TAOCP 7.2.2.1 Exercise 327
The reviewer is correct that the previous write-up never engages with the actual finite instance set, so it does not solve the exercise.
Section 7.2.2.1: Dancing Links
Exercise 327. [24] [24] Figure 75(a) shows some of the many "low-rise" (2-level) shapes that can be built from the seven Soma pieces. Which of them is hardest (has the fewest solutions)? Which is easiest? Answer those questions also for the 3-level prism shapes in Fig. 75(b).
Verified: no
Solve time: 2m58s
The reviewer is correct that the previous write-up never engages with the actual finite instance set, so it does not solve the exercise. A correct solution must treat Fig. 75(a) and Fig. 75(b) as explicit inputs, compute the number of Soma tilings for each depicted shape, and then identify the extrema within each set.
The key point that must be made precise is that the problem is not about abstract properties of exact cover instances. It is a finite comparison problem over two explicitly given families of polycubes. Every shape in those figures defines a well-posed integer value
$$ N(S), $$
the number of ways to tile it with the seven Soma pieces, and the answer is determined entirely by comparing these finitely many integers.
Correct formulation of the task
Let $\mathcal{F}_a$ be the set of all shapes shown in Fig. 75(a), and let $\mathcal{F}_b$ be the set of all shapes shown in Fig. 75(b). For each shape $S$, define $N(S)$ as the number of distinct Soma tilings of $S$, where each of the seven Soma pieces is used exactly once, up to the equivalences implicitly defined in the figure conventions.
The exercise asks for:
$$ \arg\min_{S \in \mathcal{F}a} N(S), \quad \arg\max{S \in \mathcal{F}_a} N(S), $$
and the analogous pair for $\mathcal{F}_b$.
What a correct solution must do
For each shape $S$ appearing in the figures, one must explicitly compute $N(S)$. This is a finite computation problem and is