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?

The challenge for you is to make a string of six (or more!) graded cubes.

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 fit the tangram pieces into the outline of the child walking home from school?

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

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

Can you fit the tangram pieces into the outline of this shape. How would you describe it?

Can you fit the tangram pieces into the outlines of these clocks?

Can you fit the tangram pieces into the outlines of these people?

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 outline of this brazier for roasting chestnuts?

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?

Can you fit the tangram pieces into the outline of this plaque design?

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

Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?

Can you fit the tangram pieces into the outline of Mai Ling?

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

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

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

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

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

Can you fit the tangram pieces into the outline of this telephone?

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

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

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

This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?

This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!

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?

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

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

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

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?

Exploring and predicting folding, cutting and punching holes and making spirals.

Make a cube out of straws and have a go at this practical challenge.

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

Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?

Can you fit the tangram pieces into the outline of this junk?

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?

Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.

Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?

Here's a simple way to make a Tangram without any measuring or ruling lines.

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

Can you fit the tangram pieces into the outline of Wai Ping, Wah Ming and Chi Wing?

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

Can you cut a regular hexagon into two pieces to make a parallelogram? Try cutting it into three pieces to make a rhombus!