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?

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?

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

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?

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

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?

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?

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.

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 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!

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

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

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?

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

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.

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

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?

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

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....?

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

This article gives you a few ideas for understanding the Got It! game and how you might find a winning strategy.

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

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

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

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

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

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. . . .

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.

In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?

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

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

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

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?

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?

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

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

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

An interactive activity for one to experiment with a tricky tessellation

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 to be played against the computer, or in groups. Pick a 7-digit number. A random digit is generated. What must you subract to remove the digit from your number? the first to zero wins.

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.

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.

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

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