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?

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.

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 and explain the patterns that you see from recording just the units digits of numbers in the times tables.

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?

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?

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?

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

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?

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?

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

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?

Investigate these hexagons drawn from different sized equilateral triangles.

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

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?

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

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?

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

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.

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

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

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?

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

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

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.

How many different sets of numbers with at least four members can you find in the numbers in this box?

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

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.

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

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

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

When Charlie asked his grandmother how old she is, he didn't get a straightforward reply! Can you work out how old she is?

In this section from a calendar, put a square box around the 1st, 2nd, 8th and 9th. Add all the pairs of numbers. What do you notice about the answers?

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

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

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

Investigate the different ways you could split up these rooms so that you have double the number.

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?

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!

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

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?

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

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?

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?

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?