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

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

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

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.

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

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

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?

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

Explore the triangles that can be made with seven sticks of the same length.

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

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

Investigate these hexagons drawn from different sized equilateral triangles.

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

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?

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

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

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?

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?

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

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.

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

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?

Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?

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

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

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

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?

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?

A follow-up activity to Tiles in the Garden.

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

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

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

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

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?

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?

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

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

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?

Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?

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

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?

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?

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?