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 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.

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?

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?

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.

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

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

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?

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?

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

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

Investigate the different ways you could split up these rooms so that you have double the 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?

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?

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?

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.

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

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

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.

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?

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?

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

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 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!

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?

An activity making various patterns with 2 x 1 rectangular tiles.

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

A challenging activity focusing on finding all possible ways of stacking rods.

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

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

This challenge extends the Plants investigation so now four or more children are involved.

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

How many models can you find which obey these rules?

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

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

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?

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?

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

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

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

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

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?

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

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

Ben has five coins in his pocket. How much money might he have?

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?