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.

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

How many models can you find which obey these rules?

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?

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

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.

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

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

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

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.

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

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?

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

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?

Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?

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

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.

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?

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?

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?

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

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?

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?

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?

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.

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

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

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?

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

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?

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.

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.

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?

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

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

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

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.

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

A follow-up activity to Tiles in the Garden.

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?

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

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