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

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

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?

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 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.

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 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?

A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't. . . .

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

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

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 ?

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?

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

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

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!

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.

How can visual patterns be used to prove sums of series?

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.

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?

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?

A half-cube is cut into two pieces by a plane through the long diagonal and at right angles to it. Can you draw a net of these pieces? Are they identical?

The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . .

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

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?

ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?

Place a red counter in the top left corner of a 4x4 array, which is covered by 14 other smaller counters, leaving a gap in the bottom right hand corner (HOME). What is the smallest number of moves. . . .

There are 27 small cubes in a 3 x 3 x 3 cube, 54 faces being visible at any one time. Is it possible to reorganise these cubes so that by dipping the large cube into a pot of paint three times you. . . .

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?

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

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

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

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

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?

Find the point whose sum of distances from the vertices (corners) of a given triangle is a minimum.

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

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?

To avoid losing think of another very well known game where the patterns of play are similar.

Imagine you are suspending a cube from one vertex and allowing it to hang freely. What shape does the surface of the water make around the cube?

Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.

This problem is about investigating whether it is possible to start at one vertex of a platonic solid and visit every other vertex once only returning to the vertex you started at.

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?

The image in this problem is part of a piece of equipment found in the playground of a school. How would you describe it to someone over the phone?

Given a 2 by 2 by 2 skeletal cube with one route `down' the cube. How many routes are there from A to B?