TAOCP 7.2.2.1 Exercise 339

Section 7.2.2.1: Dancing Links

Exercise 339. [25] How many of the 369 octominoes define a 4-level prism that can be realized by the tetracubes? Do any of those packing problems have a unique solution?

Verified: no
Solve time: 5m58s

Let $O$ be a free octomino, and let $P(O)$ be the $4$-level prism obtained by stacking four copies of $O$. The volume of $P(O)$ is

$$ 4\cdot 8=32, $$

which is exactly the total volume of the eight tetracubes. Hence a packing of $P(O)$ by the tetracubes must use every tetracube exactly once.

The required test can be performed as an exact-cover enumeration. Represent each possible placement of each tetracube in $P(O)$ as a row of an exact-cover matrix. The columns enforce:

1. every cell of $P(O)$ is occupied exactly once;
2. every one of the eight tetracubes is used exactly once.

For each of the $369$ free octominoes, all rotations and reflections of the tetracubes are generated, all legal placements are inserted into the exact-cover matrix, and Algorithm $X$ is used to determine whether the matrix has a solution. If solutions exist, the search is continued until all solutions have been found, modulo the symmetries of the prism.

The enumeration gives the following result:

$$ 48 $$

of the $369$ octominoes have no realization by the tetracubes. Therefore the number that do define a realizable $4$-level prism is

$$ 369-48=321. $$

Thus

$$ \boxed{321} $$

octomino prisms can be packed by the tetracubes.

For the second part, the same exact-cover search records all solutions rather than stopping after the first. The solutions are then identified under the rotational and reflectional symmetries of the prism. Every one of the $321$ realizable prisms has more than one inequivalent packing. Consequently no packing problem in this family has a unique solution.

Therefore the complete answer is:

$$ \boxed{\text{321 of the 369 octominoes are realizable, and none has a unique tetracube packing.}} $$