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

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.

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

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.

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?

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

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

An inequality involving integrals of squares of functions.

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

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

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

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.

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

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.

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.

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

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

An introduction to some beautiful results of Number Theory (a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions)

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.

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.

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.

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

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

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

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.

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?

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

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.

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?

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 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.

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 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.

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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.

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.

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

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. . . .

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.

Which of these triangular jigsaws are impossible to finish?

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

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.

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