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

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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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.

Investigate circuits and record your findings in this simple introduction to truth tables and logic.

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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.

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Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?

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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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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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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.

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

This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.

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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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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?

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

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

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.

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

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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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. . . .

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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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Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

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

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

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Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

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As a quadrilateral Q is deformed (keeping the edge lengths constnt) the diagonals and the angle X between them change. Prove that the area of Q is proportional to tanX.

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

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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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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.

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

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

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If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?

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Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

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Do you have enough information to work out the area of the shaded quadrilateral?

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

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

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Can the pdfs and cdfs of an exponential distribution intersect?

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

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.

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Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

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

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

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