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?

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

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?

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

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

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

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....?

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

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?

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!

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

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

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.

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?

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?

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting 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 game for 1 person. Can you work out how the dice must be rolled from the start position to the finish? Play on line.

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

Find out how we can describe the "symmetries" of this triangle and investigate some combinations of rotating and flipping it.

Can you find all the different triangles on these peg boards, and find their angles?

This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?

How many different triangles can you make on a circular pegboard that has nine pegs?

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

What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?

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

Place six toy ladybirds into the box so that there are two ladybirds in every column and every row.

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

Try out the lottery that is played in a far-away land. What is the chance of winning?

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?

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?

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

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?

An interactive activity for one to experiment with a tricky tessellation

Use the interactivity or play this dice game yourself. How could you make it fair?

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

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

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. . . .

These interactive dominoes can be dragged around the screen.

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?

A game for 2 people that everybody knows. You can play with a friend or online. If you play correctly you never lose!

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.

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

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