Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?

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...

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.

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?

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

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?

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

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. . . .

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. . . .

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

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.

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.

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. . . .

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. . . .

Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the. . . .

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.

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

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

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

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?

The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!

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?

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.

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.

We have exactly 100 coins. There are five different values of coins. We have decided to buy a piece of computer software for 39.75. We have the correct money, not a penny more, not a penny less! Can. . . .

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. . . .

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

What can you say about the angles on opposite vertices of any cyclic quadrilateral? Working on the building blocks will give you insights that may help you to explain what is special about them.

Start with any triangle T1 and its inscribed circle. Draw the triangle T2 which has its vertices at the points of contact between the triangle T1 and its incircle. Now keep repeating this. . . .

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

I am exactly n times my daughter's age. In m years I shall be exactly (n-1) times her age. In m2 years I shall be exactly (n-2) times her age. After that I shall never again be an exact multiple of. . . .

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. . . .

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

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

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.

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

Show that if three prime numbers, all greater than 3, form an arithmetic progression then the common difference is divisible by 6. What if one of the terms is 3?

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

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?

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

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

Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation.

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.

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.