Homes

There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?

Stairs

This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

Train Carriages

Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?

Mixed-up Socks

Mixed-up Socks

Start with three pairs of socks. Now mix them up so that no mismatched pair is the same as another mismatched pair.

Now try it with four pairs of socks. Is there more than one way to do it?

Why do this problem?

Very young children can work on this problem , as all that is needed is an awareness of same and different, and the idea of making groups of two. The problem also fits in with activities about pairs and counting by twos.

Possible approach

You could make up a simple story context in which to introduce one pair of socks, then a second pair, then mix them up, or you could use this story of The Magic Washing Machine. It would be a good idea to illustrate it with two or three pairs of real socks.

When the children have seen how two pairs can be mixed, set the problem of mixing up three pairs of socks. You could print out this sheet of six socks in four or five different colours or the children could colour them themselves. They could then use these paper socks to work with either singly or in pairs. They will also need coloured pencils and paper to record their results.

Once the task has been achieved, recorded and discussed, introduce a fourth pair of socks and challenge the children to mix them up. Stress the requirement that no two pairs can use the same combination of socks, as this is an easy mistake to make when using eight socks. For example, making a pair with a blue and red sock means that the other red and blue socks cannot be used together.

Ask the children to display and compare their solutions. One way of doing this would be for half the class to stay in their seats with their solutions and the other half to move around, talking to those who are sitting about what they did. They could then swap over. In this way, learners get a chance to question others' solutions and there is a purpose to explaining what has beeen done. You could then invite a few children to draw attention to someone else's solution for a particular reason. It may be that it is recorded very systematically, for example, or that their way of finding an answer seemed particularly efficient or understandable. You may also wish to discuss the number of different solutions the children found.

Key questions

Are all the pairs you have made different?
I wonder how you can tell if you have found all the different ways of doing it?

Possible extension

Try it with five and six pairs. Children could explore to see whether there is a pattern to the number of possible solutions.
For an extra challenge, try the problem using gloves instead of socks. This adds the complication of there being a left and right glove. How does this affect the total number of possible solutions?

Possible support

Suggest that learners use paper socks starting with just two pairs. Those with difficulties in recording could paste the different pairs of socks onto a sheet of paper.

Handouts for teachers are available here (Word documentpdf), with the problem on one side and the notes on the other.