What was the historical motivation for Hilbert's third problem? Why did Hilbert feel it was worthy of including on his published list?
Hilbert's Third problem: Say that two polyhedra are scissors congruent if one can be cut up into finitely many polyhedra, each piece rotated/translated and then reassembled into the other polyhedron. Hilbert asked whether or not the unit cube and the regular tetrahedron of unit volume were scissors congruent.
Hilbert's student Dehn answered "no," providing an invariant (now known as the Dehn invariant) which remained unchanged under scissors congruence, but yielded different values for the cube and the regular tetrahedron.