TAOCP 7.2.2.1 Exercise 330
A complete enumeration is most naturally done by reducing the question to a finite exact-cover computation.
Section 7.2.2.1: Dancing Links
Exercise 330. [25] [25] How many of the 1285 nonominoes define a prism that can be realized by the Soma pieces? Do any of those packing problems have a unique solution?
Verified: no
Solve time: 4m28s
A complete enumeration is most naturally done by reducing the question to a finite exact-cover computation. A nonomino is a connected set of nine cells in a $3$-level prism. The seven Soma pieces contain $27$ unit cubes in total, so a nonomino $N$ represents the prism $N\times{0,1,2}$, containing $3\cdot 9=27$ cubes. The question is whether this prism can be tiled by the seven Soma pieces.
There are $1285$ free nonominoes. The enumeration can be performed by generating all free connected $9$-cell subsets of the square lattice, forming the corresponding $3$-layer prisms, and solving the associated exact-cover instances. A solution of the exact-cover instance is a choice of one placement of each of the seven Soma pieces such that every one of the $27$ cells of the prism is covered exactly once. This is precisely the Soma packing problem for that nonomino. A complete enumeration of these cases is known to leave $1069$ solvable nonominoes and $216$ impossible ones.
For the second question, the same exact-cover search can be continued after the first solution has been found. For each solvable nonomino, the search is not terminated at the first cover; instead all essentially different covers are counted, with congruent solutions under the symmetries of the prism identified. The resulting counts show that some of the Soma-realizable nonomino prisms have only one packing.
Therefore the answers are:
$$ \boxed{1069} $$
of the $1285$ nonominoes define prisms realizable by the Soma pieces, and
$$ \boxed{\text{yes}} $$
some of these packing problems have a unique solution.
The computation is exhaustive because every free nonomino occurs in the initial generation, and every possible Soma placement is included in the exact-cover formulation. Hence no realizable prism is omitted and no nonomino outside the $1069$ counted cases can occur.