Four Colour Theorem 


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 subcases. 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.