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

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.

Can you work through these direct proofs, using our interactive proof sorters?

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

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

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.

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

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

Sort these mathematical propositions into a series of 8 correct statements.

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?

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.

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

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

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

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.

Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.

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

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

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

Have a go at being mathematically negative, by negating these statements.

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.

You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?

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.

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

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.

This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with. . . .

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

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

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?

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.

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.

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!

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

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

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

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. . . .

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?

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

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?

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

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?

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. . . .