An introduction to some beautiful results of Number Theory

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.

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

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. . . .

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?

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.

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.

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

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

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.

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.

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

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

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

When is it impossible to make number sandwiches?

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?

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

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.

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

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

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!

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

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?

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

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

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. . . .

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?

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

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.

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.

Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.

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

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

Starting with one of the mini-challenges, how many of the other mini-challenges will you invent for yourself?

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. . . .

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?

Sort these mathematical propositions into a series of 8 correct statements.

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

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.

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.

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

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?

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...

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.