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 $F_{n+1}F_{n-1}$.

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?

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?

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

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

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.

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

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

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 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?

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

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

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

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. . . .

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

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

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

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

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.

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.

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.

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

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

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.

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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

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. . . .

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.

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 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!

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

This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.

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.

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.

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.

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.

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

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.