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

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

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

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?

You have a set of the digits from 0 – 9. Can you arrange these in the 5 boxes to make two-digit numbers as close to the targets as possible?

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

You will need a long strip of paper for this task. Cut it into different lengths. How could you find out how long each piece is?

Kimie and Sebastian were making sticks from interlocking cubes and lining them up. Can they make their lines the same length? Can they make any other lines?

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

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

These pictures show squares split into halves. Can you find other ways?

The Man is much smaller than us. Can you use the picture of him next to a mug to estimate his height and how much tea he drinks?

Galileo, a famous inventor who lived about 400 years ago, came up with an idea similar to this for making a time measuring instrument. Can you turn your pendulum into an accurate minute timer?

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?

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?

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.

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

Can you put these shapes in order of size? Start with the smallest.

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

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?

This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

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 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?

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 make the birds from the egg tangram?

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

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

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?

Try continuing these patterns made from triangles. Can you create your own repeating pattern?

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

Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?

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?

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

A game in which players take it in turns to choose a number. Can you block your opponent?

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?

Explore the triangles that can be made with seven sticks of the same length.

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

If you have ten counters numbered 1 to 10, how many can you put into pairs that add to 10? Which ones do you have to leave out? Why?

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

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.

If you count from 1 to 20 and clap more loudly on the numbers in the two times table, as well as saying those numbers loudly, which numbers will be loud?

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?

How many models can you find which obey these rules?

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

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