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. . . .
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Take any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
A, B & C own a half, a third and a sixth of a coin collection.
Each grab some coins, return some, then share equally what they had
put back, finishing with their own share. How rich are they?
Find the largest integer which divides every member of the
following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
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.
Is the mean of the squares of two numbers greater than, or less
than, the square of their means?
An article which gives an account of some properties of magic squares.
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.
A composite number is one that is neither prime nor 1. Show that
10201 is composite in any base.
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. . . .
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
Can you use the diagram to prove the AM-GM inequality?
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.
This is the second article on right-angled triangles whose edge lengths are whole numbers.
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!
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
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.
Four identical right angled triangles are drawn on the sides of a
square. Two face out, two face in. Why do the four vertices marked
with dots lie on one line?
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?
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 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.
The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?
Prove that, given any three parallel lines, an equilateral triangle
always exists with one vertex on each of the three lines.
Can you explain why a sequence of operations always gives you perfect squares?
There are 12 identical looking coins, one of which is a fake. The
counterfeit coin is of a different weight to the rest. What is the
minimum number of weighings needed to locate the fake coin?
Explore what happens when you draw graphs of quadratic equations
with coefficients based on a geometric sequence.
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?
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?
Can you make sense of these three proofs of Pythagoras' Theorem?
Which of these roads will satisfy a Munchkin builder?
Can you rearrange the cards to make a series of correct mathematical statements?
Four jewellers share their stock. Can you work out the relative values of their gems?
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
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.
If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?
Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
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. . . .
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
If you think that mathematical proof is really clearcut and
universal then you should read this article.
What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
I am exactly n times my daughter's age. In m years I shall be ... How old am I?
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?
How many noughts are at the end of these giant numbers?
Three points A, B and C lie in this order on a line, and P is any
point in the plane. Use the Cosine Rule to prove the following
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.