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?

How many models can you find which obey these rules?

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

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

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.

Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.

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?

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

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

A follow-up activity to Tiles in the Garden.

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?

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?

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

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?

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?

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

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

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

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?

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?

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

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

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.

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?

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

Explore Alex's number plumber. What questions would you like to ask? Don't forget to keep visiting NRICH projects site for the latest developments and questions.

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

What happens when you add the digits of a number then multiply the result by 2 and you keep doing this? You could try for different numbers and different rules.

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

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?

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

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

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

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 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.

Can you find ways of joining cubes together so that 28 faces are visible?

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

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?

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

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

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?

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?