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[[[> |>> IMG:AppelandHagan.jpg _ Appel and

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

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