Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
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. . . .
Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
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?
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
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.
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. . . .
An article which gives an account of some properties of magic squares.
Can you rearrange the cards to make a series of correct mathematical statements?
Can you make sense of these three proofs of Pythagoras' Theorem?
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?
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.
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?
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
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 composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.
Is the mean of the squares of two numbers greater than, or less than, the square of their means?
What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
Four jewellers share their stock. Can you work out the relative values of their gems?
If you think that mathematical proof is really clearcut and universal then you should read this article.
The sums of the squares of three related numbers is also a perfect square - can you explain why?
Can you use the diagram to prove the AM-GM inequality?
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.
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 number...
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.
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.
Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.
If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
When is it impossible to make number sandwiches?
Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?
Investigate the sequences obtained by starting with any positive 2 digit number (10a+b) and repeatedly using the rule 10a+b maps to 10b-a to get the next number in the sequence.
Can you explain why a sequence of operations always gives you perfect squares?
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
This is the second article on right-angled triangles whose edge lengths are whole numbers.
What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?
Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .
The knight's move on a chess board is 2 steps in one direction and one step in the other direction. Prove that a knight cannot visit every square on the board once and only (a tour) on a 2 by n board. . . .
The circumcentres of four triangles are joined to form a quadrilateral. What do you notice about this quadrilateral as the dynamic image changes? Can you prove your conjecture?
Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.
Which of these roads will satisfy a Munchkin builder?
Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.
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. . . .
Can you find the areas of the trapezia in this sequence?
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!
Can you make sense of the three methods to work out the area of the kite in the square?
Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?