Editing Hilbert's third problem (section)

{{Short description|On dissections between polyhedra}}
[[File:Cube and prism from two bricks.svg|thumb|Two polyhedra of equal volume, cut into two pieces which can be reassembled into either polyhedron]]
The third of [[Hilbert's problems|Hilbert's list of mathematical problems]], presented in 1900, was the first to be solved. The problem is related to the following question: given any two [[polyhedron|polyhedra]] of equal [[volume]], is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second? Based on earlier writings by [[Carl Friedrich Gauss]],<ref>[[Carl Friedrich Gauss]]: ''Werke'', vol. 8, pp. 241 and 244</ref> [[David Hilbert]] conjectured that this is not always possible. This was confirmed within the year by his student [[Max Dehn]], who proved that the answer in general is "no" by producing a counterexample.<ref>{{cite journal |first=Max |last=Dehn |title=Ueber den Rauminhalt |journal=[[Mathematische Annalen]] |volume=55 |year=1901 |issue=3 |pages=465–478 |doi=10.1007/BF01448001 |s2cid=120068465 |url=https://zenodo.org/record/2327856 }}</ref>

The answer for the analogous question about [[polygon]]s in 2 dimensions is "yes" and had been known for a long time; this is the [[Wallace–Bolyai–Gerwien theorem]].

Unknown to Hilbert and Dehn, Hilbert's third problem was also proposed independently by Władysław Kretkowski for a math contest of 1882 by the Academy of Arts and Sciences of [[Kraków]], and was solved by [[Ludwik Birkenmajer|Ludwik Antoni Birkenmajer]] with a different method than Dehn's. Birkenmajer did not publish the result, and the original manuscript containing his solution was rediscovered years later.<ref name=":0">{{Cite journal|last1=Ciesielska|first1=Danuta|last2=Ciesielski|first2=Krzysztof|date=2018-05-29|title=Equidecomposability of Polyhedra: A Solution of Hilbert's Third Problem in Kraków before ICM 1900|journal=The Mathematical Intelligencer|volume=40|issue=2|pages=55–63|language=en|doi=10.1007/s00283-017-9748-4|issn=0343-6993|doi-access=free}}</ref>