Using your knowledge of the properties of numbers, can you fill all the squares on the board?

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?

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

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.

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

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

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

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

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. . . .

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

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

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

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

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

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?

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.

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.

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

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?

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?

How many models can you find which obey these rules?

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

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

Make a clinometer and use it to help you estimate the heights of tall objects.

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

It might seem impossible but it is possible. How can you cut a playing card to make a hole big enough to walk through?

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

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.

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?

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

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.

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

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

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?

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

Can you make the birds from the egg tangram?

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.

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?

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?

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

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?

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

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?