---
tag: 81035XQ5HE
kind: problem
name: Hilbert's third problem
msc: [52B45, 51M20, 51M25]
uses: [0325SGSJAR, S0TGRX3ARS]
about: [0325SGSJAR]
sources:
  - url: https://en.wikipedia.org/wiki/Hilbert%27s_third_problem
same_as:
  wikidata: Q2667025
  wikipedia: Hilbert's third problem
history:
  posed_by: "David Hilbert"
  first_stated: "1900"
---

## Problem

Let $P$ and $Q$ be two [[polyhedra|0325SGSJAR]] in $\R^3$ of equal volume.
Can $P$ always be cut into finitely many polyhedral pieces that can be reassembled by isometries into $Q$?

## Remarks

The answer is no.
Max Dehn proved in 1900 that the regular tetrahedron and the cube of the same volume cannot be cut into each other in this way.
His 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.
Jean-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.
In the plane the analogous question has a positive answer, by the [[Wallace–Bolyai–Gerwien theorem|5RQGCEQYK5]].

## Explanations

### L1 human

Two polygons with the same area can always be cut into pieces and rearranged into each other.
Hilbert asked whether the same is true for solids with the same volume.
It is not: no matter how you cut a regular tetrahedron into finitely many flat-faced pieces, you cannot rebuild a cube from them.
This was the first of Hilbert's 23 problems to be solved.