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?

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?

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

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?

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

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

This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

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

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

What is the best way to shunt these carriages so that each train can continue its journey?

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

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

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

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

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

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

An activity making various patterns with 2 x 1 rectangular tiles.

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

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

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

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

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

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

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....?

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

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

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

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

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

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

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

These practical challenges are all about making a 'tray' and covering it with paper.

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

Design an arrangement of display boards in the school hall which fits the requirements of different people.

If you hang two weights on one side of this balance, in how many different ways can you hang three weights on the other side for it to be balanced?

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?

Can you make a train the same length as Laura's but using three differently coloured rods? Is there only one way of doing it?

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

Investigate the different ways you could split up these rooms so that you have double the number.

Try this matching game which will help you recognise different ways of saying the same time interval.