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

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

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

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

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.

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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.

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.

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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.

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

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

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

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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.

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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.

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

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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.

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An inequality involving integrals of squares of functions.

An introduction to some beautiful results in Number Theory.

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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.

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

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

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

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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.

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

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Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

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A polite number can be written as the sum of two or more consecutive positive integers, for example 8+9+10=27 is a polite number. Can you find some more polite, and impolite, numbers?

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

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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?

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

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

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Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

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.

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.

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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.

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.

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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. . . .

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.

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

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Which of these triangular jigsaws are impossible to finish?

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

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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.

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

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Mark a point P inside a closed curve. Is it always possible to find two points that lie on the curve, such that P is the mid point of the line joining these two points?