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.

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

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

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

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

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!

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

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?

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

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?

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?

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

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.

This was a problem for our birthday website. Can you use four of these pieces to form a square? How about making a square with all five pieces?

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

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?

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. . . .

Ahmed has some wooden planks to use for three sides of a rabbit run against the shed. What quadrilaterals would he be able to make with the planks of different lengths?

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

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

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 tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

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?

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

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.

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

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.

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.

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.

How many times in twelve hours do the hands of a clock form a right angle? Use the interactivity to check your answers.

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

An interactive activity for one to experiment with a tricky tessellation

A shape and space game for 2,3 or 4 players. Be the last person to be able to place a pentomino piece on the playing board. Play with card, or on the computer.

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?

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

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?

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. . . .

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

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.

Use the interactivities to fill in these Carroll diagrams. How do you know where to place the numbers?

Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?

Can you find all the different ways of lining up these Cuisenaire rods?

Train game for an adult and child. Who will be the first to make the train?

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

Choose 13 spots on the grid. Can you work out the scoring system? What is the maximum possible score?