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

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.

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

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?

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

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

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?

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?

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

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

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?

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 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?

What happens when you add the digits of a number then multiply the result by 2 and you keep doing this? You could try for different numbers and different rules.

"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 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 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?

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

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

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.

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

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

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?

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 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?

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?

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

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?

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

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?

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

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

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?

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?

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

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

Which times on a digital clock have a line of symmetry? Which look the same upside-down? You might like to try this investigation and find out!

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

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?

Investigate these hexagons drawn from different sized equilateral triangles.

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

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

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?

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

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?

A follow-up activity to Tiles in the Garden.