Carrie makes a cube by sticking together 27 small cubes.

Can she colour the exposed faces of the original cubes red, white, blue and yellow, so that no faces with a common edge have the same colour?

Does she need the yellow?

Can she get away with just red and white?

This problem explores the shape of a cube and ways that parts of it can be coloured.

For a similar problem see __Carrie’s First Cubes__, Geometry, Level 4.

There are links here to a famous problem called the Four Colour Problem that was solved in 1976 after having been around for about 126 years. This Problem asked how many colours were needed to colour any map. The restriction here was that no two countries with a common border were allowed to be coloured in the same colour. Two mathematicians in America, Appel and Haken, were able to prove that four colours were all that were needed.

Questions such as the Four Colour Problem have important applications. Using them we are able to see when certain routing problems (for buses or milk trucks, for instance) can be solved. They are an important part of topics called graph theory and operations research.

### The Problem

Carrie makes a cube by sticking together 27 small cubes. Can she colour the exposed faces of the original cubes red, white, blue and yellow, so that no faces with a common edge have the same colour?

Does she need the yellow?

Can she get away with just red and white?

### Teaching sequence

- Introduce the problem by looking at Rubik’s cube and discussing the colours used and reasons for this.
- Pose the problem to the class.
- Give the students time to think before asking them to share their ideas for proceeding with the problem.
- As the students work on the problem ask them to share the strategy they are using and their reasons for the approach taken.
- Share solutions.

#### Solution

Here is a 3-colouring. There are lots of ways of using 4 colours. Can you find any other ways of 3-colouring the cube?

3 | 1 | 3 | |||||||||

2 | 3 | 2 | |||||||||

3 | 1 | 3 | |||||||||

1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 |

2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 |

1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 |

3 | 1 | 3 | |||||||||

2 | 3 | 2 | |||||||||

3 | 1 | 3 |

Here 1 = red, 2 = white and 3 = blue.

To find a 3-colouring it was easy first of all to just use two colours across the middle band of the net above. The difficulty was fixing up the top and bottom faces to "match" the middle band of faces.

It’s not possible to just use two colours. Look at one of the small cubes at a corner of the big cube. This will have three faces all touching each other. That means that they must all be coloured differently.