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

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?

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?

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

What shape is made when you fold using this crease pattern? Can you make a ring design?

Have you ever tried tessellating capital letters? Have a look at these examples and then try some for yourself.

Can you split each of the shapes below in half so that the two parts are exactly the same?

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

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

We can cut a small triangle off the corner of a square and then fit the two pieces together. Can you work out how these shapes are made from the two pieces?

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

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

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?

What shape is the overlap when you slide one of these shapes half way across another? Can you picture it in your head? Use the interactivity to check your visualisation.

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

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

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 ...

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

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

What does the overlap of these two shapes look like? Try picturing it in your head and then use the interactivity to test your prediction.

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

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

This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!

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?

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?

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

How will you go about finding all the jigsaw pieces that have one peg and one hole?

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

Can you find ways of joining cubes together so that 28 faces are visible?

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 shape has Harry drawn on this clock face? Can you find its area? What is the largest number of square tiles that could cover this area?

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

What is the best way to shunt these carriages so that each train can continue its journey?

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

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?

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?

What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?

How many pieces of string have been used in these patterns? Can you describe how you know?

How many loops of string have been used to make these patterns?

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?

If you can post the triangle with either the blue or yellow colour face up, how many ways can it be posted altogether?

Draw three straight lines to separate these shapes into four groups - each group must contain one of each shape.

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?

Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?