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.

If the numbers 5, 7 and 4 go into this function machine, what numbers will come out?

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?

Make a chair and table out of interlocking cubes, making sure that the chair fits under the table!

These sixteen children are standing in four lines of four, one behind the other. They are each holding a card with a number on it. Can you work out the missing numbers?

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

Benâ€™s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?

Investigate the successive areas of light blue in these diagrams.

I've made some cubes and some cubes with holes in. This challenge invites you to explore the difference in the number of small cubes I've used. Can you see any patterns?

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

Daisy and Akram were making number patterns. Daisy was using beads that looked like flowers and Akram was using cube bricks. First they were counting in twos.

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?

In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?

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

An environment which simulates working with Cuisenaire rods.

Liitle Millennium Man was born on Saturday 1st January 2000 and he will retire on the first Saturday 1st January that occurs after his 60th birthday. How old will he be when he retires?

Investigate these hexagons drawn from different sized equilateral triangles.

Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?

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

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

Can you go from A to Z right through the alphabet in the hexagonal maze?

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?

If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?

Investigate the different sounds you can make by putting the owls and donkeys on the wheel.

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?

July 1st 2001 was on a Sunday. July 1st 2002 was on a Monday. When did July 1st fall on a Monday again?

At the beginning of May Tom put his tomato plant outside. On the same day he sowed a bean in another pot. When will the two be the same height?

Three beads are threaded on a circular wire and are coloured either red or blue. Can you find all four different combinations?

What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?

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.

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.

Your challenge is to find the longest way through the network following this rule. You can start and finish anywhere, and with any shape, as long as you follow the correct order.

"Tell me the next two numbers in each of these seven minor spells", chanted the Mathemagician, "And the great spell will crumble away!" Can you help Anna and David break the spell?

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

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