An introduction to some beautiful results of Number Theory

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.

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

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 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. . . .

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.

When is it impossible to make number sandwiches?

A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.

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. . . .

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 frame.

Have a go at being mathematically negative, by negating these statements.

Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.

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?

Can you work through these direct proofs, using our interactive proof sorters?

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. . . .

Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.

Can you rearrange the cards to make a series of correct mathematical statements?

What is the largest number of intersection points that a triangle and a quadrilateral can have?

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

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.

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn

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!

Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.

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. . . .

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

If you think that mathematical proof is really clearcut and universal then you should read this article.

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. . . .

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.

The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.

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.

L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.

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.

This is the second article on right-angled triangles whose edge lengths are whole 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?

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?

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.

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.

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 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.

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?

An inequality involving integrals of squares of functions.