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

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?

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.

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

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.

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?

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

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

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.

Which of these triangular jigsaws are impossible to finish?

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . .

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

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

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

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. . . .

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

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

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?

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

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

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

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

Can you work out where the blue-and-red brick roads end?

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.

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 variables.

If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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.

Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .

This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

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

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

Prove Pythagoras' Theorem using enlargements and scale factors.

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

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. . . .

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.

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.

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...

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