Delight your friends with this cunning trick! Can you explain how it works?

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

Here is a version of the game 'Happy Families' for you to make and play.

Surprise your friends with this magic square trick.

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

The triangle ABC is equilateral. The arc AB has centre C, the arc BC has centre A and the arc CA has centre B. Explain how and why this shape can roll along between two parallel tracks.

I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?

These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?

Can you make the birds from the egg tangram?

Which of the following cubes can be made from these nets?

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

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

How can you make an angle of 60 degrees by folding a sheet of paper twice?

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?

Here are some ideas to try in the classroom for using counters to investigate number patterns.

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.

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

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?

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...

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?

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

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.

Here is a chance to create some Celtic knots and explore the mathematics behind them.

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

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?

Exploring and predicting folding, cutting and punching holes and making spirals.

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?

Time for a little mathemagic! Choose any five cards from a pack and show four of them to your partner. How can they work out the fifth?

Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.

Use the tangram pieces to make our pictures, or to design some of your own!

A game in which players take it in turns to choose a number. Can you block your opponent?

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

How many models can you find which obey these rules?

Factors and Multiples game for an adult and child. How can you make sure you win this game?

These practical challenges are all about making a 'tray' and covering it with paper.

Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?

What is the greatest number of squares you can make by overlapping three squares?

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