Numbers arranged in a square but some exceptional spatial awareness probably needed.

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

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

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

Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.

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.

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?

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

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?

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

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

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?

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

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

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.

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

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

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

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

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

This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

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?

Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.

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

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.

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

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?

How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?

I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?

Investigate the different ways you could split up these rooms so that you have double the number.

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.

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

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

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

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

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

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

Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?

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

How many models can you find which obey these rules?

This challenge extends the Plants investigation so now four or more children are involved.

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

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

Complete these two jigsaws then put one on top of the other. What happens when you add the 'touching' numbers? What happens when you change the position of the jigsaws?

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

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

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