
What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

Ever thought of playing three dimensional Noughts and Crosses? This problem might help you visualise what's involved.

Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?

You can solve frogs on the computer, using counters, or acting it out. Start with frogs in a line on one side, and toads on the other, with a space in between. They need to change places.

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?
Lyndon Baker describes how the Mobius strip and Euler's law can introduce pupils to the idea of topology.

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

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

A game for 2 players. Can be played online. One player has 1 red counter, the other has 4 blue. The red counter needs to reach the other side, and the blue needs to trap the red.

A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

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

ABC is an equilateral triangle and P is a point in the interior of the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP must be less than 10 cm.

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

Starting with four different triangles, imagine you have an unlimited number of each type. How many different tetrahedra can you make? Convince us you have found them all.

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

A game for 1 or 2 people. Use the interactive version, or play with friends. Try to round up as many counters as possible.

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

A 3x3x3 cube may be reduced to unit cubes in six saw cuts. If after every cut you can rearrange the pieces before cutting straight through, can you do it in fewer?

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

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

Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

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

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

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!

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

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

Can you fit the tangram pieces into the outlines of these clocks?
Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?

These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?

A bus route has a total duration of 40 minutes. Every 10 minutes, two buses set out, one from each end. How many buses will one bus meet on its way from one end to the other end?

The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?

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 Little Ming playing the board game?

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

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

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

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

Here is a solitaire type environment for you to experiment with. Which targets can you reach?

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

Find all the ways to cut out a 'net' of six squares that can be folded into a cube.

Can you fit the tangram pieces into the outline of the rocket?

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

Imagine you are suspending a cube from one vertex (corner) and allowing it to hang freely. Now imagine you are lowering it into water until it is exactly half submerged. What shape does the surface. . . .

Can you fit the tangram pieces into the outline of this sports car?

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

Draw some isosceles triangles with an area of 9 cm squared and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

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