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

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

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 challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

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

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

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

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?

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?

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

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

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

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

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

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

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.

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?

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 many models can you find which obey these rules?

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

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

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.

Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possible answers?

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

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

Explore ways of colouring this set of triangles. Can you make symmetrical patterns?

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

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

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?

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

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

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?

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

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?

The red ring is inside the blue ring in this picture. Can you rearrange the rings in different ways? Perhaps you can overlap them or put one outside another?

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?

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

In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?

A follow-up activity to Tiles in the Garden.

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

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

This problem is intended to get children to look really hard at something they will see many times in the next few months.

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

Why does the tower look a different size in each of these pictures?