21. The Four Colour Theorem Theorem 2 four colour theorem Every planar map with regions of simple borders can Theorem 3 four colour theorem Every loopless planar graph admits a http://db.uwaterloo.ca/~alopez-o/math-faq/mathtext/node27.html

Next: The Trisection of an Up: Famous Problems in Mathematics Previous: Famous Problems in Mathematics The Four Colour Theorem Theorem 2 [Four Colour Theorem] Every planar map with regions of simple borders can be coloured with 4 colours in such a way that no two regions sharing a non-zero length border have the same colour. An equivalent combinatorial interpretation is Theorem 3 [Four Colour Theorem] Every loopless planar graph admits a vertex-colouring with at most four different colours. This theorem was proved with the aid of a computer in 1976. The proof shows that if aprox. 1,936 basic forms of maps can be coloured with four colours, then any given map can be coloured with four colours. A computer program coloured these basic forms. So far nobody has been able to prove it without using a computer. In principle it is possible to emulate the computer proof by hand computations. The known proofs work by way of contradiction. The basic thrust of the proof is to assume that there are counterexamples, thus there must be minimal counterexamples in the sense that any subset of the graphic is four colourable. Then it is shown that all such minimal counterexamples must contain a subgraph from a set basic configurations. But it turns out that any one of those basic counterexamples can be replaced by something smaller, while preserving the need for five colours, thus contradicting minimality.

22. The Four Colour Theorem A recent simplification of the four colour theorem proof, by Robertson, Sanders, The Four Color Theorem Assault and Conquest T. Saaty and Paul Kainen. http://db.uwaterloo.ca/~alopez-o/math-faq/node56.html

Next: The Trisection of an Up: Famous Problems in Mathematics Previous: Famous Problems in Mathematics The Four Colour Theorem An equivalent combinatorial interpretation is This theorem was proved with the aid of a computer in 1976. The proof shows that if aprox. 1,936 basic forms of maps can be coloured with four colours, then any given map can be coloured with four colours. A computer program coloured these basic forms. So far nobody has been able to prove it without using a computer. In principle it is possible to emulate the computer proof by hand computations. The known proofs work by way of contradiction. The basic thrust of the proof is to assume that there are counterexamples, thus there must be minimal counterexamples in the sense that any subset of the graphic is four colourable. Then it is shown that all such minimal counterexamples must contain a subgraph from a set basic configurations. But it turns out that any one of those basic counterexamples can be replaced by something smaller, while preserving the need for five colours, thus contradicting minimality. The number of basic forms, or configurations has been reduced to 633.

23. Connotea: Merryprankster's Bookmarks Matching Tag "Four Colour Theorem" Users who used four colour theorem . Merryprankster EXPORT LIST RSS ?Merryprankster s bookmarks matching tag four colour theorem http://www.connotea.org/user/Merryprankster/tag/Four Colour Theorem

SEARCH tags users My Library Registration Home Latest News ... Contact Us Merryprankster's tags: By Usage ageing AIDS AIDS treatment ... XML feed Users who used "Four Colour Theorem": Merryprankster Merryprankster's bookmarks matching tag "Four Colour Theorem" New Scientist Breaking News - Computer generates verifiable mathematics proof info http://www.newscientist.com/article.ns?id=dn7286 Posted by Merryprankster to Four Colour Theorem mathematics New Scientist on Wed Apr 20 2005 at 13:26 UTC Toolbox Add a bookmark Create a new group Create a tag note Rename a tag ... Edit my registration

24. Connotea: Bookmarks Matching Tag "Four Colour Theorem" Users who used four colour theorem . Merryprankster EXPORT LIST RSS ?Bookmarks matching tag four colour theorem . This is a list of the articles and http://www.connotea.org/tag/Four Colour Theorem

SEARCH tags users My Library Registration Home Latest News ... Contact Us Users who used "Four Colour Theorem": Merryprankster Bookmarks matching tag "Four Colour Theorem" This is a list of articles and links that have been posted by Connotea users using the tag "Four Colour Theorem". An RSS feed of the latest entries is available by clicking on the icon above. To add resources to this collection, please register New Scientist Breaking News - Computer generates verifiable mathematics proof info http://www.newscientist.com/article.ns?id=dn7286 Posted by Merryprankster to Four Colour Theorem mathematics New Scientist on Wed Apr 20 2005 at 13:26 UTC Toolbox Add a bookmark Create a new group Create a tag note Rename a tag ... Edit my registration

25. LogBlog: Four Color Theorem Verified In Coq - Richard Zach's Logic And Philosoph Four Color Theorem Verified in Coq. Georges Gonthier (MS Research, Cambridge)has a paper up entitled A computerchecked proof of the four colour theorem. http://www.ucalgary.ca/~rzach/logblog/2005/04/four-color-theorem-verified-in-coq

@import url("http://www.blogger.com/css/blog_controls.css"); @import url("http://www.blogger.com/dyn-css/authorization.css?blogID=7108230"); LogBlog Logic. Philosophy. Other Fun Stuff. Thursday, April 21, 2005 Four Color Theorem Verified in Coq Georges Gonthier (MS Research, Cambridge) has a paper up entitled "A computer-checked proof of the Four Colour Theorem." The original proof of the theorem by Appel and Haken relied on computer programs checking a very large number of cases, and raised some important conceptual and philosophical issues (see Tymoczko, " The four-color theorem and its philosophical significance Journal of Philosophy 76 (1979) 57-83 and reprinted in his collection New Directions in the Philosophy of Mathematics ). The new work has formalized a more recent version of the proof and verified it in the proof checker Coq . Here's how Gonthier describes the contribution this is making: Formalizing and verifying theorems (by computer) seems to be a hot topic, and might raise interesting questions for the philosophy of mathematics; see Jeremy Avigad's work on formalizing mathematics in Isabelle (Via Lambda the Ultimate UPDATE: More links here and see also Brian's post posted by Richard Zach @ 10:43 AM 1 comments postCountTB('111410288876838231');

26. FourColourTheorem four colour theorem Four color theorem (English). Search for Four colourtheorem OR Four color theorem in NRICH PLUS maths.org Google http://thesaurus.maths.org/mmkb/entry.html?action=entryByConcept&id=3028

27. Four Colour Theorem The four colour theorem states that any planar (flat 2d) map needs only 4 colours to This fact leads directly to the proof of the four colour theorem http://www.users.globalnet.co.uk/~perry/maths/fourcolour/fourcolour.htm

Four Colour Theorem [Home] [Other Maths] Four Colour Theorem The four colour theorem states that any planar (flat 2d) map needs only 4 colours to be coloured in. There is only one rule, adjacent regions must be in different colours. Regions that are connected at a point are not adjacent. The above image shows a skeleton map that needs colouring, and two possible colourings. There are generally many alternatives using only 4 colours, and the Four Colour theorem asks 'Is there a map that we can create that requires 5 colours?'. The first colouring is created using an agressive technique - a colour is applied to a random region, and then, again randomly, to as many regions as possible. Once exhausted, another colour is used. The second counts the regions - 12 in this case, and then creates 3 regions of each colour. This raises an important point. We can colour the maps using 5 colours, but we only NEED four. And is it possible to create a scenario where 4 colours does not seem to be enough, but there will be an different pattern that does colour the map in four. A simple proof of this theorem has eluded mathematicians for centuries, but here is a proof that the four colour theorem is true, and that there is no planar map that needs 5 or more colours.

28. About "A New Proof Of The Four Colour Theorem" , An article presenting a proof of the four color theorem that usesgroup theory and Steiner systems, illustrated using a map of Madhya Pradesh...... http://mathforum.org/library/view/16622.html

A New Proof of The Four Colour Theorem Library Home Full Table of Contents Suggest a Link Library Help Visit this site: http://www.geocities.com/dharwadker/ Author: Ashay Dharwadker Description: An article presenting a proof of the four color theorem that uses group theory and Steiner systems, illustrated using a map of Madhya Pradesh and adjoining states in India. Introduction; Map Colouring; Steiner Systems; Eilenberg Modules; Hall Matchings; Riemann Surfaces; Main Construction; References. Levels: Research Languages: English Resource Types: Articles Math Topics: Group Theory Graph Theory Home The Math Library ... Help http://mathforum.org/

29. Math Forum - Ask Dr. Math A recent simplification of the four colour theorem proof, by Robertson, Sanders, The Four Color Theorem Assault and Conquest, T. Saaty and Paul Kainen. http://mathforum.org/dr.math/problems/satyadeeppc.8.21.96.html

Associated Topics Dr. Math Home Search Dr. Math Four-Color Theorem - sci-math faq Date: 8/21/96 at 7:2:31 From: Satyadeepcp Subject: Four-Colour Theorem Do we need more than 4 colors to color a 2-dimensional map? Date: 8/27/96 at 20:40:27 From: Doctor Jodi Subject: Re: Four-Colour Theorem Here's an answer to your question from the sci-math faq ( http://daisy.uwaterloo.ca/~alopez-o/math-faq/math-faq.html http://mathforum.org/dr.math/ Associated Topics High School Discrete Mathematics Search the Dr. Math Library: Find items containing (put spaces between keywords): Click only once for faster results: [ Choose "whole words" when searching for a word like age. all keywords, in any order at least one, that exact phrase parts of words whole words Submit your own question to Dr. Math Math Forum Home Math Library Quick Reference ... Math Forum Search Ask Dr. Math TM http://mathforum.org/dr.math/

30. New Scientist Breaking News - Computer Generates Verifiable Mathematics Proof The four colour theorem, proposed by Francis Guthrie in 1852, states that anyfour colours are the minimum needed to fill in a flat map without any two http://www.newscientist.com/article.ns?id=dn7286

31. The Four Colour Theorem is there an actual equation for the four colour map theorem? i cannot find itanywhere!! if so can you give a breif description on how it works!! http://mathcentral.uregina.ca/QQ/database/QQ.09.02/rhonda1.html

Name: rhonda Who is asking: Student Level: Secondary Question: is there an actual equation for the four colour map theorem? i cannot find it anywhere!! if so can you give a breif description on how it works!! Thank you Hi Rhonda, The theorem is just that, a theorem, and it proves that any map in the plane can be coloured properly with just 4 colours. That is not the same as saying that there is an efficient algorithm to 4-colour any planar map; it just assures us that it can actually be done. Thus unless your map is quite small and not too complicated it may be a very difficult task indeed to actually colour it properly with 4-colours. Cheers, Penny Go to Math Central To return to the previous page use your browser's back button.

32. Proof By Self-checking Software: The Four Colour Problem | The Register development tools to make computing in general more reliable and trustworthy, he said. For more on the history of the four colour theorem, click here. http://www.theregister.co.uk/2005/04/18/four_colour_self_checking/

Skip to content Biting the hand that feeds IT Channel Register Cash 'n' Carrion Events Downloads SetPromptText('q', 'Search'); Enterprise Software Personal Internet ... Science Reg Shops Reg Mobile Gadgets Reg Books Hosting Reg Offers News Services Reg Newsletters Week's Headlines Reg Mobile Reg Archive ... US Edition Top Stories Hackers are all B'stards now NZ finds Black Cocks hard to swallow BOFH: Bloody computer room tourists! Apple share of MP3 player market to ... Science Proof by self-checking software: the four colour problem By Lucy Sherriff Published Monday 18th April 2005 14:56 GMT Get breaking Reg news straight to your desktop - click here to find out how Computer scientists at Microsoft Research Cambridge (MSRC) have developed a self-checking software-based proof of the four colour theorem that they say sweeps away any remaining uncertainty surrounding earlier proofs. MRSC researcher Georges Gonthier and Benjamin Werner, a researcher at INRIA, have devised a computer program that verifies the correctness of its own calculations, and applied it to the four colour problem. It constructs a precise mathematical proof and checks that it follows the strict rules of formal logic. The theorem states that any geographical map may be drawn so that no two contiguous regions are filled in the same colour, provided a palette of four colours is available. It was proven, rather controversially, in 1976. As well as using formal logic, a pair of mathematicians from the University of Illinois, Kenneth Appel and Wolfgang Haken, used a computer to check the tenets of the theorem by brute force.

33. The Four Colour Theorem And Mapping The four colour theorem. Mathematicians and map makers share a lot of common ground.No more so than in the area of the four colour theorem. http://www.aditsite.co.uk/html/colourtheory.html

Home About Adit Limited Contact us Legal ... AditMap Mapping What is a Map Map Distortions How to build maps Using a GPS survey ... RGB and CYMK Colours The Four Colour Theorem Follow the Adit Guide to maps, mapping and cartography The Four Colour Theorem Mathematicians and map makers share a lot of common ground. No more so than in the area of the four colour theorem. This theorem simply states that any map in a single plane can be coloured using four-colours in such a way that any regions sharing a common boundary (other than a single point) do not share the same colour. The theorem was first propounded by F Guthrie in 1853. Fallacious proofs have come and gone starting with Kempe in 1879 and Tait an 1880. In 1977 K. Appel and W. Haken used computer assistance to test many different combinations to effectively prove that four colours was all that was required in all instances. Since then, it may be that a mathematical proof has, at last, been arrived at. So, if we know that we can colour any map using just four colours how to we go about it. A little though would indicate that the problem is not straightforward. Simply starting with a random colour and an arbitrary polygon would soon lead to an impasse when the process met an area bounded by more than three other areas yet to be coloured.

34. Four Color Theorem -- Facts, Info, And Encyclopedia Article The four color theorem states that every possible geographical map can be colored The four colour theorem does not arise out of and has no origin in http://www.absoluteastronomy.com/encyclopedia/f/fo/four_color_theorem.htm

Four color theorem [Categories: Theorems, Topological graph theory] The four color theorem states that every possible geographical map can be colored using no more than four (A visual attribute of things that results from the light they emit or transmit or reflect) color s in such a way that no two adjacent regions receive the same color. Two regions are called adjacent if they share a border segment, not just a point. It is obvious that three colors are inadequate, and it is not difficult, relatively speaking, to prove that five colors are sufficient to color a map. The four color theorem was the first major theorem to be proven using a computer, and the proof is not accepted by all mathematicians because it would be unfeasible for a human to verify by hand. Ultimately, one has to have faith in the correctness of the compiler and hardware executing the program used for the proof. History The conjecture was first proposed in 1852 when (Click link for more info and facts about Francis Guthrie) Francis Guthrie , while trying to color the map of counties of (A division of the United Kingdom) England , noticed that only four different colors were needed. At the time, Guthrie was a student of Augustus De Morgan at

35. Four The four colour theorem returned to being the Four Colour Conjecture in 1890 . The four colour theorem was the first major theorem to be proved using a http://library.thinkquest.org/C006364/ENGLISH/problem/four.htm

The Four Colour Conjecture The Four Colour Conjecture first seems to have been made by Francis Guthrie. He was a student at University College London where he studied under De Morgan. After graduating from London he studied law but by this time his brother Frederick Guthrie had become a student of De Morgan. Francis Guthrie showed his brother some results he had been trying to prove about the colouring of maps and asked Frederick to ask De Morgan about them. De Morgan was unable to give an answer but, on 23 October 1852, the same day he was asked the question, he wrote to Hamilton in Dublin. De Morgan wrote:- A student of mine asked me today to give him a reason for a fact which I did not know was a fact - and do not yet. He says that if a figure be anyhow divided and the compartments differently coloured so that figures with any portion of common boundary line are differently coloured - four colours may be wanted, but not more - the following is the case in which four colours are wanted. Query cannot a necessity for five or more be invented. ...... If you retort with some very simple case which makes me out a stupid animal, I think I must do as the Sphynx did.... Hamilton replied on 26 October 1852 (showing the efficiency of both himself and the postal service):- I am not likely to attempt your quaternion of colour very soon.

36. Four Color Theorem Encyclopedia Four color theorem. The four colour theorem does not arise outof and has no origin in practical cartography. According to Kenneth May, http://encyclopedie-en.snyke.com/articles/four_color_theorem.html

Four color theorem This article needs to be cleaned up to conform to a higher standard of quality. See How to Edit and Style and How-to for help, or this article's talk page Missing image FourColorMapEx.png Example of a four color map The four color theorem states that any plane separated into regions, such as a political map of the counties of a state, can be colored using no more than four colors in such a way that no two adjacent regions receive the same color. Two regions are called adjacent if they share a border segment, not just a point. Each region must be contiguous - that is it may not be partitioned as are Michigan and Azerbaijan It is obvious that three colors are inadequate, and it is not at all difficult to prove that five colors are sufficient to color a map. The four color theorem was the first major theorem to be proven using a computer, and the proof is not accepted by all mathematicians because it would be infeasible for a human to verify by hand. Ultimately, one has to have faith in the correctness of the compiler and hardware executing the program used for the proof. Contents 1 History 2 Not for map-makers 3 Formal statement in graph theory 4 Generalizations ... 8 References History The conjecture was first proposed in when Francis Guthrie , while trying to color the map of counties of England , noticed that only four different colors were needed. At the time, Guthrie was a student of

37. Sci.math FAQ: The Four Colour Theorem Subject sci.math FAQ The four colour theorem; From alopezo@neumann.uwaterloo.ca (AlexLopez-Ortiz); Date 17 Feb 2000 225204 GMT; Newsgroups sci.math http://www.uni-giessen.de/faq/archiv/sci-math-faq.fourcolour/msg00000.html

Index sci.math FAQ: The Four Colour Theorem Subject : sci.math FAQ: The Four Colour Theorem From alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Date : 17 Feb 2000 22:52:04 GMT Newsgroups sci.math news.answers sci.answers Summary : Part 17 of 31, New version http://www.cs.unb.ca/~alopez-o Assistant Professor Faculty of Computer Science University of New Brunswick Index: Index sci-math-faq.fourcolour

38. Four Colour Theorem Latest four colour theorem Latest Posted by Steve on 10.01.2005 @ 214658 in ArticlesListen to a short interview with Keith Devlin about the latest verification http://sixthform.info/maths/b2print.php?p=60

39. Mathematics Weblog - Comments On 'Four Colour Theorem Latest' Have you read “Four Colours Suffice” by Robin Wilson? Tuesday 11 January 2005 @158am. close this window leave a comment http://sixthform.info/maths/b2commentspopup.php?p=60&c=1

40. CleverCS :: News Articles :: A Computer-checked Proof Of The Four Colour Theorem The four colour theorem is famous for being the first longstanding mathematicalproblem to be resolved using a computer program. http://ideas.web.cse.unsw.edu.au/index.php?module=articles&func=display&ptid=1&a

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