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

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?

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?

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

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

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possibilities that could come up?

The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?

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?

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

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!

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

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?

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?

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

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

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?

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.

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?

This article for teachers suggests ideas for activities built around 10 and 2010.

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

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?

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

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. . . .

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.

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?

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

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

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

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

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

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

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

A description of some experiments in which you can make discoveries about triangles.

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

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

"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 ...?

How will you decide which way of flipping over and/or turning the grid will give you the highest total?

A challenging activity focusing on finding all possible ways of stacking rods.

A follow-up activity to Tiles in the Garden.

In how many ways can you stack these rods, following the rules?

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

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

It starts quite simple but great opportunities for number discoveries and patterns!