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?

I cut this square into two different shapes. What can you say about the relationship between them?

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

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

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

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

These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.

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

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

How many ways can you find of tiling the square patio, using square tiles of different sizes?

If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?

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

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

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

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

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

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.

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

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?

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

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

Investigate the different ways these aliens count in this challenge. You could start by thinking about how each of them would write our number 7.

Which times on a digital clock have a line of symmetry? Which look the same upside-down? You might like to try this investigation and find out!

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

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.

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

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?

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.

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

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?

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?

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?

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

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?

In this investigation, we look at Pascal's Triangle in a slightly different way - rotated and with the top line of ones taken off.

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

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

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

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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.

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

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

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

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.

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?