A challenging activity focusing on finding all possible ways of stacking rods.

Which way of flipping over and/or turning this grid will give you the highest total? You'll need to imagine where the numbers will go in this tricky task!

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

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.

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?

This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.

How many ways can you find of tiling the square patio, using square tiles 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 how this pattern of squares continues. You could measure lengths, areas and angles.

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

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

A follow-up activity to Tiles in the Garden.

How will you decide which way of flipping over and/or turning the grid will give you the highest total?

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

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

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

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

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?

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.

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!

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

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

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.

In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.

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

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.

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 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 the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

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?

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?

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

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

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.

We went to the cinema and decided to buy some bags of popcorn so we asked about the prices. Investigate how much popcorn each bag holds so find out which we might have bought.

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

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?

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?

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

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

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 how many ways can you stack these rods, following the rules?

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

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?

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