4 colour map method

Have you ever tried to colour in areas on a map? Here we describe the four colour map method. This a mathematical problem first posed by Francis Guthrie in 1852. He was trying to colour in a map of the counties of England with the fewest possible colours (colour printing was expensive), and found he could do it with only 4 colours. He then wondered if it was always possible to colour a map with only 4 colours.

This “4 colour map theorem” was finally proved in 1976 only through the use of computers. In fact, it was the first major mathematical theory to ever be proved with computers. But actually colouring in a map with 4 colours is still quite hard. Some algorithms exist to do it automatically but they are very complex.

I came across this problem when I was colouring in a map of the parishes of the City of London. A very nice map exists from 1907 (shown below) but it actually uses 6 colours. If you look closer you can see two different shades of blue. I think most people have experienced this. You start colouring in and then find that you need an extra colour when you are half way through. So I decided to see if I could work out a good four colour map method. I describe it in the steps below.

4 Colour map problem 1a

STEP 1 – make a wire diagram

Put a dot on every parish and then draw lines between the dots showing every shared border. Every parish which has a border with another should then connecting those two together

4 Colour Map Problem 1b

STEP 2 – hide the background map

This lets you see the abstracted map clearly with all the parishes and borders represented. You need to check carefully that you have properly captured all the borders at this stage to save much pain and effort later. It is harder than you think!

4 Colour map problem 1c

STEP 3 – destroy the triangles!

Pick two colours and start filling in the dots to make sure that no triangle with grey dots remains. You also need to make sure that no continuous loop with an odd number ( ie not even) of grey dots exists. If all the relationships between the grey dots are open and linear and not closed like a triangle then you know for sure that you can fill in all the remains grey dots with only 2 more colours

4 Colour map problem 2

STEP 4 – Fill in the rest with 2 more colours

Start filling in the grey dots with two more colours. You should find that you are following a line through the maze just alternating between those two new colours.

4 Colour map problem 3

STEP 5 – put the background map back

Now use the wire diagram as your guide to colour in the background map and …voila!

4 Colour map problem 4