The sums of the squares of three related numbers is also a perfect
square - can you explain why?
A composite number is one that is neither prime nor 1. Show that
10201 is composite in any base.
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.
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS 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
Find the largest integer which divides every member of the
following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
Four jewellers share their stock. Can you work out the relative values of their gems?
Explore what happens when you draw graphs of quadratic equations
with coefficients based on a geometric sequence.
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. . . .
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?
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?
It is impossible to trisect an angle using only ruler and compasses
but it can be done using a carpenter's square.
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 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 article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
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. . . .
This is the second article on right-angled triangles whose edge lengths are whole numbers.
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.
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
Can you see how this picture illustrates the formula for the sum of
the first six cube numbers?
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. . . .
If you think that mathematical proof is really clearcut and
universal then you should read this article.
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.
Is the mean of the squares of two numbers greater than, or less
than, the square of their means?
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?
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.
What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
Use the numbers in the box below to make the base of a top-heavy
pyramid whose top number is 200.
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. . . .
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.
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
What fractions can you divide the diagonal of a square into by simple folding?
Can you rearrange the cards to make a series of correct mathematical statements?
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?
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.
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Can you make sense of these three proofs of Pythagoras' Theorem?
This article stems from research on the teaching of proof and
offers guidance on how to move learners from focussing on
experimental arguments to mathematical arguments and deductive
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
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?
L triominoes can fit together to make larger versions of
themselves. Is every size possible to make in this way?
What is the area of the quadrilateral APOQ? Working on the building
blocks will give you some insights that may help you to work it
This shape comprises four semi-circles. What is the relationship
between the area of the shaded region and the area of the circle on
AB as diameter?
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
Can you discover whether this is a fair game?