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

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?

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?

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

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

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

How many models can you find which obey these rules?

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

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?

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.

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?

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?

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?

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?

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?

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.

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

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

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

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?

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?

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

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

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

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?

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?

Factors and Multiples game for an adult and child. How can you make sure you win this game?

Can you make a rectangle with just 2 dominoes? What about 3, 4, 5, 6, 7...?

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

Make a chair and table out of interlocking cubes, making sure that the chair fits under the table!

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

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

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

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

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

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?

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

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

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?

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?

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.

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