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

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?

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

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?

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

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

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

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?

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

How many models can you find which obey these rules?

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

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?

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 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.

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?

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

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?

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

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

Can you make the birds from the egg tangram?

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?

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

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

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.

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.

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 these clocks?

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

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 these people?

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

Can you fit the tangram pieces into the outline of these rabbits?

Can you fit the tangram pieces into the outline of the telescope and microscope?

Can you fit the tangram pieces into the outline of this goat and giraffe?

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

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

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

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

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

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

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

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