an entertaining and elegant book.
This is a very readable history of the problem, from its phrasing in the mid-nineteenth century up to its mind-boggling proof in 1976, and a bit beyond that. It captures brief bits of the lives of the mathematicians who worked on it, as well as the furor over Appel and Haken's computer-based proof. Why was this so revolutionary? Because it was the first proof with steps that could never be checked by a human reader. Some people claimed the proof was incomplete until the programs were proven correct. Others stated that, if it couldn't be proven to a human mind, then nothing was really proven at all. Yet others objected to the proof's lack of mathematical elegance. It wasn't a scalpel that cut neatly to the heart of the problem, but a bulldozer hauled away huge buckets of potential counterexamples. A non-mathematician like me has to wonder: did this pave the way for acceptance of the 15,000-page "Classification theorem"? Although that theorem might not have been proven with computer assistance, its sheer mass is certainly similar.
The book does get a bit mathematical in places. The casual (and maybe not-so-casual) reader will be tempted to skip bits, and won't really lose the narrative thread by doing so. And, since the original proof is nearly 30 years old now, some of the excitement has worn off it. Even so, it's an enjoyable history of a problem that resisted attack for so long, and the remarkable attack that finally felled it.
And it leave me wondering: do my younger colleagues live in a world richer because of the radical solution, or poorer for the absence of such a wonderful mystery?
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