The question:
http://www.c3.lanl.gov/mega-math/gloss/math/4ct.html
The answer:
This picture depicts how no more than 4 colours will be needed to draw a conventional map. As you see at the bottom, If the indigo surrounds the blue country, green has no point of contact unless indigo withdraws, either to surround yellow, or severing ties with red. In any case, boundary contact cannot be maintained by five different colours, as trying to contact them all results in various severences.
