Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some. . . .

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. . . .

Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

Start with any whole number N, write N as a multiple of 10 plus a remainder R and produce a new whole number N'. Repeat. What happens?

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

In the following sum the letters A, B, C, D, E and F stand for six distinct digits. Find all the ways of replacing the letters with digits so that the arithmetic is correct.

The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.

You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

Can you explain why a sequence of operations always gives you perfect squares?

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...

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

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .

This is the second article on right-angled triangles whose edge lengths are whole numbers.

If you think that mathematical proof is really clearcut and universal then you should read this article.

This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.

The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!

In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.

The final of five articles which containe the proof of why the sequence introduced in article IV either reaches the fixed point 0 or the sequence enters a repeating cycle of four values.

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?

Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .

Four jewellers share their stock. Can you work out the relative values of their gems?

Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.

What fractions can you divide the diagonal of a square into by simple folding?

Can you make sense of these three proofs of Pythagoras' Theorem?

Can you rearrange the cards to make a series of correct mathematical statements?

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.

An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.

This article stems from research on the teaching of proof and offers guidance on how to move learners from focussing on experimental arguments to mathematical arguments and deductive reasoning.