# The hair colour game

## Problem

Mrs Bunting's class was playing a game using interlocking cubes. All the boys took a red cube and all the girls took a green cube.

Then the children took a black cube if they had dark coloured hair and a yellow cube if they had light coloured hair. They stuck their two cubes together.

The pairs of cubes looked like this:

The next day they did the same again except that this time, they had three cubes each. As well as a having a green or red and a yellow or black, they took a brown cube if they had brown eyes and a blue cube if they had blue eyes.

## Getting Started

How will you decide which three cubes to take?

You could make the tree diagram using cubes or you could draw it on paper.

## Student Solutions

Ed from St Peter's College gave a good solution to the problem:

The first cube (or pair of cubes) means a boy with dark hair, the second cube is a boy with light coloured hair, the third is a girl with dark hair and the fourth is a girl with light coloured hair.

The next day there were eight groups because they added eye colour cubes.

If there was no one with dark coloured hair and blue eyes, you could still place it on the diagram, to show all the possibilities.

Daniel and Theo sent a clear and colourful tree diagram which you can see here.

Arjun from Vidyashilp Academy thought about the number of groups you would end up with for more cubes. He says:

For $1$ cube combination there are $2$ possibilities = $2^1$

For a $2$ cube combination there are $4$ possibilities =$2^2$

So, for a $3$ cube combination there are $8$ possibilities =$2^3$

Thus I can say that the total number of combination are $2^n$ where n is number of cubes.

I wonder how we know this for certain? Could you convince us that this is always the case, Arjun?

Well done to everyone!

## Teachers' Resources

**Why do this problem?**

This problem is designed to introduce, and practise making tree diagrams in a way practical for young children. It is primarily a group activity. You will need some interlocking cubes in various colours and some large sheets of paper. A mirror in the classroom would help the children decide on their hair and eye colour.

### Possible approach

You could start by physically doing the activity like Mrs Bunting's class in the problem. Ask the group to collect a cube according to whether they are a girl (green cube) or a boy (red cube). Then they should decide for themselves whether their hair is light or dark, getting their friends to help if they want, and taking a yellow or black cube accordingly.

### Key questions

### Possible extension

For learners who found this straightforward, you could ask them to introduce another set of criteria of their choice.

### Possible support

This task is best done in collaboration with others so encourage children to talk to each other and work together. Many children will be able to make the picture out of the cubes but may find recording on paper more difficult so you could take photos of their cube pictures.