Here the diagram says it all. Can you find the diagram?
Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.
Can you work through these direct proofs, using our interactive
To find the integral of a polynomial, evaluate it at some special
points and add multiples of these values.
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.
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 serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
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 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.
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?
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
Take a complicated fraction with the product of five quartics top
and bottom and reduce this to a whole number. This is a numerical
example involving some clever algebra.
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.
When if ever do you get the right answer if you add two fractions
by adding the numerators and adding the denominators?
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.
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.
Given any two polynomials in a single variable it is always
possible to eliminate the variable and obtain a formula showing the
relationship between the two polynomials. Try this one.
Can you see how this picture illustrates the formula for the sum of
the first six cube numbers?
If you think that mathematical proof is really clearcut and
universal then you should read this article.
Is the mean of the squares of two numbers greater than, or less
than, the square of their means?
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. . . .
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.
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?
Follow the hints and prove Pick's Theorem.
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.
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. . . .
Have a go at being mathematically negative, by negating these
Can you invert the logic to prove these statements?
Can you rearrange the cards to make a series of correct mathematical statements?
L triominoes can fit together to make larger versions of
themselves. Is every size possible to make in this way?
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?
These proofs are wrong. Can you see why?
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
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?
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. . . .
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.
In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).
Tom writes about expressing numbers as the sums of three squares.
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.
Can you discover whether this is a fair game?
Prove that, given any three parallel lines, an equilateral triangle
always exists with one vertex on each of the three lines.
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?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Take any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
An article which gives an account of some properties of magic squares.
Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.
This is an interactivity in which you have to sort into the correct
order the steps in the proof of the formula for the sum of a
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. . . .