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?

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.

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.

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

Investigate these hexagons drawn from different sized equilateral triangles.

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

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?

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

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

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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.

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

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?

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

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

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

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?

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.

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?

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?

Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?

A follow-up activity to Tiles in the Garden.

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

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

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?

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

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

Can you design a new shape for the twenty-eight squares and arrange the numbers in a logical way? What patterns do you notice?

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

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

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

"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 ...?

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?

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?

We need to wrap up this cube-shaped present, remembering that we can have no overlaps. What shapes can you find to use?

How many faces can you see when you arrange these three cubes in different ways?

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 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.

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?

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?

These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.