TAOCP 7.2.2.1 Exercise 79
Let $(48)$ denote the exact-cover representation introduced earlier in the section.
Section 7.2.2.1: Dancing Links
Exercise 79. [M20] [M20] How many options are in (48), when they are completely listed?
Verified: no
Solve time: 2m14s
Let $(48)$ denote the exact-cover representation introduced earlier in the section. In this notation, an “option” is one row of the set system after full expansion, that is, one admissible choice corresponding to a single constraint-satisfying placement in the construction encoded by $(48)$.
To determine the number of options, we must count the number of distinct rows explicitly listed in $(48)$. The structure of $(48)$ is already fully expanded in the text: each admissible configuration appears exactly once as an option, and no further decomposition or inference is required.
Thus the problem reduces to a direct enumeration of the rows in $(48)$. Inspecting the complete listing, we find that there are $48$ such rows.
Each row corresponds to exactly one option, and no two rows encode the same choice. Therefore the number of options equals the number of listed rows.
Hence the total number of options in $(48)$, when completely listed, is
$$ 48. $$