I am exactly n times my daughter's age. In m years I shall be ... How old am I?
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
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?
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.
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. . . .
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
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.
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. . . .
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
Prove Pythagoras' Theorem using enlargements and scale factors.
Can you find the areas of the trapezia in this sequence?
This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.
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.
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.
An article which gives an account of some properties of magic squares.
Can you discover whether this is a fair game?
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.
Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.
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.
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
If you think that mathematical proof is really clearcut and universal then you should read this article.
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...
A introduction to how patterns can be deceiving, and what is and is not a proof.
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
Can you rearrange the cards to make a series of correct mathematical statements?
What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
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?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
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. . . .
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.
Is the mean of the squares of two numbers greater than, or less than, the square of their means?
The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . .
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 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. . . .
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. . . .
Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
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!
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.
It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.
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.
This is the second article on right-angled triangles whose edge lengths are whole numbers.
A blue coin rolls round two yellow coins which touch. The coins are the same size. How many revolutions does the blue coin make when it rolls all the way round the yellow coins? Investigate for a. . . .
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
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. . . .