The diagram shows a regular pentagon with sides of unit length.
Find all the angles in the diagram. Prove that the quadrilateral
shown in red is a rhombus.
Find the largest integer which divides every member of the
following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
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. . . .
An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.
A composite number is one that is neither prime nor 1. Show that
10201 is composite in any base.
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.
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
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.
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. . . .
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
Take any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
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.
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.
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?
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
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.
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. . . .
This is the second article on right-angled triangles whose edge lengths are whole numbers.
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.
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 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!
An article which gives an account of some properties of magic squares.
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
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?
Can you rearrange the cards to make a series of correct mathematical statements?
Explore what happens when you draw graphs of quadratic equations
with coefficients based on a geometric sequence.
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
Can you explain why a sequence of operations always gives you perfect squares?
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?
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
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?
L triominoes can fit together to make larger versions of
themselves. Is every size possible to make in this way?
The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .
What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Which of these roads will satisfy a Munchkin builder?
The tangles created by the twists and turns of the Conway rope
trick are surprisingly symmetrical. Here's why!
Is the mean of the squares of two numbers greater than, or less
than, the square of their means?
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
Four jewellers share their stock. Can you work out the relative values of their gems?
Keep constructing triangles in the incircle of the previous triangle. What happens?
I am exactly n times my daughter's age. In m years I shall be ... How old am I?
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 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
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. . . .
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. . . .
It is impossible to trisect an angle using only ruler and compasses
but it can be done using a carpenter's square.
How many noughts are at the end of these giant numbers?
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.