Here are three 'tricks' to amaze your friends. But the really clever trick is explaining to them why these 'tricks' are maths not magic. Like all good magicians, you should practice by trying. . . .

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

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.

Write down a three-digit number Change the order of the digits to get a different number Find the difference between the two three digit numbers Follow the rest of the instructions then try. . . .

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. . . .

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...

Choose any three by three square of dates on a calendar page...

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

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?

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

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.

This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility.

This jar used to hold perfumed oil. It contained enough oil to fill granid silver bottles. Each bottle held enough to fill ozvik golden goblets and each goblet held enough to fill vaswik crystal. . . .

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

I am exactly n times my daughter's age. In m years I shall be ... How old am I?

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.

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

Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?

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.

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

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}.

A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .

From a group of any 4 students in a class of 30, each has exchanged Christmas cards with the other three. Show that some students have exchanged cards with all the other students in the class. How. . . .

Six points are arranged in space so that no three are collinear. How many line segments can be formed by joining the points in pairs?

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

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. . . .

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

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?

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

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

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.

Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?

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

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

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

In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...

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.

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

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

Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?

A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.

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. . . .

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

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