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

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?

"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 ...?

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?

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.

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?

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?

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

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?

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?

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?

Bernard Bagnall describes how to get more out of some favourite NRICH investigations.

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

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!

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?

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

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

Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?

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

Why does the tower look a different size in each of these pictures?

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.

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

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

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 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?

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?

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

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?

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

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

In this section from a calendar, put a square box around the 1st, 2nd, 8th and 9th. Add all the pairs of numbers. What do you notice about the answers?

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.

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?

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?

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?

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

What is the largest cuboid you can wrap in an A3 sheet of paper?

In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?

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

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

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

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

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.