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. . . .

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

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

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?

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. . . .

Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...

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?

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

Can you find a way to identify times tables after they have been shifted up?

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

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

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?

Investigate the sequences obtained by starting with any positive 2 digit number (10a+b) and repeatedly using the rule 10a+b maps to 10b-a to get the next number in the sequence.

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

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

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

Can you find a rule which connects consecutive triangular numbers?

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?

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

Show that all pentagonal numbers are one third of a triangular number.

Three frogs hopped onto the table. A red frog on the left a green in the middle and a blue frog on the right. Then frogs started jumping randomly over any adjacent frog. Is it possible for them to. . . .

Can you show that 1^99 + 2^99 + 3^99 + 4^99 + 5^99 is divisible by 5?

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?

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

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

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

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

Find the decimal equivalents of the fractions one ninth, one ninety ninth, one nine hundred and ninety ninth etc. Explain the pattern you get and generalise.

Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.

Imagine a strip with a mark somewhere along it. Fold it in the middle so that the bottom reaches back to the top. Stetch it out to match the original length. Now where's the mark?

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

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

Can you find a rule which relates triangular numbers to square numbers?

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

Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?

Can you puzzle out what sequences these Logo programs will give? Then write your own Logo programs to generate sequences.

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

Here are some circle bugs to try to replicate with some elegant programming, plus some sequences generated elegantly in LOGO.

An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?

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

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?

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . .

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. . . .

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.