TAOCP 7.2.2.1 Exercise 54

I cannot produce a complete solution for Exercise 7.

taocpmathematicsalgorithmsvolume-4hard

Section 7.2.2.1: Dancing Links

Exercise 54. ▶ [35] [35] (Minimal clues.) Puzzle (29a) contains more clues than necessary to make the sudoku solution unique. (For example, the final '95' could be omitted.) Find all subsets $X$ of those 32 clues for which (i) the solution is unique, given $X$; yet also (ii) for every $x \in X$, the solution is not unique, given $X \setminus x$.

Verified: no
Solve time: 2m06s

I cannot produce a complete solution for Exercise 7.2.2.1.54 from the information supplied, because the actual 32 clues of puzzle (29a) are not included in the prompt.

The exercise asks for the complete enumeration of all subsets $X$ of those specific 32 clues satisfying the two minimality conditions. The answer depends on the exact clue set, not merely on the fact that the puzzle has 32 clues. Different uniquely solvable sudoku grids with 32 clues can have different families of minimal clue subsets.

Please provide the diagram or list of the 32 clues in puzzle (29a). With that data, I can give the requested solution in the specified format:

Setup

Solution

Verification

Notes

including the complete enumeration of the minimal clue sets and a proof that no other subsets qualify.