It's easy to work out the areas of most squares that we meet, but what if they were tilted?

You can move the 4 pieces of the jigsaw and fit them into both outlines. Explain what has happened to the missing one unit of area.

A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .

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.

This rectangle is cut into five pieces which fit exactly into a triangular outline and also into a square outline where the triangle, the rectangle and the square have equal areas.

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

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

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?

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

Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .

A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

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?

What is the greatest number of squares you can make by overlapping three squares?

These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.

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 red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

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.

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.

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter.

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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

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

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?

Here is a chance to play a version of the classic Countdown Game.

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

Explore this interactivity and see if you can work out what it does. Could you use it to estimate the area of a shape?

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.

Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

Use Excel to explore multiplication of fractions.

Exchange the positions of the two sets of counters in the least possible number of moves

What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it?

An environment that enables you to investigate tessellations of regular polygons

Practise your diamond mining skills and your x,y coordination in this homage to Pacman.

Can you locate the lost giraffe? Input coordinates to help you search and find the giraffe in the fewest guesses.

This interactivity invites you to make conjectures and explore probabilities of outcomes related to two independent events.

Match pairs of cards so that they have equivalent ratios.

Can you find a reliable strategy for choosing coordinates that will locate the robber in the minimum number of guesses?

Learn how to use the Shuffles interactivity by running through these tutorial demonstrations.

A collection of our favourite pictorial problems, one for each day of Advent.

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

A tool for generating random integers.

Can you make a right-angled triangle on this peg-board by joining up three points round the edge?

Find out what a "fault-free" rectangle is and try to make some of your own.