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[[[> |>> IMG:AppelandHaken.jpg _ Appel and Haken <<| ]]] The Four Colour Theorem is a problem from Graph Theory, and along with the Bridges of Koenigsberg and the Three Utilities Problem is one of the most common examples of Pure Mathematics found in school. |>> [[[ Given any map, colour the regions _ so that regions sharing a border _ get different colours. _ _ How many colours do you need? ]]] <<| The problem was first set in 1852. A false proof was given by Kempe (1879). Kempe's proof was in fact accepted for a decade, but then Heawood showed an error using a map with 18 faces. It wasn't until 1976 that a proof was finally given by Kenneth Appel (standing) and Wolfgang Haken (seated) (1977). Even then there was, and still is, some controversy, because the proof requires a computer to check a large number of sub-cases. These then have to be combined in a clever way - the computer doesn't actually *do* the proof - but even so, it's not a proof in the traditional sense. It is interesting to compare the difficulty of the proof that four colours are sufficient to colour any map on a sphere with the almost trivial proof that seven colours are sufficient to colour any map on a torus. On the Klein Bottle and Mobius Band six colours are sufficient to colour any map on their surfaces. * http://www.google.co.uk/search?q=four+colour+theorem * http://en.wikipedia.org/wiki/Four_color_theorem