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

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

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.

Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?

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?

Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?

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?

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

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

Make a cube out of straws and have a go at this practical challenge.

Can you cut up a square in the way shown and make the pieces into a triangle?

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

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?

This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?

Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?

Make a flower design using the same shape made out of different sizes of paper.

Can you visualise what shape this piece of paper will make when it is folded?

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 ...

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

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?

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

You have been given three shapes made out of sponge: a sphere, a cylinder and a cone. Your challenge is to find out how to cut them to make different shapes for printing.

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

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

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

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

This was a problem for our birthday website. Can you use four of these pieces to form a square? How about making a square with all five pieces?

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

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?

Have a look at what happens when you pull a reef knot and a granny knot tight. Which do you think is best for securing things together? Why?

Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?

Reasoning about the number of matches needed to build squares that share their sides.

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 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?

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

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

Can you make the birds from the egg tangram?

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?

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

What shape is made when you fold using this crease pattern? Can you make a ring design?

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

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?

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

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