Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?

L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?

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

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

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.

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

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

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

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}.

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

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

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

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

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?

This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with. . . .

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

An article which gives an account of some properties of magic squares.

Carry out cyclic permutations of nine digit numbers containing the digits from 1 to 9 (until you get back to the first number). Prove that whatever number you choose, they will add to the same total.

In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.

If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.

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

Can you find the areas of the trapezia in this sequence?

Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?

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.

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.

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.

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?

Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .

A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.

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.

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. . . .

What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

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. . . .

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

Here are some examples of 'cons', and see if you can figure out where the trick is.

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

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

If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?

This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.

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. . . .

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

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

Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?