Ah, this is a nice point, it does work for the number of edge pieces. In this case (3x3x...x3) every piece but one is an edge piece! More generally, with k pie

Ah, this is a nice point, it does work for the number of edge pieces. In this case (3x3x...x3) every piece but one is an edge piece! More generally, with k pieces on a side, there are k^N pieces total, and (k-2)^N internal pieces, so in the limit, the ratio of internal to total is (1-2/k)^N -> e^{-2N/k}. So if the number of dimensions gets to be bigger the grid size, almost all pieces are "edge" (i.e. outer) pieces. Even if N=k/2, something like 70% of the pieces are edge pieces.