41. Guest Post: Vladimir Khachatryan, The Higgs Mass From The Four-Colour Theorem Ashay Dharwadkeris the founder and director of the Institute of Mathematics, Gurgaon, India. He is interested in fundamental research in mathematics, particularly in algebra http://www.science20.com/quantum_diaries_survivor/guest_post_vladimir_khachatrya

42. Notations Of The Four Colour Theorem Proof 1 Notations of the Four Colour Theorem proof This document is primarily a reading guide for the Coq proof scripts of the Four Colour Theorem proof. http://research.microsoft.com/en-us/um/people/gonthier/4colnotations.pdf

43. The Four Color Theorem (4CT) File Format Microsoft Powerpoint View as HTML http://cs.nyu.edu/courses/summer08/G22.2340-001/projects/EM_4ColorTheorem.ppt

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44. Last Doubts Removed About The Proof Of The Four Color Theorem The story of the Four Color Problem begins in October 1852, when Francis Guthrie , a young mathematics graduate from University College London, was coloring http://www.maa.org/devlin/devlin_01_05.html

Devlin's Angle January 2005 Last doubts removed about the proof of the Four Color Theorem At a scientific meeting in France last December, Dr. Georges Gonthier, a mathematician who works at Microsoft Research in Cambridge, England, described how he had used a new computer technology called a mathematical assistant to verify a proof of the famous Four Color Theorem, hopefully putting to rest any doubts about the result that had remained since the first proof of the theorem was announced in 1976. The story of the Four Color Problem begins in October 1852, when Francis Guthrie, a young mathematics graduate from University College London, was coloring in a map showing the counties of England. As he did so it occurred to him that the maximum number of colors required to color any map seemed likely to be four. The coloring has to meet the obvious requirement that no two regions (countries, counties, or whatever) sharing a length of common boundary should be given the same color. Guthrie's question became known as the Four Color Problem, and it grew to be the second most famous unsolved problem in mathematics after Fermat's last theorem.

45. 4 Color Proof 1) We assume that the 4color theorem is false (i.e., that there exist finite maps that require at least five colors), and show that that assumption leads http://www.superliminal.com/4color/4color.htm

"Proof" of the 4-Color Map Theorem by Don Hatch and Melinda Green As reconstructed from the original proof by Don Kimber Proof by Contradiction: 1) We assume that the 4-color theorem is false (i.e., that there exist finite maps that require at least five colors), and show that that assumption leads to a contradiction. This means that the assumption was incorrect and that four colors are therefore sufficient to color any finite planar map. 2) There then must exist a counterexample of smallest possible size (i.e. a map with the smallest number of regions that require 5 colors). Let M be such a "smallest" counterexample. 3) Removing any region from M therefore results in a 4-colorable map. 4) Every region must therefore have valence >= 4 (ie, have 4 or more neighbors) because if such a region existed with a valence less than 4, you could: a: remove that region from M, b: 4-color the resulting map (via step 3), and proof of this using Euler's formula is straightforward.). This fact together with step 4 determine that M must contain at least one region of valence 4 or 5. CASE 1: There exists a region of valence 4. Call it v.

46. Ashay Dharwadker's Proof Of The Four Color Theorem - A Review FOUR COLOUR THEOREM. For any subdivision of the plane into nonoverlapping regions, it is always possible to mark each of the regions with one of the http://www.freewebs.com/desargues/4ct2.htm

Ashay Dharwadker's Proof of The Four Color Theorem - A Review In 2000, Indian mathematician Ashay Dharwadker finally proved the four color theorem, one of the most famous and long outstanding conjectures in mathematics. The proof, as many mathematicians had hoped for, is extremely elegant: http://www.dharwadker.org Summary of Dharwadker's Proof FOUR COLOUR THEOREM. For any subdivision of the plane into non-overlapping regions, it is always possible to mark each of the regions with one of the numbers in such a way that no two adjacent regions receive the same number. STEPS OF THE PROOF: We shall outline the strategy of the new proof given in this paper. In section I on MAP COLOURING , we define maps on the sphere and their proper colouring. For purposes of proper colouring it is equivalent to consider maps on the plane and furthermore, only maps which have exactly three edges meeting at each vertex. Lemma 1 proves the six colour theorem using Euler�s formula, showing that any map on the plane may be properly coloured by using at most six colours. We may then make the following basic definitions. Define N to be the minimal number of colours required to properly colour any map from the class of all maps on the plane.

47. Maps, Colouring, Four Colour Theorem Guthrie 185052 Francis Guthrie's Conjecture At a lecture given by Agustus de Morgan, professor of mathematics, University College, London, 23 October 1852, one of the http://www.geog.port.ac.uk/webmap/hantsmap/hantsmap/fourcols.htm

Old Hampshire Mapped Four Colours? Guthrie 1850-52 Francis Guthrie's Conjecture At a lecture given by Agustus de Morgan, professor of mathematics, University College, London, 23 October 1852, one of the students, Frederick Guthrie, asked a simple question. De Morgan mentioned the suggested conjecture in a letter to Sir William Rowan Hamilton, mathematician and physicist; was it a fact that:- ... if a figure be 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 conjecture was first proposed by Frederick's elder brother Francis Guthrie, an earlier student of de Morgan's at University College, graduated 1850, later a professor of mathematics in South Africa. This then is the:- only four colours Four Colour Theorem Every map drawn on a plane can be coloured with at most four colours in such a way that neighbouring countries are coloured differently. Note:- Some maps can be coloured with fewer colours.

48. The Act365.com Home Page FourColour Theorem. Essentially, the Four-Colour Theorem states that it is possible to paint any two-dimensional surface in such a way that no two adjacent regions on the surface http://act365.com/

act365.com Introduction The act365.com project site aims to produce tools and provide information for the general benefit of the Web community. All of the source code used on the site is available at Sourceforge , where it has been released under the Gnu Public License Target Browsers The Avalon Project The Avalon project is dedicated to providing analytical tools for the financial community. It has hosted Government bond yield calculators since August 1998 - however, the calculator has not been updated to take into account the many recent changes in the European market, so it returns reliable figures for the US only. Cricket Tools for cricket fans are available here . Note that the Duckworth/Lewis calculator that once appeared on the site has had to be removed at the insistence of Duckworth and Lewis. Four-Colour Theorem Essentially, the Four-Colour Theorem states that it is possible to paint any two-dimensional surface in such a way that no two adjacent regions on the surface share a common colour without using any more than four colours. Four-colour darts is a simple game based upon the theorem.

49. Anchoring Expository Text In Formal Mathematics Abstract. The 150 year old Four Colour Theorem is famous for being the first important mathematical result whose proof, completed in 1976 by Appel and Haken, required computer http://www.cs.cornell.edu/Nuprl/PRLSeminar/PRLSeminar04_05/Gonthier/May20_05.htm

PRL Seminars Verifying the Four Colour Theorem Georges Gonthier May 20, 2005 Abstract Home Introduction Authors Topics ... PRL Project

50. OUP: Wilson: Graphs, Colourings And The Four-Colour Theorem - Oxford University The fourcolour theorem is one of the famous problems of mathematics, that frustrated The theorem asks whether four colours are sufficient to colour all http://ukcatalogue.oup.com/product/9780198510628.do

51. Formal Proof—The Four- Color Theorem 8G. Gonthier, A computerchecked proof of the four-colour theorem. 9G. Gonthierand A. Mahboubi, A smallscalereflec-tionextensionfor the Coqsystem , INRIA Technical report. 1392 http://www.ams.org/notices/200811/tx081101382p.pdf

52. Slitherlink, Nurikabe, Heyawake, Akari, Hitori, Kakuro And Sudoku. Enjoy Nikoli Slitherlink, Nurikabe, Heyawake, Akari, Hitori, Kakuro and Sudoku. Enjoy nikoli puzzles online. http://www.nikoli.com/en/take_a_break/four_color_problem/

53. Amateur S Guide To Proving The Four Color Theorem - Wikibooks Jun 25, 2010 Although technically the Four Color Theorem has been proven, for some � professionals and amateurs alike � attempting to discover a more http://en.wikibooks.org/wiki/Amateur's_Guide_to_Proving_the_Four_Color_Theor

54. Ideas, Concepts, And Definitions Four Color Theorem (See also The Mathematics Behind the Maps, The Most Colorful Math of All, and The Story of the Young Map Colorer.) The Four Color Problem was famous and http://www.c3.lanl.gov/mega-math/gloss/math/4ct.html

Four Color Theorem (See also: The Mathematics Behind the Maps The Most Colorful Math of All , and The Story of the Young Map Colorer The Four Color Problem was famous and unsolved for many years. Has it been solved? What do you think? The story of the problem Since the time that mapmakers began making maps that show distinct regions (such as countries or states), it has been known among those in that trade, that if you plan well enough, you will never need more than four colors to color the maps that you make. The basic rule for coloring a map is that no two regions that share a boundary can be the same color. (The map would look ambiguous from a distance.) It is okay for two regions that only meet at a single point to be colored the same color, however. If you look at a some maps or an atlas, you can verify that this is how all familiar maps are colored. Mapmakers are not mathematicians, so the assertion that only four colors would be necessary for all maps gained acceptance in the map-making community over the years because no one ever stumbled upon a map that required the use of five colors. When mathematicians picked up the thread of the conversation, they began by asking questions like: Are you sure that four colors are enough? How do you know that no one can draw a map that requires five colors? What is it about the way that regions are arranged and touch each other in a map that would make such a thing true? When the question came to the European mathematics community at the end of the 19th century, it was perceived as interesting but solvable. Prominent and experienced mathematicians who tackled the problem were surprised by their inability to solve it. Take for example, this account from

55. Kempe Chain - Wikipedia, The Free Encyclopedia In mathematics, a Kempe chain is a device used mainly in the study of the four colour theorem. http://en.wikipedia.org/wiki/Kempe_chain

Kempe chain From Wikipedia, the free encyclopedia Jump to: navigation search This article does not cite any references or sources Please help improve this article by adding citations to reliable sources . Unsourced material may be challenged and removed (December 2009) In mathematics , a Kempe chain is a device used mainly in the study of the four colour theorem Contents History Formal definition In terms of maps Application to the four colour theorem ... edit History Kempe chains were first used by Alfred Kempe five-colour theorem , a weaker form of the four-colour theorem. edit Formal definition The term "Kempe chain" is used in two different but related ways. Suppose G is a graph with vertex set V , and we are given a colouring function where S is a finite set of colours, containing at least two distinct colours a and b . If v is a vertex with colour a , then the ( a b )-Kempe chain of G containing v is the maximal connected subset of V which contains v and whose vertices are all coloured either a or b The above definition is what Kempe worked with. Typically the set S has four elements (the four colours of the four colour theorem), and

56. The Four Colour Theorem Feb 20, 1998 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 http://www.cs.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.

57. The Four Color Problem Gets A Sharp New Hue - Science News Mathematicians find new answers to the still puzzling theorem that four colors suffice to color any map. http://www.sciencenews.org/view/generic/id/41486/title/The_four_color_problem_ge

Earth Environment Humans Life ... Search http://www.sciencenews.org/view/generic/id/41486 Home News Features Blogs ... SUBSCRIBE In the November 6 Issue: TABLE OF CONTENTS Iron in the Mix: Scientists look for the secret behind high-temperature superconductors Massacre at Sacred Ridge: ... Past Issues Advertisement Home Columns Math Trek Column entry The four color problem gets a sharp new hue Mathematicians find new answers to the still puzzling theorem that four colors suffice to color any map. By Julie Rehmeyer Web edition Friday, March 6th, 2009 Text Size In 1852, botanist Francis Guthrie noticed something peculiar as he was coloring a map of counties in England. Despite the counties’ meandering shapes and varied configurations, four colors were all he needed to shade the map so that any two bordering counties were different colors. Perhaps, he speculated, four colors were enough for any map. Little did Guthrie know the load of trouble he unleashed with his innocent conjecture. It took mathematicians nearly a century and a quarter to prove him right, and even that wasn’t enough to close the Pandora’s box Guthrie had opened. Mathematicians pulled out their markers and tried to color everything in sight. The particular things mathematicians wanted to color were graphs: dots connected by lines Such graphs can be used to describe everything from friendships to the Internet to gene interactions. They can even describe maps, if the countries correspond to dots and bordering countries are connected by lines. Graphs from maps have the special property that the lines will never cross, though other graphs can form hairballs as nasty as you please. How many colors, mathematicians wondered, would it take to color any graph so that connected dots are always different colors?

58. Electronic Research Announcements by N Robertson 1996 - Cited by 104 - Related articles http://www.ams.org/era/1996-02-01/S1079-6762-96-00003-0/

59. The Four Colour Theorem: Engineering Of A Formal Proof - Microsoft Academic Sear Authors Georges Gonthier. Citations 1 The following links allow you to view and download full papers. These links are maintained by other sources not http://academic.research.microsoft.com/Paper/4353208.aspx

var SiteRoot = 'http://academic.research.microsoft.com'; SHARE Author Conference Journal Year Look for results that meet for the following criteria: since equal to before Publication The Four Colour Theorem: Engineering of a Formal Proof Edit The Four Colour Theorem: Engineering of a Formal Proof Citations: 1 Georges Gonthier Published in 2007. View or Download The following links allow you to view and download full papers. These links are maintained by other sources not affiliated with Microsoft Academic Search. www.informatik.uni-trier.de dx.doi.org Citation A Brief Overview of Agda - A Functional Language with Dependent Types ... Ulf Norell Conference: Theorem Proving in Higher Order Logics - TPHOLs Microsoft Research Asia Help Feedback ... Privacy Statement Share this on Contribute to Academic

60. The Four-colour Theorem by N Robertson 1997 - Cited by 340 - Related articles http://portal.acm.org/citation.cfm?id=271675

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