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

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

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

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?

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?

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.

The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

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?

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

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?

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?

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

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

How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?

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?

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

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?

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?

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

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

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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.

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?

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?

This challenge involves calculating the number of candles needed on birthday cakes. It is an opportunity to explore numbers and discover new things.

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

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

A follow-up activity to Tiles in the Garden.

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

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

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

How many models can you find which obey these rules?

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

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

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?

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

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

An investigation that gives you the opportunity to make and justify predictions.

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

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

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?

Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?

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