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

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

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.

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.

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

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

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

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.

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

This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.

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

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

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

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?

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

These pictures show squares split into halves. Can you find other ways?

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

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

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.

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

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

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?

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

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

Sort the houses in my street into different groups. Can you do it in any other ways?

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

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?

In this article for teachers, Bernard gives an example of taking an initial activity and getting questions going that lead to other explorations.

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

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?

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

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?

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.

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?

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

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?

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

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?

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

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

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?

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

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

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

Numbers arranged in a square but some exceptional spatial awareness probably needed.

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