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?

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

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

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.

Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

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

Investigate the number of faces you can see when you arrange three cubes in different ways.

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

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?

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

Investigate this balance which is marked in halves. If you had a weight on the left-hand 7, where could you hang two weights on the right to make it balance?

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

There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

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?

Try continuing these patterns made from triangles. Can you create your own repeating pattern?

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?

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

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

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!

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.

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

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.

We need to wrap up this cube-shaped present, remembering that we can have no overlaps. What shapes can you find to use?

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?

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

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?

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?

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?

Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?

How many models can you find which obey these rules?

What can you say about the child who will be first on the playground tomorrow morning at breaktime in your school?

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

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

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

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?

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

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

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.

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?

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?

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?

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

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.

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?