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?

What shape is the overlap when you slide one of these shapes half way across another? Can you picture it in your head? Use the interactivity to check your visualisation.

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

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?

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

Show how this pentagonal tile can be used to tile the plane and describe the transformations which map this pentagon to its images in the tiling.

This problem is about investigating whether it is possible to start at one vertex of a platonic solid and visit every other vertex once only returning to the vertex you started at.

Use the interactivity to make this Islamic star and cross design. Can you produce a tessellation of regular octagons with two different types of triangle?

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. . . .

What shaped overlaps can you make with two circles which are the same size? What shapes are 'left over'? What shapes can you make when the circles are different sizes?

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

The computer has made a rectangle and will tell you the number of spots it uses in total. Can you find out where the rectangle is?

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.

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

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

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.

An environment that enables you to investigate tessellations of regular polygons

What happens when you turn these cogs? Investigate the differences between turning two cogs of different sizes and two cogs which are the same.

How can the same pieces of the tangram make this bowl before and after it was chipped? Use the interactivity to try and work out what is going on!

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?

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.

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?

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

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

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. . . .

A game for 1 or 2 people. Use the interactive version, or play with friends. Try to round up as many counters as possible.

Can you find triangles on a 9-point circle? Can you work out their angles?

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. . . .

Can you explain the strategy for winning this game with any target?

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?

Choose the size of your pegboard and the shapes you can make. Can you work out the strategies needed to block your opponent?

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 greatest number of squares you can make by overlapping three squares?

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

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

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.

Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.

An interactive game to be played on your own or with friends. Imagine you are having a party. Each person takes it in turns to stand behind the chair where they will get the most chocolate.

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

A game for two people that can be played with pencils and paper. Combine your knowledge of coordinates with some strategic thinking.

An interactive game for 1 person. You are given a rectangle with 50 squares on it. Roll the dice to get a percentage between 2 and 100. How many squares is this? Keep going until you get 100. . . .

A game for 1 person. Can you work out how the dice must be rolled from the start position to the finish? Play on line.

Three beads are threaded on a circular wire and are coloured either red or blue. Can you find all four different combinations?

An interactive activity for one to experiment with a tricky tessellation

Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?

Can you spot the similarities between this game and other games you know? The aim is to choose 3 numbers that total 15.

Use Excel to explore multiplication of fractions.

Two circles of equal radius touch at P. One circle is fixed whilst the other moves, rolling without slipping, all the way round. How many times does the moving coin revolve before returning to P?