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  "body": "\n## Problem\n\nLet $P$ and $Q$ be two [[polyhedra|0325SGSJAR]] in $\\R^3$ of equal volume.\nCan $P$ always be cut into finitely many polyhedral pieces that can be reassembled by isometries into $Q$?\n\n## Remarks\n\nThe answer is no.\nMax Dehn proved in 1900 that the regular tetrahedron and the cube of the same volume cannot be cut into each other in this way.\nHis tool was the [[Dehn invariant|S0TGRX3ARS]], which does not change under cutting and reassembling: it is $0$ for the cube and nonzero for the regular tetrahedron.\nJean-Pierre Sydler proved in 1965 that two polyhedra are scissors-congruent if and only if they have the same volume and the same Dehn invariant.\nIn the plane the analogous question has a positive answer, by the [[Wallace–Bolyai–Gerwien theorem|5RQGCEQYK5]].\n\n## Explanations\n\n### L1 human\n\nTwo polygons with the same area can always be cut into pieces and rearranged into each other.\nHilbert asked whether the same is true for solids with the same volume.\nIt is not: no matter how you cut a regular tetrahedron into finitely many flat-faced pieces, you cannot rebuild a cube from them.\nThis was the first of Hilbert's 23 problems to be solved.\n",
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