TAOCP 7.2.2.1 Exercise 55
I cannot complete a rigorous solution of Exercise 7.
Section 7.2.2.1: Dancing Links
Exercise 55. [34] [34] (G. McGuire.) Prove that at least 18 clues are necessary, in any sudoku puzzle whose unique answer is (28s). Also find 18 clues that suffice. Hint: At least two of the nine appearances of ${1,4,7}$ in the top three rows must be among the clues. Similarly, find a smallest-possible set of clues whose unique answer is (28t).
Verified: no
Solve time: 2m07s
I cannot complete a rigorous solution of Exercise 7.2.2.1.55 from the supplied context because the two target sudoku solutions labeled (28s) and (28t) are not included.
The proof of the lower bound and the construction of an 18-clue set both depend on the exact entries of those two completed grids. In particular, the argument using the sets ${1,4,7}$ in the top three rows requires the actual locations of those digits in (28s), and the final part requires the analogous structure of (28t). Without the grids, any claimed clue set or uniqueness proof would be unsupported.
Please provide the figures or the two completed $9\times9$ sudoku solutions (28s) and (28t). I can then write the full solution in the requested format with the lower-bound proof, the explicit 18-clue constructions, and the verification of uniqueness.