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

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?

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

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 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?

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

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

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

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

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?

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.

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

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.

It starts quite simple but great opportunities for number discoveries and patterns!

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?

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?

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

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

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.

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

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

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?

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

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

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?

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!

Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?

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 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?

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

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?

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?

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

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?

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

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?

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

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.

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

How many models can you find which obey these rules?

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

A group of children are discussing the height of a tall tree. How would you go about finding out its height?

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

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

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