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.

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

A introduction to how patterns can be deceiving, and what is and is not a proof.

Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.

How many different ways can I lay 10 paving slabs, each 2 foot by 1 foot, to make a path 2 foot wide and 10 foot long from my back door into my garden, without cutting any of the paving slabs?

Make some loops out of regular hexagons. What rules can you discover?

Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.

A tower of squares is built inside a right angled isosceles triangle. The largest square stands on the hypotenuse. What fraction of the area of the triangle is covered by the series of squares?

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.

Explore this how this program produces the sequences it does. What are you controlling when you change the values of the variables?

Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.

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?

Investigate the successive areas of light blue in these diagrams.

Alison, Bernard and Charlie have been exploring sequences of odd and even numbers, which raise some intriguing questions...

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

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

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.

A story for students about adding powers of integers - with a festive twist.

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

What is the total area of the first two triangles as a fraction of the original A4 rectangle? What is the total area of the first three triangles as a fraction of the original A4 rectangle? If. . . .

There are lots of ideas to explore in these sequences of ordered fractions.

Choose any 4 whole numbers and take the difference between consecutive numbers, ending with the difference between the first and the last numbers. What happens when you repeat this process over and. . . .

Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .

Square numbers can be represented on the seven-clock (representing these numbers modulo 7). This works like the days of the week.

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

What's the greatest number of sides a polygon on a dotty grid could have?

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

"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?

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

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?

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

An environment which simulates working with Cuisenaire rods.

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?

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

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?

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?

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?

What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

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

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

Investigate these hexagons drawn from different sized equilateral triangles.

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

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?