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.

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

Ahmed is making rods using different numbers of cubes. Which rod is twice the length of his first rod?

How many models can you find which obey these rules?

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 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.

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?

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?

This project challenges you to work out the number of cubes hidden under a cloth. What questions would you like to ask?

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?

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?

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?

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

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 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?

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

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?

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

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?

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?

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

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

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

Can you make the birds from the egg tangram?

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

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.

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?

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

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

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

This activity investigates how you might make squares and pentominoes from Polydron.

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

Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?

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

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

For this activity which explores capacity, you will need to collect some bottles and jars.

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?

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

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?

In this activity focusing on capacity, you will need a collection of different jars and bottles.

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

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

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

Can you make five differently sized squares from the tangram 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?