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.

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

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.

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.

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

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.

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. . . .

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 for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.

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

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.

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.

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

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.

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

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.

Relate these algebraic expressions to geometrical diagrams.

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

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?

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.

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

Freddie Manners, of Packwood Haugh School in Shropshire solved an alphanumeric without using the extra information supplied and this article explains his reasoning.

An inequality involving integrals of squares of functions.

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.

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

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.

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

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

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

This article stems from research on the teaching of proof and offers guidance on how to move learners from focussing on experimental arguments to mathematical arguments and deductive reasoning.

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

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.

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

Explore a number pattern which has the same symmetries in different bases.

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.

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

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.

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

By proving these particular identities, prove the existence of general cases.

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

A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?

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

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?

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?

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

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