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?

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

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

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?

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?

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?

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

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?

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.

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

Have a go at making a few of these shapes from paper in different sizes. What patterns can you create?

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?

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

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

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?

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?

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?

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

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

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?

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

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

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 many models can you find which obey these rules?

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?

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

Can you describe a piece of paper clearly enough for your partner to know which piece it is?

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

Have you ever tried tessellating capital letters? Have a look at these examples and then try some for yourself.

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

Can you fit the tangram pieces into the outlines of the candle and sundial?

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 outlines of the workmen?

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

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 Little Ming playing the board game?

Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts?

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

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

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

Can you fit the tangram pieces into the outline of Little Fung at the table?

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

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 outline of this telephone?

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 chairs?

Can you fit the tangram pieces into the outlines of Mai Ling and Chi Wing?

We can cut a small triangle off the corner of a square and then fit the two pieces together. Can you work out how these shapes are made from the two pieces?