It has been observed by map-makers through the centuries that any geographical map (that is, a division of the plane into regions) can be coloured with just four colours in such a way that no two neighbouring regions have the same colour. A proof of this, the Four Colour Theorem, was sought by mathematicians from about the 1850s. In 1890 Heawood proved that five colours would suffice, but it was not until 1976 that Appel and Haken proved the Four Colour Theorem itself. Initially, some mathematicians were sceptical of the proof because it relied, in an essential way, on a massive amount of checking of configurations by computer that could not easily be verified independently. However, the proof is now generally accepted and considered a magnificent achievement.
http://serendip.brynmawr.edu/playground/fourcolor/ A Shockwave file allowing you to create your own map and colour it in, or see an automated solution with no more than four colours.