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Colouring a map using only 4 colours

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Imagine that you have a map with lots of countries on it. You want to colour the countries. It doesn't matter what colours you choose, but two countries with a common border must be coloured differently, so you can tell them apart. It doesn't matter if similarly coloured countries touch at a point, only if they share a border. How many colours will you need if you use as few as possible?

Believe it or not, however complicated the map, you will only need four colours. Have a go yourself! On the left is a fairly complicated map, even if the countries look very strange as they are square. What's more, there is one country that goes right round the edge (this is blue and you can't change that one). Try colouring in the others using only the four colours provided. You will find it possible, even if you don't manage it first time.

Ah, you say, but that is still quite a small map. What happens if there are thousands of countries, and they are irregular, not square? You still only need four colours. Try drawing some maps yourself and try!