Naked Single is another basic technique for solving Sudoku puzzles.
For an empty cell in a Sudoku puzzle, nine digits from 1 to 9 may be candidates. Based on the Sudoku rules, no digits can appear more than once in each row, each column, and each 3x3 block. So, if a digit appears in the same row, in the same column, or in the 3x3 block as the empty cell, then this digit cannot appear in that empty cell, and we can rule it out as a candidate for the cell. If eight out of the nine digits have been ruled out as candidates for an empty cell and only one digit can be a candidate, then the remaining candidate is the correct digit for the cell. In other words, for a particular empty cell, if eight out of the nine digits have been placed in other cells in the same row, in the same column, or in the same 3x3 block, then we can determine that the remaining one digit must be placed in that particular cell.
Let's exhibit this technique in the following partial Sudoku puzzle:
In Figure 1, the orange cell (R2,C5) is empty. Let's check if we can determine which digit should be filled in it.
First, we identify the related area of the orange cell. The related area of a certain cell contains all other cells falling in the same row, in the same column, or in the same 3x3 block as that certain cell. In this case, those twenty light brown cells in row R2, in column C5, or in block B2, as shown in the grid in Figure 1, form the related area of the orange cell. Notice that eight digits 1, 2, 4, 5, 6, 7, 8, and 9 have appeared in the related area (those light brown cells). According to the Sudoku rules, each digit from 1 to 9 can appear only once in each row, each column, and each 3x3 block. As a result, these eight digits cannot appear in the orange cell, and the only digit that can be filled in this orange cell is the digit 3. So, we can determine the orange cell is 3 (Figure 2).
Below is a real Sudoku puzzle. We will demonstrate this Naked Single technique in more detail with this example.
The Sudoku puzzle in Figure 3 is a valid Sudoku puzzle that has a unique solution. The orange cell (R8,c3) is empty. We will use the Naked Single technique to see if we can determine a digit for it. If there is only one candidate digit for the cell, then that digit must be the right answer for the cell.
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First, we identify the related area of the orange cell. The twenty cells in the same row, in the same column, or in the same 3x3 block form the related area of the orange cell, and they are highlighted in light brown as shown in Figure 4. Nine digits from 1 to 9 can be candidates for the orange cell. Based on the Sudoku rules that no digits can appear more than once in each row, in each column, and in each 3x3 block, we are going to rule out some digits as candidates for the orange cell. If we reach the case that there is only one candidate digit left, then this digit is the correct answer for the cell (the Naked Single technique).
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The digit 1 appears twice in the related area of the orange cell. The first one is in the cell (R7,C3), which is in the same block B7 as the orange cell. The second one is in the cell (R8,C4), which is in the same row R8 as the orange cell. So, the digit 1 cannot be a candidate for the orange cell.
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The digit 2 appears twice in the related area of the orange cell. The first one is in the cell (R4,C3), which is in the same column C3 as the orange cell. The second one is in the cell (R8,C1), which is in the same block B7 and in the same row R8 as the orange cell. So, the digit 2 cannot be a candidate for the orange cell.
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The digit 3 appears once in the related area of the orange cell. It is in the cell (R8,C9), which is in the same row R8 as the orange cell. So, the digit 3 cannot be a candidate for the orange cell.
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The digit 5 appears once in the related area of the orange cell. It is in the cell (R8,C2), which is in the same block B7 and in the same row R8 as the orange cell. So, the digit 5 cannot be a candidate for the orange cell.
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The digit 6 appears once in the related area of the orange cell. It is in the cell (R9,C3), which is in the same block B7 and in the same column C3 as the orange cell. So, the digit 6 cannot be a candidate for the orange cell.
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The digit 7 appears once in the related area of the orange cell. It is in the cell (R9,C2), which is in the same block B7 as the orange cell. So, the digit 7 cannot be a candidate for the orange cell.
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The digit 8 appears three times in the related area of the orange cell. The first one is in the cell (R2,C3), which is in the same column C3 as the orange cell. The second one is in the cell (R7,C1), which is in the same block B7 as the orange cell. The third one is in the cell (R8,C7), which is in the same row R8 as the orange cell. So, the digit 8 cannot be a candidate for the orange cell.
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The digit 9 appears once in the related area of the orange cell. It is in the cell (R6,C3), which is in the same column C3 as the orange cell. So, the digit 9 cannot be a candidate for the orange cell.
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The digit 4 does not appear in the related area of the orange cell. Now, it is the only remaining candidate for the orange cell. So, we can determine that the digit 4 is the correct answer for the orange cell.
The orange cell should be filled with the digit 4.
List of Sudoku Solving Techniques
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