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

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

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

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.

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.

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

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

How many models can you find which obey these rules?

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?

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

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?

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?

In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?

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

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

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.

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

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

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?

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

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

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.

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

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

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?

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

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

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?

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

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

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

Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?

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?

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

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.

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?

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

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

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?

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

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

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?