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

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

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

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?

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 make the birds from the egg tangram?

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

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?

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?

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.

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?

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

A game to make and play based on the number line.

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?

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?

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

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

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

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

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?

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

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?

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

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

Surprise your friends with this magic square trick.

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?

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

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

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.

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

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?

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

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

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

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

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

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

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

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.

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.

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?

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?

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