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?

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?

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.

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.

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.

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

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

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?

How many models can you find which obey these rules?

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?

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?

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

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

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

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?

Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

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

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.

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

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

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

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

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

Can you make the birds from the egg tangram?

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

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?

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

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

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?

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

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?

Ideas for practical ways of representing data such as Venn and Carroll diagrams.

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

Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

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

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?

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

Can you recreate this Indian screen pattern? Can you make up similar patterns of your own?

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

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

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

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

What do these two triangles have in common? How are they related?