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

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?

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

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

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

These caterpillars have 16 parts. What different shapes do they make if each part lies in the small squares of a 4 by 4 square?

Explore ways of colouring this set of triangles. Can you make symmetrical patterns?

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

"Ip dip sky blue! Who's 'it'? It's you!" Where would you position yourself so that you are 'it' if there are two players? Three players ...?

What can you say about the child who will be first on the playground tomorrow morning at breaktime in your school?

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?

The red ring is inside the blue ring in this picture. Can you rearrange the rings in different ways? Perhaps you can overlap them or put one outside another?

Investigate the number of faces you can see when you arrange three cubes in different ways.

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.

Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?

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?

There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?

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

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

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

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

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

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

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?

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.

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

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

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

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?

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!

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?

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?

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

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?

Take a look at these data collected by children in 1986 as part of the Domesday Project. What do they tell you? What do you think about the way they are presented?

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

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

This challenge extends the Plants investigation so now four or more children are involved.

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

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

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

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

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

Many natural systems appear to be in equilibrium until suddenly a critical point is reached, setting up a mudslide or an avalanche or an earthquake. In this project, students will use a simple. . . .

A challenging activity focusing on finding all possible ways of stacking rods.

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

How many models can you find which obey these rules?