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?
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
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
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?
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?
Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.
How many noughts are at the end of these giant numbers?
Find the largest integer which divides every member of the
following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
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. . . .
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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.
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!
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.
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 looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
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.
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?
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. . . .
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. . . .
It is impossible to trisect an angle using only ruler and compasses
but it can be done using a carpenter's square.
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?
Can you see how this picture illustrates the formula for the sum of
the first six cube numbers?
Is the mean of the squares of two numbers greater than, or less
than, the square of their means?
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. . . .
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
What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.
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. . . .
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
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.
Four jewellers share their stock. Can you work out the relative values of their gems?
Can you discover whether this is a fair game?
What fractions can you divide the diagonal of a square into by simple folding?
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
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
Can you rearrange the cards to make a series of correct mathematical statements?
The tangles created by the twists and turns of the Conway rope
trick are surprisingly symmetrical. Here's why!
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?
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
Explore what happens when you draw graphs of quadratic equations
with coefficients based on a geometric sequence.
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?
Can you make sense of the three methods to work out the area of the kite in the square?
A introduction to how patterns can be deceiving, and what is and is not a proof.