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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.

Challenge Level

An inequality involving integrals of squares of functions.

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

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

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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Explain why, when moving heavy objects on rollers, the object moves twice as fast as the rollers. Try a similar experiment yourself.

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

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Can the pdfs and cdfs of an exponential distribution intersect?

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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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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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Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

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Prove that the shaded area of the semicircle is equal to the area of the inner circle.

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Prove Pythagoras' Theorem using enlargements and scale factors.

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Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

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A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

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ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

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Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

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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.

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Can you produce convincing arguments that a selection of statements about numbers are true?

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Kyle and his teacher disagree about his test score - who is right?

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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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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?

This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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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?

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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?

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.

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.

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

Challenge Level

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.

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.

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

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

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

This is the second article on right-angled triangles whose edge lengths are whole numbers.

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

The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!

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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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It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

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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?

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