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 ?

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?

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

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

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

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?

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?

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.

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?

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

Find all the ways to cut out a 'net' of six squares that can be folded into a cube.

Can you make a tetrahedron whose faces all have the same perimeter?

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

Imagine you are suspending a cube from one vertex (corner) and allowing it to hang freely. Now imagine you are lowering it into water until it is exactly half submerged. What shape does the surface. . . .

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

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.

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. . . .

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?

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 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?

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?

Square It game for an adult and child. Can you come up with a way of always winning this game?

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. . . .

A game for 2 players. Can be played online. One player has 1 red counter, the other has 4 blue. The red counter needs to reach the other side, and the blue needs to trap the red.

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 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.

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

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. . . .

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

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

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.

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

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.

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

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 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. . . .

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

Bilbo goes on an adventure, before arriving back home. Using the information given about his journey, can you work out where Bilbo lives?

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?

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

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?

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?