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?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
How many noughts are at the end of these giant numbers?
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
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. . . .
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
I added together some of my neighbours house numbers. Can you explain the patterns I noticed?
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.
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.
Replace each letter with a digit to make this addition correct.
Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?
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. . . .
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. . . .
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. . . .
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. . . .
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...
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
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.
These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.
Which set of numbers that add to 10 have the largest product?
What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.
Choose any three by three square of dates on a calendar page...
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. . . .
Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.
The sums of the squares of three related numbers is also a perfect square - can you explain why?
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. . . .
Here are some examples of 'cons', and see if you can figure out where the trick is.
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.
Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.
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.
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?
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.
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!
This is the second article on right-angled triangles whose edge lengths are whole numbers.
Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?
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. . . .
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. . . .
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?
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?
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?
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
Nine cross country runners compete in a team competition in which there are three matches. If you were a judge how would you decide who would win?
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. . . .
What are the missing numbers in the pyramids?