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Four Colors Suffice: How the Map Problem Was Solved (2002)

av Robin WILSON

MedlemmarRecensionerPopularitetGenomsnittligt betygOmnämnanden
291889,834 (3.43)20
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.… (mer)
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Interesting, and well-presented. It's more about math and less about maps than I'd thought, but those are both interests of mine so it worked. The history of a math problem - not just "can every map be colored with no more than four colors so that no two countries that share a border have the same color" - but can that be _proved_. Chapter after chapter, he'd mention that it had been proved that the theory was true for any map with no more than...20, 50, 150 countries. But is it true for _all_ maps? The question kept getting more abstract - from maps, to geometric shapes, to graphs of connected points. There were a lot of proofs that were eventually shown not to be proofs - holes in their logic. The final answer (the question was first posed in the mid-1800s; the answer came in 1976) required a computer to work the proof, and was greeted with a good deal of skepticism thereby. Was it really a proof if a human hadn't done all the steps? I found that part particularly interesting. It's a question I'd heard of vaguely, and I'm glad I read this; I now understand the question, at least, though the details of the math began to escape me near the end. Definitely worth reading. ( )
  jjmcgaffey | Mar 3, 2021 |
Indeholder "Preface", "1. The four-colour problem", "What is The Four-colour 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", "De Morgan Writes a letter | Hotspur and the Athenaeum | Mobius and the five princes | Confusion reigns", "3. Euler's famous formula", "Euler writes a letter | From polyhedra to maps | Only five neighbours | A counting formula", "4. Cayley revives the problem ...", "Cayley's query | Knocking down dominoes | Minimal criminals | The six-colour theorem", "5. ... and Kempe solves it", "Sylvester's new journal | Kempe's paper | Kempe chains | Some variations | Back to Baltimore", "6. A chapter of accidents", "A challenge for the bishop | A visit to Scotland | Cycling around polyhedra | A voyage around the world | Wee planetoids", "7. A bombshell from Durham", "Heawood's map | A salvage operation | Colouring empires | Maps on doughnuts | Picking up the pieces", "8. Crossing the Atlantic", "Two fundamental ideas | Finding unavoidable sets | Finding reducible configurations | Colouring diamonds | How many ways?", "9. A new dawn breaks", "Doughnuts and traffic cops | Heinrich Heesch | Wolfgang Haken | Enter the computer | Colouring horseshoes", "10. Success! ...", "A Heesch-Haken partnership? | Kenneth Appel | Getting down to business | The final onslaught | A race against time | Aftermath", "11. ... but is it a proof?", " Cool reaction | What is a proof today? | Meanwhile ... | A new proof | The future ...", "Notes and references", "Chronology of events", "Glossary", "Picture credits", "Index".

Vældigt underholdende fortælling om firfarveteoremet og hvordan man kan slå det ihjel med en computer. ( )
  bnielsen | Jun 19, 2012 |
The title Four Colors Suffice refers to a simple mathematics problem that was first discussed in the 1850's. Namely, how many colors does it take to color a map so that no two bordering countries have the same color? The answer appeared to be 4 colors, but proving that took 150 years and required the use of computers. This book traces the history of the problem from the first publications to the proof of it in 1976, plus a discussion of the validity of a computer proof.

The writing overall was fine -- not brilliant, not poor, but somewhere in the middle. For some of the mathematics, the discussion was a bit unclear and hard to follow. (Yes, the methods used by various attempted proofs are difficult topics, but even having a decent mathematical background, I had to reread several pages to understand what he was trying to say.) The diagrams were terrific! The history was well researched with notes and a bibliography.

On the down side, the book lacked somewhat. As mentioned above, some areas were obtuse. Also some topics, such as why this is an important problem and not just a "let's see if we can prove it" type of problem, were alluded to but never really discussed. Wilson stated several places that the math used to solve this problem led to other important results, such as...well, he never says. Also, the final section as to whether or not the proof is valid felt like it was added later, without much enthusiasm. Proof by exhaustive computer search is a very interesting question -- is it really a proof? I understand that a thorough discussion would involve a lot of discussion of computer programming, but this book (as evidenced by the level of math when discussing the historic proofs) is aimed for a mathematical literate audience that could understand the basics of the computer issues.

Overall, the history part of the book was fine, but the "we have a proof" section was lacking. I will probably read it again at sometime, but not terribly soon. ( )
1 rösta LMHTWB | Mar 31, 2012 |
This is a relatively brief (228 pages with lots of illustrations) and coherent history of the 4-color map problem. A map is what you think it is, a surface with boundaries between regions. Other rules: the map may be on a sphere but it may not be on a torus (donut) or other 3D form with a hole, each region is independent (so not a map of the world in which some countries are split into parts that must be the same color), and the boundary is defined as more than a single point (n regions that meet in the center of a pie do not require n colors to be distinct). The problem made its appearance in 1852, when a student asked a professor, who asked a friend... and remained unsolved until 1976. It rose to notoriety because it's a simple question that was difficult to answer, and it was worth tackling because effort on any one problem can yield results that apply to other problems. One strategy was to prove the impossibility of a "minimal criminal": a minimal counterexample with a configuration of regions such that (a) the configuration _cannot_ be colored with 4 colors, but (b) any sub-configuration (the same configuration with one or more regions removed) _can_ be colored with 4 colors. A configuration might be a square (a region surrounded by four others), or a cluster of three pentagons (three regions each surrounded by five others), etc. Various mathematicians over decades contributed proofs regarding specific configurations of increasing complexity, and different methods of determining their properties. The strategy that eventually led to a proof was to find an "unavoidable set" (a set of regions one or more of which _must_ be in any map) of "reducible configurations" (configurations that may not be in a minimal criminal). The proof, by Kenneth Appel and Wolfgang Haken, consisted of nearly 2000 such configurations verified by a computer program, and was disturbing for its inelegance and non-transparency, to the extent that one math department deemed Appel and Haken a bad influence and barred them from meeting students. The proof has since been streamlined, but not fundamentally changed. The book is nicely presented in chronological order, with concepts succinctly explained and helpfully illustrated, especially in the earlier stages when things were still relatively straightforward. It becomes less clear in the later stages, but this is not the fault of the author, as the details are far too numerous for this sort of publication.

(read 27 Jun 2011)
1 rösta qebo | Jul 16, 2011 |
Excellent: On the surface the four colour theorem may seem like a dull story for a book like this. But Robin Wilson does a great job of making this a fascinating story. I was hooked from the start. It doesn't get too technical but nevertheless you do get an idea, though perhaps not a complete understanding, of how and why the proof worked. A great read for any 'popular science' fan. If you enjoy Simon Singh, Ian Stewart and the like, you'll like this too.
  euang | Sep 1, 2008 |
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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.

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