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

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

Bernard Bagnall describes how to get more out of some favourite NRICH investigations.

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

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

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

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 make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles?

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.

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?

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?

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?

"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 ...?

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.

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.

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

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?

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?

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?

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.

When Charlie asked his grandmother how old she is, he didn't get a straightforward reply! Can you work out how old she is?

Follow the directions for circling numbers in the matrix. Add all the circled numbers together. Note your answer. Try again with a different starting number. What do you notice?

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?

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

Here is your chance to investigate the number 28 using shapes, cubes ... in fact anything at all.

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

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

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

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?

48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers?

This activity asks you to collect information about the birds you see in the garden. Are there patterns in the data or do the birds seem to visit randomly?

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?

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

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

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

Investigate what happens when you add house numbers along a street in different ways.

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!

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

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?

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

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?

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?

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