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

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

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

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.

How many models can you find which obey these rules?

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

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?

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?

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?

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?

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 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.

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

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

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

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

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

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.

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

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

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

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

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

A follow-up activity to Tiles in the Garden.

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?

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.

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?

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

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

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 these hexagons drawn from different sized equilateral triangles.

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?

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

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?

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

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?

In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?

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

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?

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?

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

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