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

How many different triangles can you draw on the dotty grid which each have one dot in the middle?

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

How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?

How many trains can you make which are the same length as Matt's, using rods that are identical?

Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.

How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?

What happens when you try and fit the triomino pieces into these two grids?

Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.

Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.

Can you work out how to balance this equaliser? You can put more than one weight on a hook.

Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?

Take it in turns to make a triangle on the pegboard. Can you block your opponent?

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

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?

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?

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

Board Block game for two. Can you stop your partner from being able to make a shape on the board?

How many different rhythms can you make by putting two drums on the wheel?

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

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

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

What does the overlap of these two shapes look like? Try picturing it in your head and then use the interactivity to test your prediction.

Terry and Ali are playing a game with three balls. Is it fair that Terry wins when the middle ball is red?

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

Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?

Use your mouse to move the red and green parts of this disc. Can you make images which show the turnings described?

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

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 information about Sally and her brother to find out how many children there are in the Brown family.

Choose four of the numbers from 1 to 9 to put in the squares so that the differences between joined squares are odd.

Can you hang weights in the right place to make the equaliser balance?

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

Investigate the different sounds you can make by putting the owls and donkeys on the wheel.

Can you sort these triangles into three different families and explain how you did it?

Make one big triangle so the numbers that touch on the small triangles add to 10. You could use the interactivity to help you.

NRICH December 2006 advent calendar - a new tangram for each day in the run-up to Christmas.

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?

Our 2008 Advent Calendar has a 'Making Maths' activity for every day in the run-up to Christmas.

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

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

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?

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

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

In your bank, you have three types of coins. The number of spots shows how much they are worth. Can you choose coins to exchange with the groups given to make the same total?

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