A challenge that requires you to apply your knowledge of the properties of numbers. Can you fill all the squares on the board?

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

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

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

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?

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

You have 27 small cubes, 3 each of nine colours. Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour.

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 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?

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?

Our 2008 Advent Calendar has a 'Making Maths' activity for every day in the run-up to Christmas.

NRICH December 2006 advent calendar - a new tangram for each day in the run-up to Christmas.

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

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

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

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

Starting with four different triangles, imagine you have an unlimited number of each type. How many different tetrahedra can you make? Convince us you have found them all.

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

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

In this article for teachers, Bernard uses some problems to suggest that once a numerical pattern has been spotted from a practical starting point, going back to the practical can help explain. . . .

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?

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?

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.

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

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?

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.

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

How many models can you find which obey these rules?

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

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

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

Here is a solitaire type environment for you to experiment with. Which targets can you reach?

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

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?

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?

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

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?

You could use just coloured pencils and paper to create this design, but it will be more eye-catching if you can get hold of hammer, nails and string.

Have a go at drawing these stars which use six points drawn around a circle. Perhaps you can create your own designs?

Can you make the birds from the egg tangram?

Here's a simple way to make a Tangram without any measuring or ruling lines.

Can you fit the tangram pieces into the outline of this junk?

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

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

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

Follow the diagrams to make this patchwork piece, based on an octagon in a square.

If you'd like to know more about Primary Maths Masterclasses, this is the package to read! Find out about current groups in your region or how to set up your own.

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