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.

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . .

Prove Pythagoras' Theorem using enlargements and scale factors.

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?

If you think that mathematical proof is really clearcut and universal then you should read this article.

Four jewellers share their stock. Can you work out the relative values of their gems?

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?

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

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. . . .

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

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

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

An article about the strategy for playing The Triangle Game which appears on the NRICH site. It contains a simple lemma about labelling a grid of equilateral triangles within a triangular frame.

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

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

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.

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

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

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

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.

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.

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

A blue coin rolls round two yellow coins which touch. The coins are the same size. How many revolutions does the blue coin make when it rolls all the way round the yellow coins? Investigate for a. . . .

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 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.

A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?

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.

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

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

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 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.

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

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.

Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.

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

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.

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

A connected graph is a graph in which we can get from any vertex to any other by travelling along the edges. A tree is a connected graph with no closed circuits (or loops. Prove that every tree has. . . .

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. . . .

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin?

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

When is it impossible to make number sandwiches?

What is the largest number of intersection points that a triangle and a quadrilateral can have?

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