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Circles Ad Infinitum

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

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Climbing Powers

$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or $(r^r)^r$, where $r$ is $\sqrt{2}$?

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Flexi Quads

A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular?

Impossible Square?

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

You are given a limitless supply of right-angled triangles with hypotenuses of unit length and angles of $60$ and $30$ degrees. Can you make a square from these pieces without any gaps or overlaps? See the video clip for a discussion of this problem.

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This problem is about possibility or impossibility. Can you or can you not make a square from the triangles? You will need to think carefully about the structure of the problem: What properties do the square and triangle have? How can you relate these? What do squares and the triangles have in common? How are they different?

For a similar challenge, see the problem Impossible Triangles?

For small squares, it is easy to check all possible configurations of pieces to check whether a solution exists. For larger squares the number of combinations of pieces gets larger extremely rapidly, and quickly reaches the point at which a check of all of the combinations is impossible, even on a supercomputer. Even if we have checked a large number of square sizes and found no solution, this does not necessarily mean that we cannot find a solution for a larger square. To find a solution, you often need to mine the depths of your cunning and ingenuity. To show that a solution does not exist you often have to use the concept of proof by contradiction .

Proving that a solution does not exist is often much easier than proving that a solution does exist. Interestingly, if a solution can be found, then it is usually very quick to check that the solution is correct, although finding the solution in the first place might be exceptionally difficult. This behaviour underlies the notion of the mysteriously titled 'P vs NP' problem, the solution of which will earn the solver $1,000,000

Ideas concerning proof are discussed in the fascinating article Proof: A Brief Historical Survey Ideas concerning complexity, P vs NP and other ways to earn a million dollars mathematically are found in the PLUS article Code-breakers, Doughnuts, and Violins .