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 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?

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

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?

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

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

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

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?

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?

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?

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?

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

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?

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?

Can you find a rule which connects consecutive triangular numbers?

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

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?

Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

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.

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

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

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

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

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

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.

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?

How efficiently can various flat shapes be fitted together?

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

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

A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .

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

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 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 ?

Imagine a stack of numbered cards with one on top. Discard the top, put the next card to the bottom and repeat continuously. Can you predict the last card?

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

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?

Consider a watch face which has identical hands and identical marks for the hours. It is opposite to a mirror. When is the time as read direct and in the mirror exactly the same between 6 and 7?

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?

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?

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. . . .

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

In how many different ways can I colour the five edges of a pentagon red, blue and green so that no two adjacent edges are the same colour?