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?

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

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

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

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 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?

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

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?

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?

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

Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?

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

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

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

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

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?

How many models can you find which obey these rules?

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

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

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

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

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?

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.

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

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

What is the largest cuboid you can wrap in an A3 sheet of paper?

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

What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

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

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

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?

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.

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

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?

How many ways can you find of tiling the square patio, using square tiles of different sizes?

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?

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

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.

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

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

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

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

An investigation that gives you the opportunity to make and justify predictions.