A City of Towers

In a certain city, houses have to be built in a particular way.

There have to be two rooms on the ground floor and all other rooms have to be built on top of these.

Families are allowed to build just one room for each person living in the house.

So a house for two people would look like this:

but a house for three people could look like one of these:

What might a house for four people look like?
In how many different ways could a family of four people build their house?

What about a house for five people?
In how many different ways could a family of five people build their house?

What do you notice?

Now predict how many ways there are to build a house for a family of seven people.

Try it! Were you right?

Will your noticing always be true? Can you create an argument that would convince mathematicians?

Printable NRICH Roadshow resource.