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

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

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

The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

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

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

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

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?

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

This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

An investigation that gives you the opportunity to make and justify predictions.

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

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?

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?

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

Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.

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

Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.

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

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

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

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

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?

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.

Your challenge is to find the longest way through the network following this rule. You can start and finish anywhere, and with any shape, as long as you follow the correct order.

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

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

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?

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

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

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

This activity investigates how you might make squares and pentominoes from Polydron.

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....?

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?

Look carefully at the numbers. What do you notice? Can you make another square using the numbers 1 to 16, that displays the same properties?

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