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?

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

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

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?

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

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?

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?

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?

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

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

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.

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

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

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

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?

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

Can you fit the tangram pieces into the outlines of the workmen?

Can you fit the tangram pieces into the outlines of the candle and sundial?

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

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

Can you fit the tangram pieces into the outline of Little Ming and Little Fung dancing?

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

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?

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

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

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.

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?

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?

Can you fit the tangram pieces into the outlines of these people?

Can you fit the tangram pieces into the outline of this shape. How would you describe it?

Can you fit the tangram pieces into the outlines of these clocks?

Can you fit the tangram pieces into the outlines of the chairs?

Can you fit the tangram pieces into the outline of the child walking home from school?

Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts?

Can you fit the tangram pieces into the outlines of Mai Ling and Chi Wing?

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

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

Can you fit the tangram pieces into the outline of Little Ming playing the board game?

Can you fit the tangram pieces into the outline of Little Fung at the table?

Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?

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

How can you make a curve from straight strips of paper?

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

The challenge for you is to make a string of six (or more!) graded cubes.

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

For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...

What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?

This is a simple paper-folding activity that gives an intriguing result which you can then investigate further.