Monday Maths: Colouring In

The essence of mathematics is not to make simple things complicated, but to make complicated things simple.

~ S. Gudder

It sounds like the start of a bad joke about a Geography degree, but actually, colouring in is involved in an important mathematical theorem.  This theorem was proposed in 1852, but not actually proven until 1976.  That’s over a hundred years before anyone could prove whether the theorem was right or wrong.

It’s called the Four Colour Theorem.

A four colour map.
Source: Based on a raster image by chas zzz brown on en.wikipedia. CC-BY-SA 3.0

Simply put, it says that for any map (or design made up of geometric shapes) can be coloured in so that each country is a different colour to the ones it is next to, with only four different colours.

Any map.

Apart from bizarre ones that don’t obey laws of common sense (like ones with finite area but infinite perimeters – yes, they exist!).  And ones with countries that aren’t all in the same place, like the US and Azerbaijan.  But we don’t need to worry about those for the moment.

If you want the mathematical proof, you can look at Wikipedia, but I won’t go into it here.  They needed a computer to come up with it, so the chances of me understanding it are slim!

So how can you do this with Brownies?

Take a set of random maps, like the one to the right, or maps of actual countries.  Challenge the girls to colour them in using as few colours as possible.  Ask them before they start how many colours they think they will need.  If you make some of the maps complicated enough they will probably say some large numbers.

Afterwards, you can tell them that it’s actually been proven that they only need four – see if they believe you, and see how many people guessed right.  Tell them how long it took mathematicians to prove it so they don’t feel bad about guessing wrong!

You could use the theory to create interestingly patterned greetings cards, or stained glass windows.  For bonus points, sell them and give the money to a charity of the girls’ choice.


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