The knight's move on a chess board is 2 steps in one direction and one step in the other direction. Prove that a knight cannot visit every square on the board once and only (a tour) on a 2 by n board. . . .

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.

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. . . .

How many tours visit each vertex of a cube once and only once? How many return to the starting point?

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

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.

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

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. . . .

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

Keep constructing triangles in the incircle of the previous triangle. What happens?

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

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.

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

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

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.

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.

How many noughts are at the end of these giant numbers?

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

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

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

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. . . .

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

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

The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!

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?

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

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

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

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

When is it impossible to make number sandwiches?

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

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.

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. . . .

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.

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?

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

Use this interactivity to sort out the steps of the proof of the formula for the sum of an arithmetic series. The 'thermometer' will tell you how you are doing

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

Prove Pythagoras' Theorem using enlargements and scale factors.

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.

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.

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

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

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.