The person who solved this problem went on to propose an algorithm to solve the word problem for groups, and also showed that if an ordered pair on a plane is selected from the hyperreal numbers, the parallel postulate doesn't hold. Dupont and Sah used the central relation of this problem to derive a few results in K-theory
The person who solved this problem went on to propose an algorithm to solve the word problem for groups, and also showed that if an ordered pair on a plane is selected from the hyperreal numbers, the parallel postulate doesn't hold. Dupont and Sah used the central relation of this problem to derive a few results in K-theory,  and algebraic K-theory was partly developed due to this problem's solution. The solution of this problem relied on the lemma that if the cosine of alpha is equal to one-third, then alpha divided by pi is irrational. Sydler's theorem gives cases in which the answer to this problem is "yes", and the Bolyai-Gerwin theorem  solves the equivalent problem in two dimensions. An important quantity used to solve this problem is an element of the tensor product of R with the quotient space R divided by Q pi. That quantity is the Dehn invariant, which is equal for two figures if and only if they are scissors-congruent and have the same volume.  For 10 points, name this problem, which asks whether a polyhedron can be cut up and reassembled into any other polyhedron of equal volume.
The person who solved this problem went on to propose an algorithm to solve the word problem for groups, and also showed that if an ordered pair on a plane is selected from the hyperreal numbers, the parallel postulate doesn't hold.
{Hilbert's Third} Problem (accept anything {involving cutting up} a {polyhedron} and {reassembling} it {into another} one)