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?

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

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?

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

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

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

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?

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

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?

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

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

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?

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?

How will you go about finding all the jigsaw pieces that have one peg and one hole?

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

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

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

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?

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

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?

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?

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

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

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

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?

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

How many trapeziums, of various sizes, are hidden in this picture?

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?

How many models can you find which obey these rules?

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

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

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

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?

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

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

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?

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?

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

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

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. . . .

Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

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

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