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?

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

NRICH December 2006 advent calendar - a new tangram for each 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?

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

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?

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

Can you make the birds from the egg tangram?

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?

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

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

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?

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

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

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

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

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

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?

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?

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.

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

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

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

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.

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

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

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?

How many models can you find which obey these rules?

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

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

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

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

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

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

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

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

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

Here are some ideas to try in the classroom for using counters to investigate number patterns.

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

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?

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

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