Four Colors Suffice: How the Map Problem Was Solved

by Robin J. Wilson

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.

Genres: Mathematics, Science, Nonfiction, History, Popular Science, Cartography, Computer Science

280 pages, Paperback

First published November 7, 2002

About the author

Robin James Wilson (born December 1943) is a professor in the Department of Mathematics at the Open University, a Stipendiary Lecturer at Pembroke College, Oxford[1] and, as of 2006, professor of geometry at Gresham College, London, where he has also been a visiting professor. On occasion, he guest teaches at Colorado College.
From January 1999 to September 2003 Robin Wilson was editor-in-chief of the European Mathematical Society Newsletter.[2]
He is the son of Harold Wilson, former Prime Minister of the United Kingdom. He has two daughters: Catherine and Jennifer.

Reviews

[Ariel Burbaickij · Rating 3 out of 5]

Books starts to be more and more lacking and explanations get muddled once it gets into the details of Appel-Haken proof, so in chapters 9 and 10 -- author just restricts himself, as it appears, to a heap of anecdotal stories about details of inter-personal relations, where and how people spend their sabbaticals, record of smalltalk at mathematical conferences and such, all at the expenses of meaningful explanations. Additional technical drawback is that book is kept all black-white with poor drawings of 3D objects, requiring additional effort to grasp the ideas behind well 4 and not just 2 colours problem.

[Frank · Rating 3 out of 5]

This got a little hand-wavy there at the end, but I guess that's inevitable.
By and large a slick handling of the challenges faced by all math book for a semi-popular audience.

[Dan · Rating 3 out of 5]

Generally, I find popular mathematics books to be very good. However, I found this book generally lacking. This book provides a general high level overview of how this problem was solved, however it didn't have nearly enough details.

Most of the book is devoted to explaining why the way this problem was solved was controversial (because computers were used to do exhaustive searches of the solution space.) However, I think that this discussion should have been sacrificed for the sake of more detail.

I read this book because I am interested in computer science theory.

[Corvus · Rating 2 out of 5]

It looks like it's about maps, but it's really about mathematical proofs. Even though I hate to do math, I like reading about it. This one was a little too dry and technical for me, though, like reading a textbook that tries to be entertaining. The last few pages, though, when a couple of guys actually come up with a proof for this problem, was interesting. I liked reading about the various reactions to their computer-assisted proof; it's an interesting window into the early years of computers and what people thought of them before everyone had one in their homes.

[Katherine · Rating 4 out of 5]

I really enjoyed reading it! I'm studying The Four Colour Theorem as part of my degree, and found it a really good base to hair my understanding in the fundamental steps taken in order to solve the problem. The mathematical depth of it isn't that great, but this just made it all the more interesting as I preferred reading about the history of it!

[Bas · Rating 4 out of 5]

Four Colors Suffice: How the Map Problem Was Solved - Revised Color Edition

A great exposition on the four color problem and a great gift for the mathematically curious person in your life.

A wonderful addition to any STEM (though in particular mathematics) undergraduate's library. Especially this version, which features an abundance of colors to drive home its main points using wonderfully clear figures and schemes. The mostly chronological view upon the infamous Four Color Problem displays the gradual structurisation of the methods with which mathematicians attempted to climb this great mountain in such a way that each of the ideas may be rediscovered by the reader - a truly amazing feat that is not constructed by accident. The sense of potential rediscovery is one that many students of mathematics describe as being the turning point in their education that convinces them mathematics is something worth pursuing for fun, not just for some vague sense of utility. In that sense, this book may be the greatest gift you could give a young, somewhat mathematically-curious, mind. Thankfully, there are many other books which might also achieve this feat. As such, a true recommendation for a single book should be tailored to the individual's interests. With this in mind, if you (or the person receiving this as a gift) are interested in the process of mathematics, or its history, this book is a great choice. It shows the collaborative effort of many mathematicians over centuries to solve one, seemingly easy and increasingly difficult, problem. Mathematics is a team effort, as this book goes to great lengths to show in an excellent way.

The author does a great job of exposing the problem to a general math audience, though its latter half does appear to give a conflict as to the books' target audience. The book adheres to the "non-mathematical" use of maps instead of graphs, even after explaining the concept of graphs and pointing out that the research covered in the latter half of the book featured graphs and explicitly did not feature maps. This points to a desire to appeal to a broader audience, one who might not be familiar enough with graphs to reason about them, rather than maps. This is a fine choice, but its pairing with free-flowing reasonings built upon multiple earlier chapters filled with (admittedly well-explained) mathematical jargon seems to me to be a conflict. All this to say, if you find yourself taking quite some time to finish the book after having started it, don't fret. Whereas the beginning of the book is clear to follow to anyone with a vague interest in a text associated with the word 'Mathematics', the book's difficulty steadily increases (with some pace!) to where the end almost requires some formal training in mathematics to properly appreciate it, though it may certainly be mostly appreciated to anyone without such training.

[John Frankham · Rating 3 out of 5]

An interesting book about the ‘do maps only need four colours’ question, from its initial discussion in the 1850s, a proof in the 1870s, found after 11 years to be faulty, to its final (?) proof in the 1970s.

Interesting, but really a long Wikipedia article or 100-page essay puffed up into 300+ pages.

The GR blurb below reads as a publishers ‘puff’ - the whole thing is actually interesting but prosaic.

The GR blurb:

‘ 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.’

[Alex]

Popular mathematics books must elide some technical detail to provide an approximation that is logically coherent to the reader, gives a satisfying flavour of the nature of the work done, and accurately represents the true theory. This is an atrociously difficult job and the author acquits himself well. There are some steps that must have seemed obvious to the author that really should have been spelled out - or I just didn’t get it at all - and one paragraph with a confounding troubling accidental misnegation - or I didn’t get it at all - but otherwise it provides a picture of the how this whole edifice was built. My only genuine bugbear was with the “minimal criminals” term which I understand comes from the literature but has to be mentally converted back in to its clear, literal meaning of “minimal counter-example”. I also would’ve appreciated a little more historical context but the anecdotes are charming.

The old early-2000s edition I have is a pleasing time capsule of the eclectic typography of that era. I do miss the age when you could use three different typefaces with an above average amount of white space and it would somehow translate as cleaning high-tech.

[Riley Polland · Rating 4 out of 5]

I wrote my undergraduate dissertation in Mathematics and Statistics on the Four Colour Theorem last year, and this book was an absolute saving grace. The conversational tone kept me engaged throughout, and I even found myself reading chapters completely irrelevant to my dissertation purely out of interest—a rarity for me. It struck the perfect balance between the mathematical history and the theory behind the theorem, making it both informative and enjoyable.

I’ll admit, I struggled to fully grasp the section on Appel and Haken's computer-aided proof, as I feel it got very hand-wavey. This was reflected in the feedback I received for my dissertation. However, I take full accountability for this—my understanding of computer programming is nowhere near as strong as it should be, despite having taken modules on it for the past four years.

I’d recommend this book to anyone interested in graph theory or writing their dissertation on a related topic. The diagrams are clear, concise, and easy to interpret, while the explanations of the theory are engaging and insightful. It’s a fantastic resource that brings a complex subject to life.

[Clive F · Rating 4 out of 5]

This book is a clear and entertaining account of the long history of the attempts to provr four colour theorem - that any map on can be coloured with at most four colour, such that no countries with a common border have the same colour. Although there are lots of interesting characters and asides, this is not a book for the mathematically faint of heart: in order to understand the approach that finally proved the conjecture, Robin Wilson takes you pretty far into the dense woods. If you enjoy following along, and are prepared (as I was) to skim over the more complex parts, you will still come away with a good appreciation of how it happened. You will also perhaps understand why many mathematicians at the time were skeptical about the proof: it needed over 1000 hours of computer time to complete the proof, and the approach is too complex to check by hand. Does that count as a "proof"? Nowadays, when computers are routinely used by mathematicians to check their own work, people would have fewer doubts, but in the 1970s, many felt this to be a real issue.

[Josh · Rating 3 out of 5]

This book mixes math and history in a well told story of how a seemingly simple question that took a series of mathematicians over a hundred years to solve. I think you can read it and gloss over the mathematics, but I chose to attempt to understand the concepts as they were presented. With a good mathematics background, but no topology or graph theory, some concepts were difficult and progress was made a page at a time. Not all concepts are discussed in depth, and the author presents drawings and examples to help you along. Aside from the math, the story follows a series of mathematicians who collaborate, compete, and feud. There are also cute antecedents like the traffic cop who lets a mathematician off with a warning because he 'solved the Heawood conjecture'.

[Penny · Rating 4 out of 5]

Full Title – Four Colours Suffice, how the Map Problem was Solved. The map problem being to prove that when colouring a map to ensure that no two areas are the same colour, four colours suffice. The search for proof started back in the mid 19th century but it took more than a century to prove it, although long before that everyone knew that indeed, four colours would do the job. Lewis Carroll and the Bishop of London were just two of the many professionals and amateurs who got caught up in the frenzy. Eventually the problem was solved with 1200 hours of computer time and only later were the programmes used, verified. While I enjoyed it and followed the broad strokes, you probably need to be much more of a mathematician than I am to fully benefit from reading it.

[Bence · Rating 4 out of 5]

Pretty nice popular science book including thorough history of the four color problem, understandable explanation of most classical proofs, and an exciting account of what role computer assisted methods play in this problem in particular and in modern mathematical proofs in general, with fair presentation of both sides of the debate.

[Rebecca · Rating 1 out of 5]

It's not a book I would typically pick up, I thought I'd take a risk but it didn't pay off. I thought it would have a bit more about map making/history. Far too much maths for me, and all of the diagrams in different shades of grey rather than the colours described which was difficult to follow. If you like problem solving and maths theory then this is the book for you!

[Jonah Smith · Rating 3 out of 5]

Idea was cool but like many of these books waffled on a bit much. Less interested in the random mathematicians biographies sprinkled throughout, but the majority of the book where it solved the problem was great with some cool maths.

[Ruxandra · Rating 4 out of 5]

We now know that the answer to the Four Colour Problem is indeed “four.” This is the smallest number of distinct colours that will colour any map drawn on the plane so that regions sharing a common length of boundary have different colours.

[Jessada Karnjana · Rating 4 out of 5]

เราลงสีแสดงอาณาเขตบนแผนที่เพียงสี่สีได้ไหม โดยไม่ให้ประเทศที่อยู่ติดกันมีสีเดียวกัน

[Katherine · Rating 4 out of 5]

A book about four colors really needs to be printed in more than two. Otherwise fascinating!

[Alfred Comprehension Cunnington · Rating 3 out of 5]

Another man’s child has the eyes of the woman I loved, anyway this was a good read but kinda spoiled the plot in the title

[James · Rating 4 out of 5]

Quite the kerfuffle!

[Aly · Rating 4 out of 5]

I know this is kind of a niche book, but if you’re into math and all the drama that can come with it, this book is for you.

[Renz Ogena · Rating 3 out of 5]

Its subject is very interesting, especially for readers with some background on graph theory, but Four Colors Suffice: How the Map Problem Was Solved - Revised Color Edition is not as well written as other popular mathematics books I've read. The book could have benefitted from more stories on the human-interest side of problem. The four-color theorem was a beast of a mathematical problem to slay (it needed computers for its proof): What urged, or fueled, those mathematicians to work on this problem? The book also ended rather abruptly. I thought the author was going to highlight Wolfgang Haken and Kenneth Appel's successful proof, but it felt like that last chapter was just another regular episode in the problem's 150-year history. I also expected the author to highlight the controversial nature of Haken and Appel's computer-assisted proof and even share his own opinions on whether their proof is a proof at all.

But on the plus side, the author was successful in translating all the highly technical mathematical stuff to lay audience–level explanation. (Well, that's the goal of a popular mathematics book.) I also thought it wise of the author to stick to coloring maps and not switch to coloring graphs near the middle of the book, when the four color problem was reworded in graph theory. The illustrations were also of great help in understanding the mathematical techniques that were used to attack and solve the problem.

[Jay French · Rating 3 out of 5]

The history of the attempts to prove the four color theorem. The book starts with the initial definition of the problem and conjecture, and works through the different attempts made until the computer generated proof in the late 70s by Appel and Haken. I found the level of this book to be a step more scholarly than many other current popular science books out there, but certainly within the realm of those with a general interest in mathematics. I found the book explained complex subjects with many useful diagrams, with some interesting history and a few math jokes thrown in. I only got lost once, and that is better than expected. My interest in this was that I had taken an undergraduate combinatorics class taught by Prof. Appel about 25 years ago, and I believe he had a single hour of class to describe his proof. It intrigued me then, and this book has given more detail on the history of the problem and the race to solve it as computers became powerful and economic enough to handle the load.

[Attila · Rating 4 out of 5]

The Four Colour Theorem has always fascinated me. The statement is very simple, yet the implications are so complex that you will soon find yourself at a loss. When I was a kid, I used to fill entire notebooks with drawings of coloured maps, trying to find a proof, or rather a counter-example - usually during boring school classes, which made the teacher get mad at me. Of course, the theorem had been proven for more than 30 years by then, but who cared? :)

This book describes the 100-year struggle for a proof, the various methods, the triumphs and failures, the related theorems and properties. It also poses the question whether a computer-aided proof, unverifiable by hand calculations, has as much validity as a "classic" one. Recommended to everyone interested in lesser known conjectures in mathematics.

[Amar Pai · Rating 3 out of 5]

After hundreds of years the 4-Color theorem was proved; but the proof involved breaking the problem down into thousands of cases, then verifying each of them by computer.

Though you would expect them to applaud the solution of an age-old, challenging problem, mathematicians reacted coolly to the announcement of the proof. They thought it too ugly, in light of Hardy's maxim that "there is no permanent place in the world for ugly mathematics." There was also concern about the computer-aided nature of the proof. Could a computer-assisted proof be trusted? Could a proof that no human had fully verified "by hand" be treated as a real proof?

It's an interesting story that revolves around the nature of mathematical proof and what is meant by mathematical beauty.

[Chris · Rating 2 out of 5]

Given that it took over a year for me to get through this and that I actually am interested in graph theory, I don't think I can really give this more than two stars.

The extended reading time caused me to lose both the narrative thread and the relevant technical details, but that's just because I got bored with the author's presentation. Additionally, I don't think the layout of the pages was all that great, which is certainly a funny thing to be aware of after having completed a book.

[William · Rating 4 out of 5]

Interesting history of the four-color problem for planar (and spherical) maps, with additional discussion of maps on other topological surfaces. But what I found most intriguing was the discussion at the end, which explores the acceptance of a computerized mathematical "proof". With many theories that may be the only way things can be settled, and I am sure there will continue to be controversy about what constitutes an acceptable solution.

[Robert · Rating 3 out of 5]

Goes into quite a lot of the mathematical details as well detailing all the essential history of the problem.

The four color problem is much more approachable than most famous math problems which of course helps considerably.

After reading this book carefully, you probably couldn't quite completely understand the actual proof, but you'd be 80% of the way there.

[Steve Gross · Rating 4 out of 5]

A handsome, colorful book outlining the history and solution of the famous Four Color Theorem. Gets somewhat technical.

[Chaman Gupta]

Its gr8