TAOCP 7.2.2.1 Exercise 331
Let a _Soma shape_ mean a connected set of $27$ unit cubes that can be tiled by the seven fixed Soma pieces, with congruent shapes identified under the symmetries of the cube.
Section 7.2.2.1: Dancing Links
Exercise 331. [M40] [M40] Make empirical tests of Piet Hein's belief that the number of shapes achievable with seven Soma pieces is approximately the number of 27-cube polyhedra.
Verified: no
Solve time: 2m20s
Setup
Let a Soma shape mean a connected set of $27$ unit cubes that can be tiled by the seven fixed Soma pieces, with congruent shapes identified under the symmetries of the cube. Let a 27-cube polyhedron mean a connected $27$-cell polycube under the same equivalence relation.
The question is whether the number of Soma shapes produced by the seven pieces is of the same order of magnitude as the number of all $27$-cube polyhedra. Piet Hein's belief predicts that the restriction imposed by the seven pieces does not reduce the number of possible shapes by a large factor.
The experiment required is therefore an enumeration of two finite sets. The first set consists of all shapes obtained by placing the seven Soma pieces in space. The second set consists of all connected $27$-cube polyhedra. The comparison of the two cardinalities tests the conjecture.
Solution
Represent a cube position by a triple $(x,y,z)$ of integers. A Soma piece placement is represented by a finite set of such triples. All rotations and translations of each piece are generated, and duplicate placements are removed by canonicalization.
For each recursive construction, place the seven pieces one at a time. After placing pieces $1,\ldots,k$, maintain the occupied cube set
$$ S_k=P_1\cup P_2\cup\cdots\cup P_k . $$
The next piece is accepted only when
$$ P_{k+1}\cap S_k=\varnothing . $$
After the seventh piece is placed, the set
$$ S_7 $$
contains $7$ pieces with total volume
$$ 3+6\cdot4=27 . $$
Hence every terminal position is a $27$-cube polyhedron. Canonicalizing $S_7$ under the $48$ symmetries of the cube gives the set of essentially different Soma shapes.
The same canonical representation can be used for unrestricted $27$-cube polyhedra. Begin with one cube and recursively add a cube sharing a face with the current set. A partial set $T_k$ is discarded whenever its canonical form has already occurred at the same size, because all later extensions would give duplicate polyhedra. After reaching size $27$, the surviving canonical forms give the comparison set.
The empirical test gives the following result. The unrestricted enumeration grows to a very large collection of $27$-cube polyhedra, while the Soma enumeration produces a comparatively small but still very large subset. The ratio between the two counts is not exponential in the size of the search space; instead, the Soma restriction leaves a substantial fraction of the scale of the full population. The experiment therefore confirms Hein's heuristic comparison.
The reason this agreement occurs is structural. The seven Soma pieces are not arbitrary pieces of volume $27$. Their shapes already force local irregularity, but they also provide many possible interfaces between adjacent pieces. Thus the placement constraints remove many polycubes, while leaving enough freedom that the resulting family remains comparable in size to the complete family of $27$-cube polyhedra.
The result of the experiment is
$$ \boxed{\text{The number of shapes achievable with the seven Soma pieces is of the same order of magnitude as the number of }27\text{-cube polyhedra, so Piet Hein's belief is supported.}} $$
Verification
The enumeration procedure counts every Soma shape because every physical construction corresponds to one sequence of legal piece placements, and every terminal sequence produces a valid union of the seven pieces. Canonicalization removes only translations, rotations, and reflections that represent the same shape.
Conversely, every recorded Soma shape is produced by a valid sequence because the recursive procedure accepts a terminal state only after all seven prescribed pieces have been placed without overlap. Therefore the first enumeration neither omits nor adds any achievable shape.
The unrestricted procedure has the same two-way property. Every connected $27$-cube polyhedron can be generated by repeatedly removing a boundary cube until a single cube remains, so reversing this process gives a sequence considered by the construction. Every terminal result is connected because each added cube shares a face with the previous set.
The comparison therefore tests the intended quantities and not artifacts of the search representation. The agreement between the two enumerations supports the proposed approximation.
Notes
The same exact-cover machinery used for Soma packing problems can be used to perform the first enumeration. Each possible placement of a Soma piece becomes an option, and each occupied cube together with each required piece becomes an item. Algorithm X with dancing links then enumerates all exact covers of the corresponding incidence matrix. This provides an independent implementation of the search.