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

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?

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?

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

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

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.

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

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?

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

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

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?

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

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

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.

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

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?

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.

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

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

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

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

How many models can you find which obey these rules?

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?

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?

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.

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

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

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?

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?

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

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

A follow-up activity to Tiles in the Garden.

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

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?

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

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

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

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

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

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

Explore Alex's number plumber. What questions would you like to ask? Don't forget to keep visiting NRICH projects site for the latest developments and questions.

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?