Hilbert's third problem (or a modern formulation thereof) asks whether two polyhedra $P,Q$ of equal volume are equidecomposable by cutting $P$ into finitely many polyhedral pieces and rearranging them to obtain $Q$. (In the two-dimensional case, this can always be done with polygons of equal area.)
With the use of Dehn invariants, one can show that this is impossible; however, the proof focuses on specific equivalence classes of dihedral angles, and does not seem to rule out "approximate" solutions.
To be more precise: Given polyhedra $P,Q$ of identical volume, here are some notions of a "close" solution to Hilbert's third problem:
For all $\epsilon>0$, $P$ may be cut into finitely many polyhedra which can be reassembled to form a polyhedron which contains a copy of $Q$ scaled down by $1-\epsilon$ and is contained in a copy of $Q$ scaled up by $1+\epsilon$.
For all $\epsilon>0$, $P$ may be cut into finitely many polyhedra which can be reassembled to form a polyhedron whose boundary differs from that of $Q$ over a region of surface area less than $\epsilon$.
For all $\epsilon>0$ and points $x$ on the boundary of $Q$, $P$ may be cut into finitely many polyhedra which can be reassembled to form a polyhedron which is identical to $Q$ except within a ball of radius $\epsilon$ centered at $x$.
The first condition is easily satisfied by chopping up $P$ into very very small cubes; I do not see how to show the latter two conditions, though I expect they are probably true. Is it known whether dissections meeting the latter two conditions can always be performed?
