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

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?

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. . . .

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.

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.

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

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?

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

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?

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

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.

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

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

Can you set the logic gates so that the number of bulbs which are on is the same as the number of switches which are on?

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. . . .

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

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.

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

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.

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?

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

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.

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?

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

A metal puzzle which led to some mathematical questions.

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.

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.

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

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.

You have 27 small cubes, 3 each of nine colours. Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour.

Meg and Mo need to hang their marbles so that they balance. Use the interactivity to experiment and find out what they need to do.

What are the coordinates of the coloured dots that mark out the tangram? Try changing the position of the origin. What happens to the coordinates now?

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.

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?

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

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

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

Can you beat the computer in the challenging strategy game?

Match pairs of cards so that they have equivalent ratios.

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

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

Can you work out what is wrong with the cogs on a UK 2 pound coin?

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

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

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?