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.

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.

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

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

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

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

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.

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 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 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?

Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?

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

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

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. . . .

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

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

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.)

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

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

Starting with one of the mini-challenges, how many of the other mini-challenges will you invent for yourself?

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?

The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?

Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.

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.

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

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

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

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?

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

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.

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.

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.

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

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

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

Some diagrammatic 'proofs' of algebraic identities and inequalities.

When is it impossible to make number sandwiches?

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

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

A introduction to how patterns can be deceiving, and what is and is not a proof.

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

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

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