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

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

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

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?

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?

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

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

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

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?

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?

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

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

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

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

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

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?

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

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

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

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

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

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

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

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

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

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

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

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

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.

A follow-up activity to Tiles in the Garden.

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 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?

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

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

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

Bernard Bagnall looks at what 'problem solving' might really mean in the context of primary classrooms.

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

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?

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?

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

How many ways can you find of tiling the square patio, using square tiles of different sizes?

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