Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

Can you each work out the number on your card? What do you notice? How could you sort the cards?

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

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

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?

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

If these balls are put on a line with each ball touching the one in front and the one behind, which arrangement makes the shortest line of balls?

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

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

What happens when a procedure calls itself?

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?

More Logo for beginners. Learn to calculate exterior angles and draw regular polygons using procedures and variables.

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

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

Can you make the birds from the egg tangram?

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

Write a Logo program, putting in variables, and see the effect when you change the variables.

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

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

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

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?

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?

Can you puzzle out what sequences these Logo programs will give? Then write your own Logo programs to generate sequences.

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

Turn through bigger angles and draw stars with Logo.

Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?

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.

Build a scaffold out of drinking-straws to support a cup of water

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

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

How many models can you find which obey these 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?

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

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

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

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 deduce the pattern that has been used to lay out these bottle tops?

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?

Learn to write procedures and build them into Logo programs. Learn to use variables.

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?

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

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

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