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?

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?

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.

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

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

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

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?

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

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?

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

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?

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

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

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

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 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?

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

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?

Can you make the birds from the egg tangram?

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

Can you split each of the shapes below in half so that the two parts are exactly the same?

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

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?

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

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

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

Can you put these shapes in order of size? Start with the smallest.

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 fit the tangram pieces into the outline of this junk?

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

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?

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

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?

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

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

Can you fit the tangram pieces into the outline of Little Ming playing the board game?

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 the child walking home from school?

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

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 these people?

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

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

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 Fung at the table?

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?