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DanielVarga comments on Explained: Gödel's theorem and the Banach-Tarski Paradox - Less Wrong Discussion
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Okay. In the original formulation of the paradox, the task is to cut a ball into pieces, and assemble two balls from the pieces. If I am not mistaken, you have solved a slightly easier task: cut a ball into pieces, and covered two balls with the pieces (with overlaps). A part of the "nice exercise" is to bridge this gap.