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?
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
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.
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.
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 smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
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?
In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.
Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?
How many triangles can you make on the 3 by 3 pegboard?
Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.
How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?
In how many ways can you stack these rods, following the rules?
How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?
What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?
In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?
Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?
Investigate the different ways you could split up these rooms so that you have double the number.
I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?
Have a go at this 3D extension to the Pebbles problem.
How many models can you find which obey these rules?
This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
An activity making various patterns with 2 x 1 rectangular tiles.
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.
This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
This challenge involves calculating the number of candles needed on birthday cakes. It is an opportunity to explore numbers and discover new things.
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?
Can you find ways of joining cubes together so that 28 faces are visible?
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?
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?
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?
Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?
Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?
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.
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 ways can you find of tiling the square patio, using square tiles of different sizes?
This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.
Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.
48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.
Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possible answers?
What do these two triangles have in common? How are they related?
How many faces can you see when you arrange these three cubes in different ways?
What is the largest cuboid you can wrap in an A3 sheet of paper?
Can you design a new shape for the twenty-eight squares and arrange the numbers in a logical way? What patterns do you notice?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?