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

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.

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?

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

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

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.

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 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 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.

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?

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?

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

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.

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

Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?

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

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.

Start with four numbers at the corners of a square and put the total of two corners in the middle of that side. Keep going... Can you estimate what the size of the last four numbers will be?

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

Which times on a digital clock have a line of symmetry? Which look the same upside-down? You might like to try this investigation and find out!

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

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?

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.

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?

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?

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

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?

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.

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.

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 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?

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

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?

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?

Investigate the different ways these aliens count in this challenge. You could start by thinking about how each of them would write our number 7.

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

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?

How many models can you find which obey these rules?

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

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

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

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

This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.