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?

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?

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?

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

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

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

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?

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?

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?

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.

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

The reader is invited to investigate changes (or permutations) in the ringing of church bells, illustrated by braid diagrams showing the order in which the bells are rung.

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?

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

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

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

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

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

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

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

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 ?

Two intersecting circles have a common chord AB. The point C moves on the circumference of the circle C1. The straight lines CA and CB meet the circle C2 at E and F respectively. As the point C. . . .

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

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

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

A circular plate rolls inside a rectangular tray making five circuits and rotating about its centre seven times. Find the dimensions of the tray.

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?

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?

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.

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. . . .

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. . . .

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?

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

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.

A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

This article outlines the underlying axioms of spherical geometry giving a simple proof that the sum of the angles of a triangle on the surface of a unit sphere is equal to pi plus the area of the. . . .

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?

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?

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

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 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 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.

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?

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

Use the diagram to investigate the classical Pythagorean means.

The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?