From the introductory section: "This article is an attempt to put some algebraic order into the business of solving cubelike puzzles. Ideally, an algebraic theory would unfold that would, in an elementary way, yield highly efficient algorithms for all such puzzles. The work here is a step in that direction." Included in the applications section are the tetrahedron, octahedron, dodecahedron, icosahedron, and the general n X n X n cube. --Rodney Hoffman