Sudoku Solving Techniques

Example: Hidden Single in Block

This example is to demonstrate how to apply the Hidden Single technique in a 3x3 block.

C1
C2
C3
C4
C5
C6
C7
C8
C9
R1
R2
R3
R4
R5
R6
R7
R8
R9
4
6
2
3
9
3
1
2
4
2
5
3
8
4
1
7
8
6
3
9
3
9
4
1
6
?
?
2
?
3
?
?
1
?
6
2
4
9
3
4
3
5
8
2
7
5
7
3
2
6
4
Figure 1: Six Empty Cells in Block B6

In block B6 of this puzzle, there are six empty cells, which are (R4,C7), (R4,C8), (R5,C7), (R5,C9), (R6,C7), and (R6,C9). The digit 6 is not in this block yet. We want to decide which of these six empty cells should be filled with the digit 6. The Hidden Single technique is to look for the cells that are possible to accommodate the digit 6 in the block. If there is only one such cell, then that cell must be the digit 6.

C1
C2
C3
C4
C5
C6
C7
C8
C9
R1
R2
R3
R4
R5
R6
R7
R8
R9
4
6
2
3
9
3
1
2
4
2
5
3
8
4
1
7
8
6
3
9
3
9
4
1
6
|
?
2
|
3
?
|
1
?
6
2
4
9
3
4
3
5
8
2
7
5
7
3
2
6
4
Figure 2: Three Cells (R4,C7), (R5,C7), and (R6,C7) Excluded for Digit 6

As shown in Figure 2, the cell (R9,C7) in column C7 is 6. Since each column can only contain the digit 6 once, three cells (R4,C7), (R5,C7), and (R6,C7) in the same column cannot be 6. These three cells are eliminated as possible positions for the digit 6. Now three empty cells (R4,C8), (R5,C9), and (R6,C9) remain for the digit 6 in block B6.

C1
C2
C3
C4
C5
C6
C7
C8
C9
R1
R2
R3
R4
R5
R6
R7
R8
R9
4
6
2
3
9
3
1
2
4
2
5
3
8
4
1
7
8
6
3
9
3
9
4
1
6
|
?
2
|
3
|
1
?
6
2
4
9
3
4
3
5
8
2
7
5
7
3
2
6
4
Figure 3: One More Cell (R5,C9) Excluded for Digit 6

As shown in Figure 3, the cell (R5,C2) in row R5 is 6. Since each row can only contain the digit 6 once, the cell (R5,C9) in the same row cannot be 6. It is eliminated as a possible position for the digit 6. Now two empty cells (R4,C8) and (R6,C9) remain for the digit 6 in block B6.

C1
C2
C3
C4
C5
C6
C7
C8
C9
R1
R2
R3
R4
R5
R6
R7
R8
R9
4
6
2
3
9
3
1
2
4
2
5
3
8
4
1
7
8
6
3
9
3
9
4
1
6
|
?
2
|
3
|
1
6
2
4
9
3
4
3
5
8
2
7
5
7
3
2
6
4
Figure 4: Another Cell (R6,C9) Excluded for Digit 6

As shown in Figure 4, the cell (R6,C4) in row R6 is 6. Since each row can only contain the digit 6 once, the cell (R6,C9) in the same row cannot be 6, and it is eliminated as a possible position for the digit 6. Now only one empty cell (R4,C8) remains for the digit 6 in block B6.

C1
C2
C3
C4
C5
C6
C7
C8
C9
R1
R2
R3
R4
R5
R6
R7
R8
R9
4
6
2
3
9
3
1
2
4
2
5
3
8
4
1
7
8
6
3
9
3
9
4
1
6
|
2
|
3
|
1
6
2
4
9
3
4
3
5
8
2
7
5
7
3
2
6
4
Figure 5: Cell (R4,C8) - a Possible Position for Dight 6

The cell (R4,C8) is located in row R4, column C8, and block B6. No digit 6 appears in the same row, the same column, or the same 3x3 block. So the cell (R4,C8) can be a possible position for the digit 6 (Figure 5).

C1
C2
C3
C4
C5
C6
C7
C8
C9
R1
R2
R3
R4
R5
R6
R7
R8
R9
4
6
2
3
9
3
1
2
4
2
5
3
8
4
1
7
8
6
3
9
3
9
4
1
6
6
2
3
1
6
2
4
9
3
4
3
5
8
2
7
5
7
3
2
6
4
Figure 6: Digit 6 Filled in Cell (R4,C8)

As we can see in Figure 5, the only possible position for the digit 6 in block B6 is the cell (R4,C8). Since each 3x3 block must contain the digit 6 once, we can conclude that the cell (R4,C8), as the only possible position for the digit 6 in block B6, must be 6. The cell (R4,C8) should be filled with the digit 6, as shown in Figure 6 above.

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