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

What statements can you make about the car that passes the school gates at 11am on Monday? How will you come up with statements and test your ideas?

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

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?

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

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?

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?

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

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?

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

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

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

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

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

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?

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

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. . . .

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

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?

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?

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

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.

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?

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

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?

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?

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

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

All types of mathematical problems serve a useful purpose in mathematics teaching, but different types of problem will achieve different learning objectives. In generalmore open-ended problems have. . . .

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?

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?

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.

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?

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?

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

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?

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

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

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

In this investigation, you are challenged to make mobile phone numbers which are easy to remember. What happens if you make a sequence adding 2 each time?

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?

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

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

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

Numbers arranged in a square but some exceptional spatial awareness probably needed.