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.

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

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

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?

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

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?

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 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?

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

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

A follow-up activity to Tiles in the Garden.

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?

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

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.

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?

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

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

Investigate these hexagons drawn from different sized equilateral triangles.

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

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?

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

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

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?

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

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

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?

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?

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

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

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.

How many models can you find which obey these rules?

"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 ...?

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

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

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?

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?

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?

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

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?

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

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 faces can you see when you arrange these three cubes in different ways?

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

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 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.

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