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?

Investigate these hexagons drawn from different sized equilateral triangles.

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

This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.

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?

How many models can you find which obey these rules?

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

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

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

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

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

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

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

Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?

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

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

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.

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

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

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

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?

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

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?

Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?

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

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

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?

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?

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

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?

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

How many faces can you see when you arrange these 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?

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

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

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?

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?

Can you find ways of joining cubes together so that 28 faces are visible?

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

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

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

A follow-up activity to Tiles in the Garden.

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

Sort the houses in my street into different groups. Can you do it in any other ways?

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.