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?

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

This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.

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 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?

Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?

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

A follow-up activity to Tiles in the Garden.

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

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

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?

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

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

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.

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

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

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

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

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

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

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

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.

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?

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

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

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

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

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

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

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?

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

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

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

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

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?

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

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

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

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

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?

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

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

A description of some experiments in which you can make discoveries about 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 ...?