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.

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

Can you find the link between these beautiful circle patterns and Farey Sequences?

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

In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

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?

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

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

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?

A triangle PQR, right angled at P, slides on a horizontal floor with Q and R in contact with perpendicular walls. What is the locus of P?

Glarsynost lives on a planet whose shape is that of a perfect regular dodecahedron. Can you describe the shortest journey she can make to ensure that she will see every part of the planet?

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design...

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?

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.

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 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. . . .

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

This article explores ths history of theories about the shape of our planet. It is the first in a series of articles looking at the significance of geometric shapes in the history of astronomy.

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?

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?

An irregular tetrahedron has two opposite sides the same length a and the line joining their midpoints is perpendicular to these two edges and is of length b. What is the volume of the tetrahedron?

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

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

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?

The second in a series of articles on visualising and modelling shapes in the history of astronomy.

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 ?

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?

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?

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 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.

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.

Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.

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.

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

Some treasure has been hidden in a three-dimensional grid! Can you work out a strategy to find it as efficiently as possible?

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!

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

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

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

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

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?