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?

What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?

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

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

Make a cube out of straws and have a go at this practical challenge.

Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?

Imagine a 3 by 3 by 3 cube. If you and a friend drill holes in some of the small cubes in the ways described, how many will have holes drilled through them?

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

Here are the six faces of a cube - in no particular order. Here are three views of the cube. Can you deduce where the faces are in relation to each other and record them on the net of this cube?

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

Make a flower design using the same shape made out of different sizes of paper.

Exploring and predicting folding, cutting and punching holes and making spirals.

Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?

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

Have a look at what happens when you pull a reef knot and a granny knot tight. Which do you think is best for securing things together? Why?

A game has a special dice with a colour spot on each face. These three pictures show different views of the same dice. What colour is opposite blue?

Can you cut up a square in the way shown and make the pieces into a triangle?

This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?

For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...

Imagine a 4 by 4 by 4 cube. If you and a friend drill holes in some of the small cubes in the ways described, how many will not have holes drilled through them?

Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?

You have been given three shapes made out of sponge: a sphere, a cylinder and a cone. Your challenge is to find out how to cut them to make different shapes for printing.

Reasoning about the number of matches needed to build squares that share their sides.

Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?

Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.

Imagine a 3 by 3 by 3 cube made of 9 small cubes. Each face of the large cube is painted a different colour. How many small cubes will have two painted faces? Where are they?

Eight children each had a cube made from modelling clay. They cut them into four pieces which were all exactly the same shape and size. Whose pieces are the same? Can you decide who made each set?

Can you work out what is wrong with the cogs on a UK 2 pound coin?

Each of the nets of nine solid shapes has been cut into two pieces. Can you see which pieces go together?

A hundred square has been printed on both sides of a piece of paper. What is on the back of 100? 58? 23? 19?

What happens when you turn these cogs? Investigate the differences between turning two cogs of different sizes and two cogs which are the same.

Have a go at making a few of these shapes from paper in different sizes. What patterns can you create?

Can you visualise what shape this piece of paper will make when it is folded?

Can you fit the tangram pieces into the outline of the plaque design?

An extension of noughts and crosses in which the grid is enlarged and the length of the winning line can to altered to 3, 4 or 5.

This article looks at levels of geometric thinking and the types of activities required to develop this thinking.

Can you arrange the shapes in a chain so that each one shares a face (or faces) that are the same shape as the one that follows it?

This article for teachers describes a project which explores the power of storytelling to convey concepts and ideas to children.

These points all mark the vertices (corners) of ten hidden squares. Can you find the 10 hidden squares?

Can you work out what shape is made when this piece of paper is folded up using the crease pattern shown?

Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

Which of these dice are right-handed and which are left-handed?

A game for 2 players. Given a board of dots in a grid pattern, players take turns drawing a line by connecting 2 adjacent dots. Your goal is to complete more squares than your opponent.

Can you fit the tangram pieces into the outlines of the convex shapes?

Can you fit the tangram pieces into the outlines of Mah Ling and Chi Wing?

This second article in the series refers to research about levels of development of spatial thinking and the possible influence of instruction.

Can you describe a piece of paper clearly enough for your partner to know which piece it is?