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

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

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

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

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.

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.

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 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?

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?

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.

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?

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

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

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

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

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?

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

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

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

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

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

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

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

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 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 models can you find which obey these rules?

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

In my local town there are three supermarkets which each has a special deal on some products. If you bought all your shopping in one shop, where would be the cheapest?

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

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?

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

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

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?

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

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?

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

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?

Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?

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

Bernard Bagnall describes how to get more out of some favourite NRICH investigations.

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?