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

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

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

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

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 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?

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

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

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

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

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

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

A follow-up activity to Tiles in the Garden.

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

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

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.

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?

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

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

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

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

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

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

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

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

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

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?

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?

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?

Complete these two jigsaws then put one on top of the other. What happens when you add the 'touching' numbers? What happens when you change the position of the jigsaws?

Place this "worm" on the 100 square and find the total of the four squares it covers. Keeping its head in the same place, what other totals can you make?

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?

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?

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

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?

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?

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.

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

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

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

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?

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

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