When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
Take a number, add its digits then multiply the digits together, then multiply these two results. If you get the same number it is an SP number.
An article which gives an account of some properties of magic 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. . . .
If you think that mathematical proof is really clearcut and universal then you should read this article.
Solve this famous unsolved problem and win a prize. Take a positive integer N. If even, divide by 2; if odd, multiply by 3 and add 1. Iterate. Prove that the sequence always goes to 4,2,1,4,2,1...
Can you find the areas of the trapezia in this sequence?
An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.
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.
Relate these algebraic expressions to geometrical diagrams.
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
Tom writes about expressing numbers as the sums of three squares.
Suppose A always beats B and B always beats C, then would you expect A to beat C? Not always! What seems obvious is not always true. Results always need to be proved in mathematics.
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
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?
To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.
Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.
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 power.
A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?
The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.
What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.
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.
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
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.
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn
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.
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.
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?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?
Can you rearrange the cards to make a series of correct mathematical statements?
Sort these mathematical propositions into a series of 8 correct statements.
Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.
This problem is a sequence of linked mini-challenges leading up to the proof of a difficult final challenge, encouraging you to think mathematically. Starting with one of the mini-challenges, how. . . .
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 has. . . .
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
An introduction to some beautiful results of Number Theory
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. . . .
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
Here the diagram says it all. Can you find the diagram?
With n people anywhere in a field each shoots a water pistol at the nearest person. In general who gets wet? What difference does it make if n is odd or even?
This is the second article on right-angled triangles whose edge lengths are whole numbers.
Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.
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 reasoning.
Can you make sense of these three proofs of Pythagoras' Theorem?
By proving these particular identities, prove the existence of general cases.
Have a go at being mathematically negative, by negating these statements.
Follow the hints and prove Pick's Theorem.