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

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

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?

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

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

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?

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

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

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

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.

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

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?

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

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

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

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?

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?

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!

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

How many models can you find which obey these rules?

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?

Kaia is sure that her father has worn a particular tie twice a week in at least five of the last ten weeks, but her father disagrees. Who do you think is right?

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?

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?

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

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

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?

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

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 greatest number of squares you can make by overlapping three squares?

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

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

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

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

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

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

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?

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

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.

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

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

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

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

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?

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.