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?

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?

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

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?

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?

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.

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?

How many models can you find which obey these rules?

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?

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

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

If you had 36 cubes, what different cuboids could you make?

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?

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.

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

Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?

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.

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?

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

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

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

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?

Penta people, the Pentominoes, always build their houses from five square rooms. I wonder how many different Penta homes you can create?

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?

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

Take three differently coloured blocks - maybe red, yellow and blue. Make a tower using one of each colour. How many different towers can you make?

How long does it take to brush your teeth? Can you find the matching length of time?

These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.

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

These activities focus on finding all possible solutions so if you work in a systematic way, you won't leave any out.

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

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

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?

In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?

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

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.

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

These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.

Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.

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

These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.

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. . . .

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

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

Find all the numbers that can be made by adding the dots on two dice.

These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.