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?

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

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

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

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

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

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

I cut this square into two different shapes. What can you say about the relationship between them?

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

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

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.

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

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?

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

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

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?

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 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.

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?

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?

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

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

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

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

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.

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

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?

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

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?

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?

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

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?

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

This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?

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?

It starts quite simple but great opportunities for number discoveries and patterns!

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

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

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

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

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?

These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.