This article for teachers describes the exchanges on an email talk list about ideas for an investigation which has the sum of the squares as its solution.

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?

These caterpillars have 16 parts. What different shapes do they make if each part lies in the small squares of a 4 by 4 square?

Here are some ideas to try in the classroom for using counters to investigate number patterns.

Investigate and explain the patterns that you see from recording just the units digits of numbers in the times tables.

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?

Using only the red and white rods, how many different ways are there to make up the other colours of rod?

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.

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

A case is found with a combination lock. There is one clue about the number needed to open the case. Can you find the number and open the case?

Make an estimate of how many light fittings you can see. Was your estimate a good one? How can you decide?

Use the interactivity to sort these numbers into sets. Can you give each set a name?

A hundred square has been printed on both sides of a piece of paper. What is on the back of 100? 58? 23? 19?

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

Look carefully at the numbers. What do you notice? Can you make another square using the numbers 1 to 16, that displays the same properties?

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

Look at the squares in this problem. What does the next square look like? I draw a square with 81 little squares inside it. How long and how wide is my 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.

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 are the coordinates of this shape after it has been transformed in the ways described? Compare these with the original coordinates. What do you notice about the numbers?

Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.

This number has 903 digits. What is the sum of all 903 digits?

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

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

Libby Jared helped to set up NRICH and this is one of her favourite problems. It's a problem suitable for a wide age range and best tackled practically.

Can you find any perfect numbers? Read this article to find out more...

Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create.