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.

Investigate the sequences obtained by starting with any positive 2 digit number (10a+b) and repeatedly using the rule 10a+b maps to 10b-a to get the next number in the sequence.

Which of these triangular jigsaws are impossible to finish?

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

An inequality involving integrals of squares of functions.

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

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.

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.

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

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.

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

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

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

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

When is it impossible to make number sandwiches?

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.

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 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 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.

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.

Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.

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

Can you rearrange the cards to make a series of correct mathematical statements?

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

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn

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

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?

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?

The sums of the squares of three related numbers is also a perfect square - can you explain why?

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.

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

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

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

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

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.

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

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

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

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!

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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.

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

Kyle and his teacher disagree about his test score - who is right?

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

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.