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?

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

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 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.

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

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

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?

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

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

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

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

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?

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?

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

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

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

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

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?

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?

Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?

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

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?

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

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?

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

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?

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.

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?

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

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

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

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

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

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.

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

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?

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

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?

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?

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

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"?

In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?