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?

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.

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.

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?

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

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.

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.

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?

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

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?

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

How many models can you find which obey these rules?

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?

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?

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

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

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

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

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

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

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?

When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.

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?

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

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

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

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?

How many ways can you find of tiling the square patio, using square tiles of different sizes?

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 two group activities use mathematical reasoning - one is numerical, one geometric.

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.

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?

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

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

In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?

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.

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

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

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

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.

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

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

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.

Can you draw a square in which the perimeter is numerically equal to the area?

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

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?

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....?

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?

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. . . .