Box-packing puzzles

A box-packing puzzle, or “polycube” puzzle, consists of a number of polycubes (assemblies of unit cubes) that are to be packed into a cubical box. Typically, the total number of unit cubes used to form all the polycubes of a particular puzzle is equal to the volume of the box to be packed. These types of puzzles have been around for at least 50 years and excellent insightful discussion has been given by Gardner, Golomb and Berlekamp et al..