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.

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