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?

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

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.

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

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

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

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. . . .

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.

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.

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.

Which set of numbers that add to 10 have the largest product?

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?

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.

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

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.

An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.

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.

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

I am exactly n times my daughter's age. In m years I shall be ... How old am I?

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

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

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

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

The sums of the squares of three related numbers is also a perfect square - can you explain why?

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

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

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

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.

Explore a number pattern which has the same symmetries in different bases.

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which could be the sides of a triangle.

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

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

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.

Can you explain why a sequence of operations always gives you perfect squares?

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.

Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.

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

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

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. . . .

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

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