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.

These pictures show squares split into halves. Can you find other ways?

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

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

Investigate these hexagons drawn from different sized equilateral triangles.

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?

Explore the triangles that can be made with seven sticks of the same length.

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?

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

If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?

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

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

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

Here is your chance to investigate the number 28 using shapes, cubes ... in fact anything at all.

How many models can you find which obey these rules?

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

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?

Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?

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.

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?

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

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

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

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?

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?

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

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

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?

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

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

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?

Here are many ideas for you to investigate - all linked with the number 2000.

Investigate the numbers that come up on a die as you roll it in the direction of north, south, east and west, without going over the path it's already made.

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

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

A follow-up activity to Tiles in the Garden.

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

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

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

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

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?

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?