Sudoku Solving Techniques

Coloring (candidate confirmed)

The example below demonstrates the Coloring technique that leads to confirming a candidate digit in a cell.

C1
C2
C3
C4
C5
C6
C7
C8
C9
R1
R2
R3
R4
R5
R6
R7
R8
R9

3

4

6

9

3

4

7

1

3

4

7

5
8

3

4

6

2

3

4

6

3

4

6

7

8
9

2

3

4

7

2

3

5

3

4

6

1
1
2
5
8

3

4

6
9

3

4

7

3

4

9

3

4

8

2

4

6

4

6

7

4

7

2

3

9

5
1
7
5

3

4

9

8
1

2

4

3

6

2

3

6

2

3

9

2
6
1
3
5
9
7
8
4
5
8
6
7
2

3

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3

4

1
9
2
9
7
1

3

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6

3

4

8
5
4
1
3
5
9
8
6
7
2
Figure 1a: Two Possible Positions for Digit 1 in Block B2

In row R2 of this puzzle, the digit 2 is a candidate for exactly two cells (R2,C5) and (R2,C6). That means there are two possible positions for the digit 2 in this row, and one of these two cells (R2,C4) and (R2,C6) must be 2. Let's see what conclusion we can draw.

Case 1: Cell (R2,C5) is 2
C1
C2
C3
C4
C5
C6
C7
C8
C9
R1
R2
R3
R4
R5
R6
R7
R8
R9

3

4

6

9

3

4

7

1

3

4

7

5
8

3

4

6

2

3

4

6

3

4

6

7

8
9

2

3

4

7

2

3

5

3

4

6

1
1
2
5
8

3

4

6
9

3

4

7

3

4

9

3

4

8

2

4

6

4

6

7

4

7

2

3

9

5
1
7
5

3

4

9

8
1

2

4

3

6

2

3

6

2

3

9

2
6
1
3
5
9
7
8
4
5
8
6
7
2

3

4

3

4

1
9
2
9
7
1

3

4

6

3

4

8
5
4
1
3
5
9
8
6
7
2
Figure 2a: Suppose Cell (R2,C5) is 2

In the first case, suppose that the cell (R2,C5) is 2.

C1
C2
C3
C4
C5
C6
C7
C8
C9
R1
R2
R3
R4
R5
R6
R7
R8
R9

3

4

6

9

3

4

7

1

3

4

7

5
8

3

4

6

2

3

4

6

3

4

6

7

8
9

2

3

4

7

2

3

5

3

4

6

1
1
2
5
8

3

4

6
9

3

4

7

3

4

9

3

4

8

2

4

6

4

6

7

4

7

2

3

9

5
1
7
5

3

4

9

8
1

2

4

3

6

2

3

6

2

3

9

2
6
1
3
5
9
7
8
4
5
8
6
7
2

3

4

3

4

1
9
2
9
7
1

3

4

6

3

4

8
5
4
1
3
5
9
8
6
7
2
Figure 2b: Cell (R2,C5) Cannot be 7

Since the cell (R2,C5) is 2, it cannot be 7. The candidate digit 7 in cell (R2,C5) is eliminated.

C1
C2
C3
C4
C5
C6
C7
C8
C9
R1
R2
R3
R4
R5
R6
R7
R8
R9

3

4

6

9

3

4

7

1

3

4

7

5
8

3

4

6

2

3

4

6

3

4

6

7

8
9

2

3

4

7

2

3

5

3

4

6

1
1
2
5
8

3

4

6
9

3

4

7

3

4

9

3

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8

2

4

6

4

6

7

4

7

2

3

9

5
1
7
5

3

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8
1

2

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6

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6

2

3

9

2
6
1
3
5
9
7
8
4
5
8
6
7
2

3

4

3

4

1
9
2
9
7
1

3

4

6

3

4

8
5
4
1
3
5
9
8
6
7
2
Figure 2c: Cell (R1,C5) Must be 7

The cell (R1,C5) must be 7 since it is the only possible position for the digit 7 in block B2.

Case 2: Cell (R2,C6) is 2
C1
C2
C3
C4
C5
C6
C7
C8
C9
R1
R2
R3
R4
R5
R6
R7
R8
R9

3

4

6

9

3

4

7

1

3

4

7

5
8

3

4

6

2

3

4

6

3

4

6

7

8
9

2

3

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7

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3

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1
1
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5
8

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9

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5
1
7
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2
6
1
3
5
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4
5
8
6
7
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3

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3

4

1
9
2
9
7
1

3

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6

3

4

8
5
4
1
3
5
9
8
6
7
2
Figure 3a: Suppose Cell (R2,C6) is 2

In the second case, suppose that the cell (R2,C6) is 2.

C1
C2
C3
C4
C5
C6
C7
C8
C9
R1
R2
R3
R4
R5
R6
R7
R8
R9

3

4

6

9

3

4

7

1

3

4

7

5
8

3

4

6

2

3

4

6

3

4

6

7

8
9

2

3

4

7

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3

5

3

4

6

1
1
2
5
8

3

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9

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8

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5
1
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9

2
6
1
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7
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1
9
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9
7
1

3

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6

3

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8
5
4
1
3
5
9
8
6
7
2
Figure 3b: Cell (R5,C6) Must be 4

The cell (R5,C6) cannot be 2 since the cell (R2,C6) in the same column C6 is 2. As a result, the cell (R5,C6) must be 4, as it turns out to be the only candidate for the cell.

C1
C2
C3
C4
C5
C6
C7
C8
C9
R1
R2
R3
R4
R5
R6
R7
R8
R9

3

4

6

9

3

4

7

1

3

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5
8

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6

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6
1
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7
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9
2
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7
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8
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9
8
6
7
2
Figure 3c: Cell (R5,C3) Must be 7

The cell (R5,C3) cannot be 4 since the cell (R5,C6) in the same row R5 is 4. As a result, the cell (R5,C3) must be 7, as it turns out to be the only candidate for the cell.

C1
C2
C3
C4
C5
C6
C7
C8
C9
R1
R2
R3
R4
R5
R6
R7
R8
R9

3

4

6

9

3

4

7

1

3

4

7

5
8

3

4

6

2

3

4

6

3

4

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7

8
9

2

3

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7

2

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5

3

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6

1
1
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5
8

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9

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7

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5
1
7
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8
1

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2
6
1
3
5
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8
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5
8
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7
2

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3

4

1
9
2
9
7
1

3

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6

3

4

8
5
4
1
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5
9
8
6
7
2
Figure 3d: Cell (R1,C3) Cannot be 7

The cell (R1,C3) cannot be 7 since the cell (R5,C3) in the same column C3 is 7.

C1
C2
C3
C4
C5
C6
C7
C8
C9
R1
R2
R3
R4
R5
R6
R7
R8
R9

3

4

6

9

3

4

7

1

3

4

7

5
8

3

4

6

2

3

4

6

3

4

6

7

8
9

2

3

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7

2

3

5

3

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6

1
1
2
5
8

3

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6
9

3

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7

3

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9

3

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8

2

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6

4

6

7

4

7

2

3

9

5
1
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5

3

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8
1

2

4

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6

2

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2

3

9

2
6
1
3
5
9
7
8
4
5
8
6
7
2

3

4

3

4

1
9
2
9
7
1

3

4

6

3

4

8
5
4
1
3
5
9
8
6
7
2
Figure 3e: Cell (R1,C5) Must be 7

The cell (R1,C5) must be 7 since it is the only possible position for the digit 7 in row R1.

As we can see in both cases, the cell (R1,C5) must be 7. We can conclude that the cell (R1,C5) can be confirmed to be 7, as shown in Figure 4.

C1
C2
C3
C4
C5
C6
C7
C8
C9
R1
R2
R3
R4
R5
R6
R7
R8
R9

3

4

6

9

3

4

7

1

3

4

7

5
8

3

4

6

2

3

4

6

7
8
9

2

3

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7

2

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5

3

4

6

1
1
2
5
8

3

4

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9

3

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7

3

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9

3

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8

2

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7

2

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5
1
7
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8
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6

2

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6

2

3

9

2
6
1
3
5
9
7
8
4
5
8
6
7
2

3

4

3

4

1
9
2
9
7
1

3

4

6

3

4

8
5
4
1
3
5
9
8
6
7
2
Figure 4: The Cell (R1,C5) Confirmed to be 7

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