Colouring Cubes
A whole family of mathematical problems begin: How many ways are there…? Some of these have cropped up in earlier issues of SYMmetry. This time we are going to look at the ways of colouring the six faces of a cube. We start with a problem posed in an Oxford University Entrance Examination in Mathematics a few years ago.
Problem 1 A toy maker makes cubes from coloured squares. How many distinguishable cubes can he make using all of six different colours?
If you have never tackled a problem like this before, try the practical approach first. Make cubes with coloured squares and see how many different cubes you can make. If you have some plastic cubes (or even toy bricks), you could try colouring these in different ways – using colours that you can wipe off afterwards or coloured labels that you can peel off.
Take care when you compare cubes. One method is to check the colours of opposite faces – but beware! There are, for instance, two possible cubes with opposite faces coloured red/green, blue/orange and purple/yellow. One is the mirror image of the other.
You might decide to explore this problem by drawing out nets of cubes on squared paper. This can work quite well, but there are difficulties. For example, it is not easy to see at a glance whether or not the diagrams below are all nets for the same cube.
To use nets to solve cube colouring problems we clearly need a system for drawing nets, which will help us decide quickly whether or not two nets represent different cubes.
This is how one system works using the six colours: yellow, blue, red, green, purple, orange. The net is coloured in the same order every time according to the following set of rules:
| (1) | Yellow; |
| (2) | One of the five colours left; |
| (3) | Blue, if still available, - otherwise red; |
| (4) | One of the 3 colours left; |
| (5) | 1st choice – red, 2nd choice – green, 3rd choice purple. |
If you are good at juggling with mental pictures of 3-D objects, you can probably come up with an organised way of counting cubes without drawing nets. You could, for example, start by deciding how many cubes have the yellow square opposite to a green square. This number will help you to find the total number of different cubes.
Now you may be ready to tackle some even harder cube colouring questions.
Problem 2 How many different cubes can you make using six different colours, when you have seven colours to choose from?
Problem 3 How many different cubes can you make when you have only four colours to choose from? You must use all the colours at least once, but will need to use one or two colours more than once.
You can probably see that it would be possible to make up lots of colouring problems. In Problem 2 you could replace ‘seven’ by a larger number. In Problem 3 you could remove hte condition that each colour should be used at least once. You could investigate the colouring of other 3-D shapes such as the tetrahedron, the octahedron etc. and write to the Editor about your discoveries.
