TAOCP 7.2.2.1 Exercise 47

Section 7.2.2.1: Dancing Links

Exercise 47. [19] [19] What hidden singles are present in (32), after '3' is placed in cell (2,3)?

Verified: no
Solve time: 3m30s

The previous solution failed because it never engaged with the actual candidate structure of chart (32). A correct solution must restart from the updated state of chart (32) and explicitly test every unit.

Let $C(32)$ be the candidate structure of chart (32), which is fixed data in the exercise. After placing $3$ in cell $(2,3)$, form the reduced candidate structure $C'$ by standard constraint propagation:

• remove $3$ from all other cells in row $2$,
• remove $3$ from all other cells in column $3$,
• remove $3$ from all other cells in the $3\times 3$ block containing $(2,3)$.

A hidden single is defined as a pair $(U,d)$, where $U$ is a unit (row, column, or block) and digit $d$ appears as a candidate in exactly one cell of $U$ under $C'$.

The correction is to actually perform the required finite scan of $C'$:

1. For each row $r \in {1,\dots,9}$, count occurrences of each digit $d \in {1,\dots,9}$ among candidates in that row. Record $(\text{row } r, d)$ whenever the count equals $1$.
2. Repeat the same procedure for each column.
3. Repeat for each $3\times 3$ block.

Every such unique occurrence is a hidden single move.

At this point the crucial issue is the same one identified by the reviewer: the concrete candidate table of chart (32) is not present in the problem statement as provided here. Without the explicit contents of $C(32)$, the above procedure cannot be executed to produce the actual list of hidden singles, because the result depends entirely on the specific candidate eliminations already present in chart (32).

Thus, the only mathematically valid correction is to separate:

• the correct method, which is the unit-wise uniqueness test described above, and
• the missing data requirement, namely the explicit candidate state of chart (32) after inserting $x_{2,3}=3$.

Conclusion

The hidden singles in chart (32) after placing $3$ in cell $(2,3)$ are exactly the unit-digit pairs $(U,d)$ such that $d$ occurs in exactly one candidate cell of $U$ in the updated structure $C'$. Determining the explicit set of such moves requires the full candidate table of chart (32), which is not included in the information available here.

Therefore, no concrete enumeration can be produced without that chart, but the corrected solution reduces the problem to a complete and well-defined mechanical scan of $C'$ as specified above.