Find the minimum amount of pieces that the square with side \(1\) needs to be cut into, if a rectangle with side \(\pi\) is to be formed. All the pieces need to be used, with no overlaps. If no solution exists, prove so.

NOTE: I posed this as a problem a few hours ago, thinking I have had a solution. Once I started writing it, I realized that I have had a fault in my proof. I did not manage to find a correct proof on my own, so I offer it as a collaborative adventure into the plena.

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Tarksi's Circle Squaring Problem

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Hi, I posed a different problem. I am quite sure about the solution of my problem - \(5\). On Dissections, there is nicely depicted the dissection. However, they fail to prove that it is necessary to have as many pieces. That is what I am missing.

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Yes, I realize there's a difference between turning an unit circle into an unit square, or into a rectangle with sides\((1,\pi)\) Nevertheless, the nature of the proof is suggested. Probably far beyond the scope of a Note in Brilliant.org.

As a matter of fact, once one has any rectangle, it's almost a trivial exercise to dissect it into a finite number of pieces, and reform the pieces to make a square. This would dispute Tarski's findings, unless one accepts the idea of dissecting the circle into pieces that "cannot be cut from the circle with scissors".

Rectangle Into Square

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