Try to picture these buildings of cubes in your head. Can you make them to check whether you had imagined them correctly?

How can you paint the faces of these eight cubes so they can be put together to make a 2 x 2 cube that is green all over AND a 2 x 2 cube that is yellow all over?

This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.

Here are more buildings to picture in your mind's eye. Watch out - they become quite complicated!

Make a chair and table out of interlocking cubes, making sure that the chair fits under the table!

Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

I've made some cubes and some cubes with holes in. This challenge invites you to explore the difference in the number of small cubes I've used. Can you see any patterns?

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

How many models can you find which obey these rules?

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

Kimie and Sebastian were making sticks from interlocking cubes and lining them up. Can they make their lines the same length? Can they make any other lines?

When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.

Here are some pictures of 3D shapes made from cubes. Can you make these shapes yourself?

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?

Take three differently coloured blocks - maybe red, yellow and blue. Make a tower using one of each colour. How many different towers can you make?

We need to wrap up this cube-shaped present, remembering that we can have no overlaps. What shapes can you find to use?

In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?

Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?

Penta people, the Pentominoes, always build their houses from five square rooms. I wonder how many different Penta homes you can create?

Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?

Ahmed is making rods using different numbers of cubes. Which rod is twice the length of his first rod?

This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

These two group activities use mathematical reasoning - one is numerical, one geometric.

Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?

This project challenges you to work out the number of cubes hidden under a cloth. What questions would you like to ask?

This activity is best done with a whole class or in a large group. Can you match the cards? What happens when you add pairs of the numbers together?

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

Use cubes to continue making the numbers from 7 to 20. Are they sticks, rectangles or squares?

If you had 36 cubes, what different cuboids could you make?

This article takes a closer look at some of the toys and games that can enhance a child's mathematical learning.