Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?

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

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

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?

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?

These pictures show squares split into halves. Can you find other ways?

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?

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

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?

Try continuing these patterns made from triangles. Can you create your own repeating pattern?

Use your mouse to move the red and green parts of this disc. Can you make images which show the turnings described?

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

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

Explore the triangles that can be made with seven sticks of the same length.

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

There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?

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

A group of children are discussing the height of a tall tree. How would you go about finding out its height?

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?

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.

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

How many models can you find which obey these rules?

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

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?

Sort the houses in my street into different groups. Can you do it in any other ways?

Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?

Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

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

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

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?

I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

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

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

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

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?

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.

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.

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?

How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?

Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?

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

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

Place this "worm" on the 100 square and find the total of the four squares it covers. Keeping its head in the same place, what other totals can you make?

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?

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?