What statements can you make about the car that passes the school gates at 11am on Monday? How will you come up with statements and test your ideas?

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

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?

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.

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!

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?

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

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?

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

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.

Work with numbers big and small to estimate and calculate various quantities in biological contexts.

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

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.

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?

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 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!

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

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?

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?

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?

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?

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.

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

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?

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.

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

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

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?

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 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?

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?

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?

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?

Formulate and investigate a simple mathematical model for the design of a table mat.

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

Many natural systems appear to be in equilibrium until suddenly a critical point is reached, setting up a mudslide or an avalanche or an earthquake. In this project, students will use a simple. . . .

What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?

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?

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

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

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?

A description of some experiments in which you can make discoveries about triangles.

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

All types of mathematical problems serve a useful purpose in mathematics teaching, but different types of problem will achieve different learning objectives. In generalmore open-ended problems have. . . .

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

How many different sets of numbers with at least four members can you find in the numbers in this box?