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.
This new edition features many color illustrations. It also includes a new foreword by Ian Stewart on the importance of the map problem and how it was solved.
"The simplicity of the four-color conjecture is deceptive. Just how deceptive is made clear by Robin Wilson's delightful history of the quest to resolve it... Four Colors Suffice is strewn with good anecdotes, and the author ... proves himself skillful at making the mathematics accessible."--Jim Holt, New York Review of Books "Wilson's lucid history weaves together lively anecdotes, biographical sketches, and a non-technical account of the mathematics."--Science "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 "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 "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 "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 "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 "Wilson gives a clear account of the proof ... enlivened by historical tales."--Alastair Rae, Physics World "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
Foreword by Ian Stewart xi Preface to the Revised Color Edition xiii Preface to the Original Edition xv 1 The Four-Color Problem 1 What Is the Four-Color Problem? | Why Is It Interesting? | Is It Important? | What Is Meant by "Solving" It? | Who Posed It, and How Was It Solved? | Painting by Numbers | Two Examples 2 The Problem Is Posed 12 De Morgan Writes a Letter | Hotspur and the Athenaeum | Mobius and the Five Princes | Confusion Reigns 3 Euler's Famous Formula 28 Euler Writes a Letter | From Polyhedra to Maps | Only Five Neighbors | A Counting Formula 4 Cayley Revives the Problem ... 45 Cayley's Query | Knocking Down Dominoes | Minimal Criminals | The Six-Color Theorem 5 ... and Kempe Solves It 55 Sylvester's New Journal | Kempe's Paper | Kempe Chains | Some Variations | Back to Baltimore 6 A Chapter of Accidents 71 A Challenge for the Bishop | A Visit to Scotland | Cycling around Polyhedra | A Voyage around the World | Wee Planetoids 7 A Bombshell from Durham 86 Heawood's Map | A Salvage Operation | Coloring Empires | Maps on Bagels | Picking Up the Pieces 8 Crossing the Atlantic 105 Two Fundamental Ideas | Finding Unavoidable Sets | Finding Reducible Configurations | Coloring Diamonds | How Many Ways? 9 A New Dawn Breaks 124 Bagels and Traffic Cops | Heinrich Heesch | Wolfgang Haken | Enter the Computer | Coloring Horseshoes 10 Success! 139 A Heesch-Haken Partnership? | Kenneth Appel | Getting Down to Business | The Final Onslaught | A Race against Time | Aftermath 11 Is It a Proof? 157 Cool Reaction | What Is a Proof Today? | Meanwhile ... | A New Proof | Into the Next Millennium | The Future Chronology of Events 171 Notes and References 175 Glossary 187 Picture Credits 193 Index 195