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.

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?

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

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

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

This article for teachers discusses examples of problems in which there is no obvious method but in which children can be encouraged to think deeply about the context and extend their ability to. . . .

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.

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?

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 Hamiltonian circuit is a continuous path in a graph that passes through each of the vertices exactly once and returns to the start. How many Hamiltonian circuits can you find in these graphs?

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

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?

In this problem, we have created a pattern from smaller and smaller squares. If we carried on the pattern forever, what proportion of the image would be coloured blue?

When dice land edge-up, we usually roll again. But what if we didn't...?

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

A 3x3x3 cube may be reduced to unit cubes in six saw cuts. If after every cut you can rearrange the pieces before cutting straight through, can you do it in fewer?

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.

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?

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?

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

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

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. . . .

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?

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

A package contains a set of resources designed to develop pupils' mathematical thinking. This package places a particular emphasis on “visualising” and is designed to meet the needs. . . .

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?

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

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

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?

Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . .

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

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 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. . . .

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. . . .

Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?

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. . . .

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

Anne completes a circuit around a circular track in 40 seconds. Brenda runs in the opposite direction and meets Anne every 15 seconds. How long does it take Brenda to run around the track?

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?

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

Every day at noon a boat leaves Le Havre for New York while another boat leaves New York for Le Havre. The ocean crossing takes seven days. How many boats will each boat cross during their journey?

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

Can you mentally fit the 7 SOMA pieces together to make a cube? Can you do it in more than one way?

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

Take a line segment of length 1. Remove the middle third. Remove the middle thirds of what you have left. Repeat infinitely many times, and you have the Cantor Set. Can you picture it?

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

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?

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

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?