Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

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

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?

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.

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

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

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.

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?

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?

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

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.

Find out what a "fault-free" rectangle is and try to make some of your own.

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

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

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?

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

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

This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards.

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?

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

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. . . .

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 challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

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?

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

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.

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

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

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?

There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it?

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

You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how?

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?

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....?

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

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?

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?

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!

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

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.

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

Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it?

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

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?

There are 44 people coming to a dinner party. There are 15 square tables that seat 4 people. Find a way to seat the 44 people using all 15 tables, with no empty places.

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

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