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. . . .
Show that if three prime numbers, all greater than 3, form an
arithmetic progression then the common difference is divisible by
6. What if one of the terms is 3?
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
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
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.
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?
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.
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?
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.
A quadrilateral inscribed in a unit circle has sides of lengths s1, s2, s3 and s4 where s1 ≤ s2 ≤ s3 ≤ s4.
Find a quadrilateral of this type for which s1= sqrt2 and show s1 cannot. . . .
Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the. . . .
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.
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.
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. . . .
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.
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.
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. . . .
L triominoes can fit together to make larger versions of
themselves. Is every size possible to make in this way?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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?
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?
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?
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 article which gives an account of some properties of magic squares.
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.
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.
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.
Take any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
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?
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 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?
A connected graph is a graph in which we can get from any vertex to
any other by travelling along the edges. A tree is a connected
graph with no closed circuits (or loops. Prove that every tree. . . .
Clearly if a, b and c are the lengths of the sides of a triangle and the triangle is equilateral then
a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true, and if so can you prove it? That is if. . . .
Can you make sense of these three proofs of Pythagoras' Theorem?
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
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
An equilateral triangle is sitting on top of a square.
What is the radius of the circle that circumscribes this shape?
This is the second article on right-angled triangles whose edge lengths are whole numbers.
From a group of any 4 students in a class of 30, each has exchanged
Christmas cards with the other three. Show that some students have
exchanged cards with all the other students in the class. How. . . .
Nine cross country runners compete in a team competition in which
there are three matches. If you were a judge how would you decide
who would win?
I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?
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?
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
Six points are arranged in space so that no three are collinear.
How many line segments can be formed by joining the points in