"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 ...?

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

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.

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?

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

Investigate the different ways these aliens count in this challenge. You could start by thinking about how each of them would write our number 7.

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

In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?

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?

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?

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

Explore ways of colouring this set of triangles. Can you make symmetrical patterns?

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?

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

The red ring is inside the blue ring in this picture. Can you rearrange the rings in different ways? Perhaps you can overlap them or put one outside another?

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

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

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?

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?

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

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

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

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 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.

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?

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?

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?

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

If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?

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

Take a look at these data collected by children in 1986 as part of the Domesday Project. What do they tell you? What do you think about the way they are presented?

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 these hexagons drawn from different sized equilateral triangles.

Here is your chance to investigate the number 28 using shapes, cubes ... in fact anything at all.

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

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

This problem is intended to get children to look really hard at something they will see many times in the next few months.

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?

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?

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

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?

How many models can you find which obey these rules?

Bernard Bagnall looks at what 'problem solving' might really mean in the context of primary classrooms.