Hilbert's Third Problem and Dehn's Invariant

At the beginning of the twentieth century, David Hilbert published a list of 23 open problems which were considered by many to be the most significant open questions facing mathematicians at the time. The third problem on his list was related to the following question: given any two polyhedra of equal volume, is it possible to cut the first one into finitely many polyhedral pieces so that they can be reassembled to give the second? This was the first of Hilbert's problems to be solved and the solution belongs to his student, Max Dehn, who introduced a numeric ``invariant" in a rather ingenious way. In this talk we will not only discuss Hilbert's third problem and Dehn's solution, but also take time to review some of the rich history behind Hilbert's question which dates back to Euclid.