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?

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

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?

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

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.

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

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

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

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?

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

"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 ...?

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

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

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

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?

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

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

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?

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.

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?

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

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.

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

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

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

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

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?

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?

Investigate these hexagons drawn from different sized equilateral triangles.

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

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

Start with four numbers at the corners of a square and put the total of two corners in the middle of that side. Keep going... Can you estimate what the size of the last four numbers will be?

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

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?

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

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

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?

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

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

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

The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

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

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

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?

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?

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

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

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