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?

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

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?

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.

Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?

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?

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 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?

Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?

EWWNP means Exploring Wild and Wonderful Number Patterns Created by Yourself! Investigate what happens if we create number patterns using some simple rules.

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

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.

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.

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!

Explore Alex's number plumber. What questions would you like to ask? Don't forget to keep visiting NRICH projects site for the latest developments and questions.

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.

Take ten sticks in heaps any way you like. Make a new heap using one from each of the heaps. By repeating that process could the arrangement 7 - 1 - 1 - 1 ever turn up, except by starting with it?

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?

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?

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

Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possibilities that could come up?

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?

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!

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?

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?

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?

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

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

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?

The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?

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?

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

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

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

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?

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

Follow the directions for circling numbers in the matrix. Add all the circled numbers together. Note your answer. Try again with a different starting number. What do you notice?

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

What is the largest cuboid you can wrap in an A3 sheet of paper?

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?

In this article for teachers, Bernard gives an example of taking an initial activity and getting questions going that lead to other explorations.

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.

I cut this square into two different shapes. What can you say about the relationship between them?

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

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

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