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

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

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.

How many models can you find which obey these rules?

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

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?

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.

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?

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?

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

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

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?

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?

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

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

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?

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?

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

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?

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

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.

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

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?

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?

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?

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

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

It might seem impossible but it is possible. How can you cut a playing card to make a hole big enough to walk through?

You could use just coloured pencils and paper to create this design, but it will be more eye-catching if you can get hold of hammer, nails and string.

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

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

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

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

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

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?

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?

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

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

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

Can you deduce the pattern that has been used to lay out these bottle tops?

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

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?

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

A brief video looking at how you can sometimes use symmetry to distinguish knots. Can you use this idea to investigate the differences between the granny knot and the reef knot?

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