Your seventh grade teacher may have taught you how to cut a triangle into pieces and glue them together to form a rectangle. She did not teach you the analogous trick in three dimensions -- and with good reason. In 1900, Hilbert conjectured (and Max Dehn proved) that it is impossible to cut a tetrahedron into a finite number of pieces and glue them together to form a rectangular box.

I shall discuss the history of this problem, which extends back to Euclid, and present a modified form of Dehn's proof, accessible to anyone who knows what a group is.