A cheap and simple toy with lots of mathematics. Can you interpret the images that are produced? Can you predict the pattern that will be produced using different wheels?

For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.

A bicycle passes along a path and leaves some tracks. Is it possible to say which track was made by the front wheel and which by the back wheel?

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

A 10x10x10 cube is made from 27 2x2 cubes with corridors between them. Find the shortest route from one corner to the opposite corner.

Imagine a rectangular tray lying flat on a table. Suppose that a plate lies on the tray and rolls around, in contact with the sides as it rolls. What can we say about the motion?

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

Small circles nestle under touching parent circles when they sit on the axis at neighbouring points in a Farey sequence.

A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be?

The net of a cube is to be cut from a sheet of card 100 cm square. What is the maximum volume cube that can be made from a single piece of card?

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

A blue coin rolls round two yellow coins which touch. The coins are the same size. How many revolutions does the blue coin make when it rolls all the way round the yellow coins? Investigate for a. . . .

A box of size a cm by b cm by c cm is to be wrapped with a square piece of wrapping paper. Without cutting the paper what is the smallest square this can be?

A visualisation problem in which you search for vectors which sum to zero from a jumble of arrows. Will your eyes be quicker than algebra?

What can you see? What do you notice? What questions can you ask?

A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square?

A right-angled isosceles triangle is rotated about the centre point of a square. What can you say about the area of the part of the square covered by the triangle as it rotates?

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

It is known that the area of the largest equilateral triangular section of a cube is 140sq cm. What is the side length of the cube? The distances between the centres of two adjacent faces of. . . .

I found these clocks in the Arts Centre at the University of Warwick intriguing - do they really need four clocks and what times would be ambiguous with only two or three of them?

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

See if you can anticipate successive 'generations' of the two animals shown here.

In this problem we see how many pieces we can cut a cube of cheese into using a limited number of slices. How many pieces will you be able to make?

Two angles ABC and PQR are floating in a box so that AB//PQ and BC//QR. Prove that the two angles are equal.

Have you got the Mach knack? Discover the mathematics behind exceeding the sound barrier.

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.

Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?

In a three-dimensional version of noughts and crosses, how many winning lines can you make?

This is a simple version of an ancient game played all over the world. It is also called Mancala. What tactics will increase your chances of winning?

Takes you through the systematic way in which you can begin to solve a mixed up Cubic Net. How close will you come to a solution?

A game for 2 people. Take turns joining two dots, until your opponent is unable to move.

Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?

A cube is made from smaller cubes, 5 by 5 by 5, then some of those cubes are removed. Can you make the specified shapes, and what is the most and least number of cubes required ?

Can you find a rule which relates triangular numbers to square numbers?

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

This is an interactive net of a Rubik's cube. Twists of the 3D cube become mixes of the squares on the 2D net. Have a play and see how many scrambles you can undo!

A square of area 3 square units cannot be drawn on a 2D grid so that each of its vertices have integer coordinates, but can it be drawn on a 3D grid? Investigate squares that can be drawn.

On the 3D grid a strange (and deadly) animal is lurking. Using the tracking system can you locate this creature as quickly as possible?

Show that all pentagonal numbers are one third of a triangular number.

Can you find a rule which connects consecutive triangular numbers?

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

What 3D shapes occur in nature. How efficiently can you pack these shapes together?

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?

We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?

Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?

Can you describe this route to infinity? Where will the arrows take you next?