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.