1 The Four Colour Problem Louis Lim MATH 5400 Monday, January 22, 2007

2 The Problem What is the maximum number of colours needed to colour any map such that neighbouring regions are coloured differently? –Does a map with more regions require more colours?

3 Elaboration… Map is in one piece Use different colours to distinguish between neighbouring regions and to separate boundaries Regions can share a common point but not common boundary; for example, –A and C can be coloured the same –A and B are coloured differently

4 … Elaboration We may omit exterior regions –Do not require an additional colour

5 Your Turn… Map of the United States:

6 Beginnings Four Colour Problem first proposed in 1852 by Francis Guthrie, graduate student at the University College in London Francis’ brother, Frederick, communicated with mathematician Augustus de Morgan, who became intrigued De Morgan shared problem with mathematician colleagues and students; published the problem in 1860 in Athenaeum 1878: Arthur Cayley asked at the London Mathematical Society if anyone solved the problem (i.e., find a general argument to show a maximum of 4 colours are needed to colour any map) – nobody did

7 Famous “Proof” Alfred Kempe’s 1879 proof stood as correct until 1890 Kempe elected as a Fellow of the Royal Society; served as treasurer Percy Heawood published a major flaw in Kempe’s proof – not positively received; Heawood did not receive much recognition

8 Progress 1922: Philip Franklin proved 4 colour problem for up to 25 regions; 1926: Clarence Reynolds extended 4 colour problem for up to 27 regions; 1936: Heawood estimated the probability that the 4 Colour Problem is false to be 1 in a million billion 1938: Franklin to 31 regions; 1940: C. E. Winin to 35 regions; 1968: Oystein Ore & Joel Stemple to 40 regions 1970s: extended to 52 and then 96 regions

9 Difficult to Solve “Problem so infuriating because it appears so simple to prove, yet no mathematician has ever found a proof with traditional methods of logic and math” (Peirce, in Danesi, 2004) “Supposing a system of n areas coloured according to the theorem with four colours only, if we add an ( n + 1)th colour, it by no means follows that we can without altering the original colouring colour this with one of the four colours” (Bigg, Lloyd, & Wilson, 1976, p. 93) G. D. Birkhoff believes all mathematicians at some point have attempted to solve the problem (Bigg, Lloyd & Wilson) Some mathematicians devoted professional lives to problem (e.g., Charles Peirce; Percy Heawood)

10 Controversial Proof 1976: Kenneth Appel & Wolfgang Haken, University of Illinois 1200 hours of computer time; billions of calculations; configurations (computer printout 4 feet in height), eventually reduced to st computer proof Published in Illinois Journal of Mathematics in 2 parts; microfiche had 450 diagrams and explanations Would require thousands of years to verify by hand

11 Mixed Reactions  “If no human being can ever hope to check a proof, is it really a proof?” (Wagon, 2006) “A good mathematical proof is like a poem – this is a mathematical directory!” (Thomas, 1995) “Our pursuit is not the accumulation of facts about the world or even facts about mathematical objects. The mission of mathematics is understanding” (Hersh, 1999) “The real thrill of mathematics is to show that as a feat of pure reasoning it can be understood why four colors suffice. Admitting the computer shenanigans of Appel and Haken to the ranks of mathematics would only leave us intellectually unfulfilled” (Daniel Cohen, in Wilson, 2002)

12 Refinement 1996: Neil Robertson, Daniel Sanders, Paul Seymour, & Robin Thomas published an alternative proof, using 633 configurations, with a computer 2004: Georges Gonthier verified each step of 1996 proof using a software called a “mathematical assistant”: –“a new kind of computer program that a human mathematician uses in an interactive fashion, with the human providing ideas and proof steps and the computer carrying out the computations and verification” (Devlin, 2005)

13 Famous! Four Colour Problem often thought as most famous problem in mathematics (Wilson, 2002) Others think it as second most famous problem, after Fermat’s Last Theorem (Devlin, 2005) Still others view it as 1 of the top 10 math puzzles (Danesi, 2004) Accessible since easy to state yet so difficult to prove: “Problem so infuriating because it appears so simple to prove, yet no mathematician has ever found a proof with traditional methods of logic and math” (Peirce, in Danesi, 2004)

14 Importance of Problem New branch of applied mathematics: Graph Theory –Within Graph Theory, progress made to network problems in 1950s (e.g., road and rail networks; communication networks) Discussion of what is mathematical proof? Contemporary problem – progress still being made e.g., Mathematical Assistant self-checking software: “to make computer in general more reliable and trustworthy” (Andrew, Herbert, in Sherriff, 2005)

15 Martin Gardner’s 1975 April Fool’s Day Hoax Published in Scientific American : map with 110 regions coloured with 5 colours:

16 Applicability to High School Mathematics MDM4U Data Management: –Specific expectation: “Solve network problems (e.g., scheduling problems, optimum-path problems, critical path problems), using introductory graph theory”.

17 An Example… To determine the number of colours, we can use graph theory: 1.) Place a vertex in each region (these represent the regions or countries); 2.) Join 2 vertices with an edge if they share a common border (i.e., adjacent to each other) In our example, 4 colours are needed:

18 Art of Justin Mullins Mathematics Photography Exhibit, in London, England, February 2006 Ugliness (i.e., not neat solution such as a paper-pencil proof) – all 663 configurations are on the art Available for purchase as a limited edition art work by contacting Justin Mullins,

19

20 Thank you!