In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).
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. . . .
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.
Tom writes about expressing numbers as the sums of three squares.
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. . . .
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.
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. . . .
By considering powers of (1+x), show that the sum of the squares of
the binomial coefficients from 0 to n is 2nCn
An article which gives an account of some properties of magic squares.
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.
An article about the strategy for playing The Triangle Game which
appears on the NRICH site. It contains a simple lemma about
labelling a grid of equilateral triangles within a triangular
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?
An inequality involving integrals of squares of functions.
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.
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.
Which of these triangular jigsaws are impossible to finish?
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.
Is the mean of the squares of two numbers greater than, or less
than, the square of their means?
Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you
notice when successive terms are taken? What happens to the terms
if the fraction goes on indefinitely?
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?
This is the second article on right-angled triangles whose edge lengths are whole numbers.
Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.
Follow the hints and prove Pick's Theorem.
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!
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 point moves around inside a rectangle. What are the least and the
greatest values of the sum of the squares of the distances from the
When if ever do you get the right answer if you add two fractions
by adding the numerators and adding the denominators?
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
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...
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.
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 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.
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 a connection between the shape of a special ellipse and an
infinite string of nested square roots.
Take any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
Sort these mathematical propositions into a series of 8 correct
Which of these roads will satisfy a Munchkin builder?
Can you work out where the blue-and-red brick roads end?
Given that a, b and c are natural numbers show that if sqrt a+sqrt
b is rational then it is a natural number. Extend this to 3
L triominoes can fit together to make larger versions of
themselves. Is every size possible to make in this way?
Can you work through these direct proofs, using our interactive
An introduction to some beautiful results of Number Theory
Can you invert the logic to prove these statements?
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?
Here the diagram says it all. Can you find the diagram?
Can you rearrange the cards to make a series of correct mathematical statements?
Explore a number pattern which has the same symmetries in different bases.