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The four colour problem

By David, 4 June 2013 Activity

Colouring in is for kids? Well, it’s also for mathematicians! Grab some pencils and have a go at one of the most interesting mathematical problems around!

You will need

a map of Europe. Each country is coloured in and somoene is currently finishing colouring Bosnia.

Colour each region on the map in one colour.

  • At least five different coloured pencils or textas
  • A copy of a map of Europe
  • Paper

The four colour problem

The four colour problem is famous in mathematics. You are given a map divided up into different regions and need to colour it in, following some rules:

  • Every region needs to be coloured in a single colour.
  • Two regions that share an edge must be coloured different colours. Two regions that only meet at a corner can be coloured the same colour.
    South-west Europe. Both Spain and France are coluored orange, and there is a cross on the border.

    Two regions that share an edge must be different colours

  • You must use as few colours as possible.

What to do

Use the four colour problem rules to colour a map of Europe. The ocean is all one region, but you might have to colour some countries the same colour as the ocean.

How many colours did you need? Do you think you could do better if you tried again?

What’s happening?

If you're using four colours, then Luxembourg must be the same colour as the ocean!

If you’re using four colours, then Luxembourg must be the same colour as the ocean!

Around 150 years ago, Francis Guthrie was colouring a map of the counties of England. He noticed that he only needed four colours to make sure adjacent counties were different colours. For over 100 years, mathematicians puzzled over this observation – could you colour any map with only four colours?

In 1976, Wolfgang Haken and Kenneth Appel used a computer to show that every map can be coloured with only four colours. Since then, other mathematicians have also written proofs, but every proof so far has used a computer.

More questions

Although mathematicians have found an answer, there are still many interesting questions you can look into. Can you do the following?

  • Draw a map that only needs two colours.
  • Create a map with only four regions that needs four different colours.

You can colour any map, on a flat sheet of paper or the surface of a globe, with four colours. However, you might run into problems with maps on more complicated shapes. Maybe you could try drawing a map on an inflatable ring. Can all three-dimensional ring-shaped maps be coloured with four colours? If not, how many colours does it take?

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