### Four Triangles Puzzle

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

### Three Squares

What is the greatest number of squares you can make by overlapping three squares?

### 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?

# Building with Cubes

## Building with Cubes

A. Imagine you have five cubes. They are blue, red, yellow, black and orange:

Now imagine making this building.

Put the black cube just behind it.
Put the orange cube on top of the black cube.
Put the red cube on the left of the yellow cube.
Put the blue cube on the right of the black cube.

What does your building look like?

Try using cubes to check whether you had pictured it correctly.
Please draw or photograph your building and send it in to us.

B. Now imagine you have six cubes. They are orange, yellow, black, brown, pink and red:

Put the orange cube on the right hand side of the black cube.
Put the yellow cube just in front of the orange cube.
Put the pink cube on the right of the yellow cube.
Put the brown cube on top of the pink cube.

What does your building look like?

Try using cubes to check whether you had pictured it correctly.
Once again, please draw or photograph your building and send it in to us.

If you've enjoyed this problem, you might like to have a go at More Buiding with Cubes.

### Why do this problem?

This problem is designed to encourage children to begin to visualise - in particular, to see an image in their head. There are two visualising aspects in this proble- imagining the act of adding a new cube, but also keeping in your head the picture of what is already there.

The activity will also help to familiarise your class with positional language (in front, behind, to the right of, to the left of, on top, underneath etc.).

Being able to visualise is a useful skill which children are often not used to drawing on. By explicitly talking about visualisation, and offering opportunities like this for your class to practise visualising, you will be encouraging children to use this skill in their independent problem solving.

### Possible approach

To introduce the activity, you might like to give each child just two or three cubes. Ask them all to close their eyes and slowly describe an arrangement of these cubes to them, inviting them to picture what they hear in their heads. Then, give them time to make the "building" with their cubes and compare their models with their neighbour's.

You could also invite a child to imagine their own arrangement of cubes and describe it to the rest of the class.

The larger the number of cubes, the harder it might become for children to draw their own "buildings", so a digital camera might prove useful. In fact, drawing any number of cubes in 3D is rather tricky! If they are used to using it, isometric paper would help.