Show that the arithmetic mean, geometric mean and harmonic mean of
a and b can be the lengths of the sides of a right-angles triangle
if and only if a = bx^3, where x is the Golden Ratio.
Can you use the diagram to prove the AM-GM inequality?
Is the mean of the squares of two numbers greater than, or less
than, the square of their means?
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
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...
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. . . .
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.
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.
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?
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 article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).
Peter Zimmerman, a Year 13 student at Mill Hill County High School
in Barnet, London wrote this account of modulus arithmetic.
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.
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.
Peter Zimmerman from Mill Hill County High School in Barnet, London
gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is
divisible by 33 for every non negative integer n.
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.
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.
An article which gives an account of some properties of magic squares.
Tom writes about expressing numbers as the sums of three squares.
When if ever do you get the right answer if you add two fractions
by adding the numerators and adding the denominators?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
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.
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.
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?
Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.
Prove that you cannot form a Magic W with a total of 12 or less or
with a with a total of 18 or more.
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. . . .
Follow the hints and prove Pick's Theorem.
This is the second article on right-angled triangles whose edge lengths are whole numbers.
We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.
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!
It is impossible to trisect an angle using only ruler and compasses
but it can be done using a carpenter's square.
A blue coin rolls round two yellow coins which touch. The coins are
the same size. How many revolutions does the blue coin make when it
rolls all the way round the yellow coins? Investigate for a. . . .
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.
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
Can you discover whether this is a fair game?
Can you make sense of these three proofs of Pythagoras' Theorem?
Have a go at being mathematically negative, by negating these
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
What fractions can you divide the diagonal of a square into by simple folding?
Can the pdfs and cdfs of an exponential distribution intersect?
Sort these mathematical propositions into a series of 8 correct
Can you work through these direct proofs, using our interactive
Can you invert the logic to prove these statements?
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. . . .