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

Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

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

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

We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

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

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?

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.

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.

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.

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

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

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

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?

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

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. . . .

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

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

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...

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

What can you say about the common difference of an AP where every term is prime?

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?

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

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. . . .

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.

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.

Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

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

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.

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.

In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).

Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.

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

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. . . .

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

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

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

The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .

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.