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

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?

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

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

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?

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

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?

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?

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

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

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?

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

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

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.

Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

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

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

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?

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

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

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.

Well now, what would happen if we lost all the nines in our number system? Have a go at writing the numbers out in this way and have a look at the multiplications table.

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.

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

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. . . .

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 out how the 6-triangle shape is transformed in these tessellations? Will the tessellations go on for ever? Why or why not?

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

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?

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

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

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

A follow-up activity to Tiles in the Garden.

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?

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

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

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

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

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

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

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

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

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?