As this sign shows there are to be 6 homes built on a new development site. Houses are often put into three groups. Perhaps you live in one of these:-

- Semi-detached houses
- Detached Houses
- Terraced houses

You probably know that detached houses are built separately with no joining to another house. Semi-detached houses are ones in which two are joined together so all the houses appear in pairs. When there are three or more houses together they're called terraced. You are asked to decide on how many of these you would use. So you might suggest:- 1 detached + 2 semi-detached (you have to have an even number of these since they come in pairs) + 3 terraced ( you have to have 3 or more of these). The order does not matter. We are just counting up what we've got. Suppose you use a short hand to record this and have the following key:-

D : Detached S : Semi-detached T : Terraced

you might record this first arrangement as:- D S S T T T and this is the same as S S T T T D for this challenge.

You don't have to use all three kinds each time, you might have:-

T T T T T T or S S D D D D etc

Well, what different collections of homes can you find?

Have you been using a system for this?

So, once you've got that, what about the question, "I wonder what would happen if there were 7 ( 8, 9 or 10 ) homes?"

Try these and see how you get on.

You might like to ask "I wonder what would happen if S S S S D D is taken to be different from S S D D S S ?" What does this produce when you're trying to find all these arrangements? You might like to try this alternative with a smaller number of homes first. So you might try 4 homes, and start off:-

D D D D S S D D D S S D D D S S

but you cannot have S D D S as the two semi-detached have to be next to each other.

Well have fun with these and good luck !