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[[[> IMG:AppelandHagan.jpg ]]]

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 accepted for a decade until 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 Hagan (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.

The Klein Bottle and Mobius Band both require six colours to colour any map on their surfaces.

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