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. . . .

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!

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. . . .

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

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

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?

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

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.

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.

On the 3D grid a strange (and deadly) animal is lurking. Using the tracking system can you locate this creature as quickly as possible?

Can you find a rule which connects consecutive triangular numbers?

Show that all pentagonal numbers are one third of a triangular number.

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?

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

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

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.

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

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

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

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?

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?

Can you describe this route to infinity? Where will the arrows take you next?

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. . . .

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design. Coins inserted into the machine slide down a chute into the machine and a drink is duly. . . .

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

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. . . .

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

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?

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. . . .

We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?

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?

Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.

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

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?

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. . . .

Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

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

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

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.

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

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?

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.

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?