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.

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

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.

Well now, what would happen if we lost all the nines in our number system? Have a go at writing the numbers out in this way and have a look at the multiplications table.

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?

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

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

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

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

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?

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?

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.

In this section from a calendar, put a square box around the 1st, 2nd, 8th and 9th. Add all the pairs of numbers. What do you notice about the answers?

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.

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?

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

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!

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.

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

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

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

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

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

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?

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?

Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a 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?

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

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 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?

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.

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?

How will you decide which way of flipping over and/or turning the grid will give you the highest total?

In how many ways can you stack these rods, following the 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 ...?

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

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 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?

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

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?

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.