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

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.

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

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?

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.

How many noughts are at the end of these giant numbers?

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. . . .

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

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?

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...

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...

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

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. . . .

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

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. . . .

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 sums of the squares of three related numbers is also a perfect square - can you explain why?

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

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

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.

I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

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

I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?

What is the largest number of intersection points that a triangle and a quadrilateral can have?

When is it impossible to make number sandwiches?

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.

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.

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

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

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!

I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?

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.

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

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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.

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

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

Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .

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.

Kyle and his teacher disagree about his test score - who is right?

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

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.

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?

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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?

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

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. . . .