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.

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

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.

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?

Investigate and explain the patterns that you see from recording just the units digits of numbers in the times tables.

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?

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

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

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

Complete these two jigsaws then put one on top of the other. What happens when you add the 'touching' numbers? What happens when you change the position of the jigsaws?

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

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

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

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?

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

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?

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

In this investigation, we look at Pascal's Triangle in a slightly different way - rotated and with the top line of ones taken off.

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?

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?

Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

We went to the cinema and decided to buy some bags of popcorn so we asked about the prices. Investigate how much popcorn each bag holds so find out which we might have bought.

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

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?

How many models can you find which obey these rules?

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

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 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?

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

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?

"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?

What is the largest number of circles we can fit into the frame without them overlapping? How do you know? What will happen if you try the other shapes?

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.

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?

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?

I cut this square into two different shapes. What can you say about the relationship between them?

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

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.