When Charlie asked his grandmother how old she is, he didn't get a straightforward reply! Can you work out how old she is?

"Ip dip sky blue! Who's 'it'? It's you!" Where would you position yourself so that you are 'it' if there are two players? Three players ...?

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?

How many different sets of numbers with at least four members can you find in the numbers in this box?

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?

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?

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

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?

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

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?

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?

Well now, what would happen if we lost all the nines in our number system? Have a go at writing the numbers out in this way and have a look at the multiplications table.

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

What happens when you add the digits of a number then multiply the result by 2 and you keep doing this? You could try for different numbers and different rules.

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

There are ten children in Becky's group. Can you find a set of numbers for each of them? Are there any other sets?

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?

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?

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?

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

This challenge involves calculating the number of candles needed on birthday cakes. It is an opportunity to explore numbers and discover new things.

This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?

This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?

In how many ways can you stack these rods, following the rules?

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

Start with four numbers at the corners of a square and put the total of two corners in the middle of that side. Keep going... Can you estimate what the size of the last four numbers will be?

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.

Can you design a new shape for the twenty-eight squares and arrange the numbers in a logical way? What patterns do you notice?

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

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?

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 this balance which is marked in halves. If you had a weight on the left-hand 7, where could you hang two weights on the right to make it balance?

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!

What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

Can you find out how the 6-triangle shape is transformed in these tessellations? Will the tessellations go on for ever? Why or why not?

Investigate the numbers that come up on a die as you roll it in the direction of north, south, east and west, without going over the path it's already made.

Explore Alex's number plumber. What questions would you like to ask? Don't forget to keep visiting NRICH projects site for the latest developments and questions.

We need to wrap up this cube-shaped present, remembering that we can have no overlaps. What shapes can you find to use?

Follow the directions for circling numbers in the matrix. Add all the circled numbers together. Note your answer. Try again with a different starting number. What do you notice?

Investigate what happens when you add house numbers along a street in different ways.

Can you make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles?

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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.

This activity asks you to collect information about the birds you see in the garden. Are there patterns in the data or do the birds seem to visit randomly?

In my local town there are three supermarkets which each has a special deal on some products. If you bought all your shopping in one shop, where would be the cheapest?

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

Here are many ideas for you to investigate - all linked with the number 2000.

Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?