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.

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?

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?

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

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?

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

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?

The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

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.

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

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

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

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

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

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

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

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

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?

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 different ways you could split up these rooms so that you have double the number.

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

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

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

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.

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.

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?

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.

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?

Can you find ways of joining cubes together so that 28 faces are visible?

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

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

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!

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

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

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?

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?

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

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?

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

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?

This challenge involves calculating the number of candles needed on birthday cakes. It is an opportunity to explore numbers and discover new things.

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?

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?

The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?

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?

Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?