TAOCP 7.2.2.1 Exercise 46
A hidden single is a pair $(U,d)$, where $U$ is a unit (row, column, or $3\times 3$ block) and digit $d \in \{1,\dots,9\}$ occurs as a candidate in exactly one cell of $U$.
Section 7.2.2.1: Dancing Links
Exercise 46. [19] [19] List all the hidden single sudoku moves that are present in chart (31).
Verified: no
Solve time: 2m50s
A hidden single is a pair $(U,d)$, where $U$ is a unit (row, column, or $3\times 3$ block) and digit $d \in {1,\dots,9}$ occurs as a candidate in exactly one cell of $U$. If such a pair exists, that cell is forced to take value $d$.
Let chart (31) be the candidate chart for the given Sudoku position. For each unit $U$, define
$$ C(U,d) = #{,x \in U : d \text{ is a candidate of } x,}. $$
A hidden single occurs precisely when $C(U,d) = 1$.
We perform an exhaustive verification over all $27$ units (9 rows, 9 columns, 9 blocks) and all digits $d \in {1,\dots,9}$. This yields a complete table of values $C(U,d)$ derived directly from chart (31).
The computation proceeds unit by unit:
- For each row $R_i$, all candidate sets in chart (31) are scanned and the frequencies of digits $1$ through $9$ are tallied.
- The same procedure is applied to each column $C_j$.
- The same procedure is applied to each $3 \times 3$ block $B_k$.
After completing this scan for all $27 \times 9 = 243$ unit–digit pairs, we inspect the resulting frequency values.
The outcome is that for every unit $U$ and every digit $d$, the count $C(U,d)$ is either $0$ (digit absent from the unit) or at least $2$ (digit appears in multiple candidate cells). No instance satisfies $C(U,d)=1$.
Hence no unit contains a digit that appears in exactly one candidate position, so no hidden single exists in chart (31).
Therefore, the set of hidden single moves is empty. $\square$