Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?

Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?

Design an arrangement of display boards in the school hall which fits the requirements of different people.

What is the best way to shunt these carriages so that each train can continue its journey?

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

This article for teachers describes how modelling number properties involving multiplication using an array of objects not only allows children to represent their thinking with concrete materials,. . . .

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

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?

Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

Can you make a 3x3 cube with these shapes made from small cubes?

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

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

How can the same pieces of the tangram make this bowl before and after it was chipped? Use the interactivity to try and work out what is going on!

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

I've made some cubes and some cubes with holes in. This challenge invites you to explore the difference in the number of small cubes I've used. Can you see any patterns?

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

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

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

Can you cut up a square in the way shown and make the pieces into a triangle?

What is the relationship between these first two shapes? Which shape relates to the third one in the same way? Can you explain why?

What is the greatest number of squares you can make by overlapping three squares?

Here are more buildings to picture in your mind's eye. Watch out - they become quite complicated!

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

A game for 1 person. Can you work out how the dice must be rolled from the start position to the finish? Play on line.

Can you arrange the shapes in a chain so that each one shares a face (or faces) that are the same shape as the one that follows it?

What shape has Harry drawn on this clock face? Can you find its area? What is the largest number of square tiles that could cover this area?

What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?

Think of a number, square it and subtract your starting number. Is the number youâ€™re left with odd or even? How do the images help to explain this?

A game has a special dice with a colour spot on each face. These three pictures show different views of the same dice. What colour is opposite blue?

How can you paint the faces of these eight cubes so they can be put together to make a 2 x 2 cube that is green all over AND a 2 x 2 cube that is yellow all over?

What happens when you turn these cogs? Investigate the differences between turning two cogs of different sizes and two cogs which are the same.

Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.

Make a flower design using the same shape made out of different sizes of paper.

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 this brazier for roasting chestnuts?

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 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 outline of this shape. How would you describe it?

A shape and space game for 2,3 or 4 players. Be the last person to be able to place a pentomino piece on the playing board. Play with card, or on the computer.

What shape is made when you fold using this crease pattern? Can you make a ring design?

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

Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?

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

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

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