KöMaL 2006 K80

Polyminoes are like dominoes: they are made up of small squares that are joined together along whole sides. Define the ``boundary number'' of a polymino $P$ as the number of different ways to place congruent copies of $P$ around its boundary. (There must be a polymino touching it along each of its edges, and touching edges touch along their full length.) For example the boundary number of the cross is 2, since it can be bounded by its copies in the following two ways: (Arrangements that are mirror images of each other are considered different.) What is the boundary number of a 3×3 square (i.e. one consisting of 9 small squares)?