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

This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.

In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.

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

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

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?

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

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

Can you make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles?

In this investigation, we look at Pascal's Triangle in a slightly different way - rotated and with the top line of ones taken off.

This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.

We went to the cinema and decided to buy some bags of popcorn so we asked about the prices. Investigate how much popcorn each bag holds so find out which we might have bought.

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.

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

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?

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

Investigate the numbers that come up on a die as you roll it in the direction of north, south, east and west, without going over the path it's already made.

Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

What do these two triangles have in common? How are they related?

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

Can you find out how the 6-triangle shape is transformed in these tessellations? Will the tessellations go on for ever? Why or why not?

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

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

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

Investigate these hexagons drawn from different sized equilateral triangles.

How many models can you find which obey these rules?

How many faces can you see when you arrange these three cubes in different ways?

What is the largest number of circles we can fit into the frame without them overlapping? How do you know? What will happen if you try the other shapes?

Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?

Bernard Bagnall looks at what 'problem solving' might really mean in the context of primary classrooms.

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

An activity making various patterns with 2 x 1 rectangular tiles.

A follow-up activity to Tiles in the Garden.

In how many ways can you stack these rods, following the rules?

Investigate the different ways you could split up these rooms so that you have double the number.

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.