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.

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

Prove Pythagoras' Theorem using enlargements and scale factors.

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

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

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

The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?

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

An equilateral triangle is constructed on BC. A line QD is drawn, where Q is the midpoint of AC. Prove that AB // QD.

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.

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

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

Can you correctly order the steps in the proof of the formula for the sum of a geometric series?

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.

Can you find the areas of the trapezia in this sequence?

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.

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

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

Do you have enough information to work out the area of the shaded quadrilateral?

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

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

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

Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.

Can you make sense of the three methods to work out the area of the kite in the square?

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

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

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

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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?

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. . . .

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.

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.

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.

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.

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

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!

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.

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.

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

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

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.

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.

When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?

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

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.

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. . . .