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 models can you find which obey these rules?

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

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?

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 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?

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?

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?

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

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

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

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

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?

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?

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.

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

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

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

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

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

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

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

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.

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

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

This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.

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

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?

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.

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?

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

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

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

We went to the cinema and decided to buy some bags of popcorn so we asked about the prices. Investigate how much popcorn each bag holds so find out which we might have bought.

This activity investigates how you might make squares and pentominoes from Polydron.

Can you make the birds from the egg tangram?

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

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?

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

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?

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

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

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?

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.

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

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

Can you fit the tangram pieces into the outline of Wai Ping, Wah Ming and Chi Wing?

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