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?

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

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

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.

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.

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?

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

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?

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.

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

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. . . .

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

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

Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create.

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

Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?

Which of the following cubes can be made from these nets?

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

Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .

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

A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .

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

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

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. . . .

What is the shape of wrapping paper that you would need to completely wrap this model?

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?

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

Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?

These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?

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?

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. . . .

Lyndon Baker describes how the Mobius strip and Euler's law can introduce pupils to the idea of topology.

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.

Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

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?

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 bus route has a total duration of 40 minutes. Every 10 minutes, two buses set out, one from each end. How many buses will one bus meet on its way from one end to the other end?

Here is a solitaire type environment for you to experiment with. Which targets can you reach?

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

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

Can you mark 4 points on a flat surface so that there are only two different distances between them?

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

Can you find a rule which relates triangular numbers to square numbers?

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

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?

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

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

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

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.