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.
You could draw the houses on squared paper or make them with cube bricks.
How will you know you have found all the different houses?
You could start by finding all the houses for four people, then five people etc.
Thank you to everybody who sent us their solutions to this problem. You agreed that there are six different ways of building a house for seven people.
Chris from Holyport C of E Primary, Yasi from Canberra Grammar and James from St Woolos, Newport all sent in pictures of the houses. Here is James':
I like the way you have done this in a very logical order, James, moving one 'room' from the left to the right each time. Jordan also wrote to us and described the same method as James', but in words:
I started with seven cubes because I had to house seven people.
I drew one stack and reversed it, just moving the odd cube across.
I used the same method two more times, changing the height of the stack and moving it from left to right.
My answer is six different types of house.
Alice from Henrietta Barnett and Patricia from Chongfu Primary School both described the six solutions too.
Callum from Canberra Grammar School went a step further and decided to look at different combinations of rooms if they were different colours. So, for example, you may decide you have two red rooms, two yellow rooms and three blue rooms. You could then investigate the number of different ways you could arrange the coloured rooms for each of James' pictures. This makes a nice extension to the problem, Callum - well done!
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.
Why do this problem?
This problem is essentially an investigation into combinations of a number of cubes. It is a practical activity which involves visualising and relating 3D shapes to their representation on paper. Young children are often introduced to sets of regular polyhedra and similar sorts of shapes, less often do they systematically explore shapes made up from cubes.
Possible approach
You could start with this story as an introduction to the problem. Alternatively you could simply talk through the problem as it is written. Ideally, it would be good to supply interlocking cubes or other cube bricks to represent the rooms and 2cm squared paper or plain paper for recording. It might help to begin the challenge all together before asking children to work in pairs on the problem so that they are able to talk through their ideas and compare their results with a partner.
Key questions
Possible extension
What would happen if the rooms were different colours? Some children could create their own rules for building houses.
Possible support
Having practical resources will help all learners access this task.