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.

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

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

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!

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

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

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.

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?

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

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?

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?

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

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 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. . . .

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

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?

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

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?

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.

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

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?

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?

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?

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.

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.

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.

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?

Use the Cuisenaire rods environment to investigate ratio. Can you find pairs of rods in the ratio 3:2? How about 9:6?

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

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

Work out the fractions to match the cards with the same amount of money.

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?

An interactive activity for one to experiment with a tricky tessellation

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

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.

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

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?

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.

Try to stop your opponent from being able to split the piles of counters into unequal numbers. Can you find a strategy?

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

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.

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.

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

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

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. . . .