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?

Can you find the value of this function involving algebraic fractions for x=2000?

Four jewellers possessing respectively eight rubies, ten saphires, a hundred pearls and five diamonds, presented, each from his own stock, one apiece to the rest in token of regard; and they. . . .

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

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?

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 geometric series.

In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!

By proving these particular identities, prove the existence of general cases.

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

An article which gives an account of some properties of magic 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.

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

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.

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.

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. . . .

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.

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.

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.

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.

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

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.

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...

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.

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

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.

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 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 vertices?

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.

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.

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 prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

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.

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?

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. . . .

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.

What fractions can you divide the diagonal of a square into by simple folding?

Can you make sense of these three proofs of Pythagoras' Theorem?

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

Can the pdfs and cdfs of an exponential distribution intersect?