A city of towers
In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?
Problem
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.
Getting Started
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.
Teachers' Resources
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.