TAOCP 7.2.2.1 Exercise 45

Section 7.2.2.1: Dancing Links

Exercise 45. [11] [11] List the sequence of naked single moves by which Algorithm X cruises to the solution of (29a). (If several such $p_{ij}$ are possible, choose the smallest $i$ at each step.)

Verified: no
Solve time: 2m47s

The exercise, as presented in the prompt, cannot be solved because the essential input data are missing.

Exercise $7.2.2.1.45$ asks for the explicit sequence of naked single moves made by Algorithm X while solving the Sudoku puzzle labeled $(29a)$, with the rule that whenever several naked singles $p_{ij}$ are available, the one with the smallest row index $i$ is chosen.

That sequence is uniquely determined by the initial Sudoku grid in $(29a)$. Since the grid itself is not included in the prompt, there is no way to determine

• which cells are naked singles initially;
• which move is selected first under the prescribed tie-breaking rule;
• how each placement changes the candidate sets; or
• the resulting ordered sequence of subsequent naked single moves.

Therefore no explicit sequence of moves can be derived from the information provided.

If the Sudoku instance $(29a)$ is supplied, the required solution is obtained by repeatedly performing the following steps:

1. Compute the current candidate set for every unfilled cell.
2. Identify every naked single $p_{ij}$, that is, every cell whose candidate set contains exactly one value.
3. If several naked singles are available, choose the one with the smallest row index $i$, exactly as specified in the exercise.
4. Record that move, update the Sudoku state and all affected candidate sets, and repeat until the puzzle is solved.

Because the defining Sudoku puzzle $(29a)$ is absent from the prompt, the explicit sequence of naked single moves requested by the exercise cannot be determined. The missing Sudoku grid is indispensable, and there is no mathematically valid way to reconstruct the required sequence without it.