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

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

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 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?

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

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

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

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?

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

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

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?

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

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

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

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

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

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?

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 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?

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

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

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

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

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?

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

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

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?

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

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?

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?

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?

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!

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?

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

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

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?