Worked solution: Dehn's invariant: the tetrahedron and cube are not scissors-congruent (1900)

Dehn's 1900 work only establishes the 'only if' direction: scissors-congruent polyhedra have equal volume and equal $\operatorname{Dehn}$ invariant. Jean-Pierre Sydler proved the converse in 1965 (published in Commentarii Mathematici Helvetici): any two polyhedra $P, Q \subset \mathbb{R}^3$ with $\operatorname{vol}(P) = \operatorname{vol}(Q)$ and $\operatorname{Dehn}(P) = \operatorname{Dehn}(Q)$ are in fact scissors-congruent. Together, Dehn's invariant and volume form a complete invariant for three-dimensional scissors-congruence — the precise three-dimensional analogue of how area alone classifies polygons in the Wallace–Bolyai–Gerwien theorem, just with one more number needed.

Sydler's proof is a highly technical existence argument and, unlike the Bolyai–Gerwien construction, does not supply an explicit cutting procedure; later work (Børge Jessen in 1968, extending the framework to four dimensions, and subsequent reinterpretations via algebraic K-theory and group homology by Dupont, Sah, and others) has re-derived and generalised Sydler's theorem, tying Hilbert's third problem to substantially more advanced areas of modern algebra than Dehn's original elementary construction required.