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

How many models can you find which obey these rules?

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

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

This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.

Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?

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?

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.

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?

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?

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

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

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?

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

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

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?

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?

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?

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

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?

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

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?

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

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

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?

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

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?

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

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?

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

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

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?

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

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

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

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

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?

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

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

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

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?

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

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

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?

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

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

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

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.