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

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

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?

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?

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

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?

Here are many ideas for you to investigate - all linked with the number 2000.

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

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?

This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.

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.

Investigate these hexagons drawn from different sized equilateral triangles.

This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.

Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?

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

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.

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

A follow-up activity to Tiles in the Garden.

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

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?

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

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

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?

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.

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

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.

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

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

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?

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

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?

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?

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

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?

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

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

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

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

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?

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

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

How many models can you find which obey these rules?

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