Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.

This article for teachers suggests ideas for activities built around 10 and 2010.

Can you design a new shape for the twenty-eight squares and arrange the numbers in a logical way? What patterns do you notice?

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?

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

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?

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

EWWNP means Exploring Wild and Wonderful Number Patterns Created by Yourself! Investigate what happens if we create number patterns using some simple rules.

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.

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.

Which times on a digital clock have a line of symmetry? Which look the same upside-down? You might like to try this investigation and find out!

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

There are ten children in Becky's group. Can you find a set of numbers for each of them? Are there any other sets?

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

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?

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!

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

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.

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?

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

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?

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

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

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?

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

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?

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

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

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

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?

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

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?

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

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?

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?

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.

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?

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.

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

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

"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 ...?

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?

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

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