In this article for teachers, Bernard gives an example of taking an initial activity and getting questions going that lead to other explorations.

In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.

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

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

A group of children are discussing the height of a tall tree. How would you go about finding out its height?

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

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

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

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.

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

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?

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 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.

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 ...?

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.

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?

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?

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

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?

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?

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

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.

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?

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?

What do these two triangles have in common? How are they related?

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

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

These caterpillars have 16 parts. What different shapes do they make if each part lies in the small squares of a 4 by 4 square?

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?

Investigate the number of faces you can see when you arrange three cubes in different ways.

In this investigation, you are challenged to make mobile phone numbers which are easy to remember. What happens if you make a sequence adding 2 each time?

Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?

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

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

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?

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

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?

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!

Sort the houses in my street into different groups. Can you do it in any other ways?

An activity making various patterns with 2 x 1 rectangular tiles.

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?

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?

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

Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?

Try continuing these patterns made from triangles. Can you create your own repeating pattern?