Challenge Level

A brand new road has been built on the edge of a village, and six new houses are being designed. The houses can be any of these three types:

- Detached houses
- Semi-detached houses
- Terraced houses

Have a think about what each of these might mean. Do you live in one of these types of house?

Detached houses are built separately and aren't joined to another house. Semi-detached houses are joined to one other house, so these types of house come in pairs. When there are three or more houses joined together they're called terraced houses.

The developers are trying to decide how many of each type of house to include. They could build houses that are all the same type, or they could build a mixture of types of houses. Don't worry about the order of the houses for the moment - we're just interested in how many of each type we can build.

Can you think of a combination of houses that could be built?

Are there any combinations that aren't possible? Why?

Here is one possible combination of houses:

This picture shows three terraced houses, two semi-detached houses and one detached house.

How could we record this combination? Rather than drawing the houses or writing the types of house, is there an easier way of keeping track of the combinations we've tried?

Once you've decided how to record your ideas, see if you can find all the possible combinations for the types of house that can be built. How do you know that you've found them all?

Once you've had a go at that, have a think about how we could change this question. You might think, "I wonder what would happen if...?"

For example, what would happen if more than six houses were being built? Or less than six? Does anything interesting happen if the total number of houses is odd instead of even?

What if we did care about the order of the houses? Try this with a smaller number of houses at first - if we wanted to build four houses, what are the different possibilities?

See if you can think of any more ideas for ways to change this question. Good luck!