How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

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

How many different symmetrical shapes can you make by shading triangles or squares?

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

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 the telescope and microscope?

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

Can you work out what kind of rotation produced this pattern of pegs in our pegboard?

Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?

Can you find ways of joining cubes together so that 28 faces are visible?

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?

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

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

Find a way to cut a 4 by 4 square into only two pieces, then rejoin the two pieces to make an L shape 6 units high.

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

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

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

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

The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . .

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?

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

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

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 the child walking home from school?

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

Can you fit the tangram pieces into the outline of Granma T?

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 telephone?

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

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

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

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

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

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

In how many ways can you fit all three pieces together to make shapes with line symmetry?

A triangle ABC resting on a horizontal line is "rolled" along the line. Describe the paths of each of the vertices and the relationships between them and the original triangle.

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

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

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 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?

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 these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

Can you fit the tangram pieces into the outline of these convex shapes?

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?