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?

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!

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!

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

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.

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.

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?

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

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.

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.

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?

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?

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?

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?

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?

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?

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?

Place this "worm" on the 100 square and find the total of the four squares it covers. Keeping its head in the same place, what other totals can you make?

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?

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?

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.

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

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?

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

This article for teachers suggests ideas for activities built around 10 and 2010.

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 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?

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

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

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

What do these two triangles have in common? How are they related?

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.

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

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?

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

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?

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?

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

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?

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

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

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?

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

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.

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.