The two basic techniques, Hidden Single and Naked Single, alone cannot solve all Sudoku puzzles. For somewhat more difficult puzzles, you may encounter a situation where you cannot apply any of these two basic techniques. You cannot find a unique cell position for a digit in a 9-cell unit (a row, a column, or a 3x3 block), so you cannot apply the Hidden Single technique. Also, you cannot find a unique possible digit for an empty cell in the grid, so you cannot apply the Naked Single technique. In this situation, you need some advanced techniques to reduce the possibilities so that you can apply the two basic techniques again. The Locked Candidates technique is one of the most commonly used techniques to reduce some possibilities when solving a Sudoku puzzle.
In some cases, you cannot find a unique cell position for a digit, but the digit is locked in a few cells as candidates. That means you can determine that the digit must be in one of a few cells. If these cells happen to fall in the same row, in the same column, or in the same block, it may lead to eliminating some possibilities in other cells. This technique is called Locked Candidates. We will explain this in more detail with examples.
The Sudoku grid in Figure 1 is partially filled with digits. It is not a real Sudoku puzzle. We only use it to demonstrate how to apply the Locked Candidates technique and the consequences after applying this technique.
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In block B1 of this grid, there are five empty cells, and we try to find the possible cell positions for the digit 3. We observe that the green cell (R2,C6) in row R2 is 3, so the pink cells in block B1 cannot be 3 since they are in the same row as the green cell. No repeated digit can be in a single row. Now two yellow cells (R1,C1) and (R3,C1) are still possible for the digit 3, and we do not have further information to rule out any of them. In other words, the digit 3 is locked in the two yellow cells in block B1. One of the two yellow cells must be 3.
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The digit 3 is locked in the two yellow cells in block B1. At the same time, the two yellow cells happen to fall in the same column C1 of this grid. Since one of the two yellow cells must be 3 and no digit can be repeated in the same column, other cells in column C1 cannot be 3. That means all orange cells (R4,C1), (R5,C1), (R6,C1), (R7,C1), (R8,C1), and (R9,C1) in column C1 cannot be 3. As we can see now, the Locked Candidates technique can eliminate some possibilities in some cells, which can help us advance the progress of solving the puzzle.
Let's go further with this puzzle.
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In block B7 of this grid, there are four empty cells, and the digit 3 is not in this block yet. After applying the Locked Candidates technique to rule out the possibilities of the digit 3 in three cells (R7,C1), (R8,C1), and (R9,C1), there is only one possible cell (R8,C2) left for the digit. As a result, the cell (R8,C2) must be 3 (Hidden Single). Without applying the Locked Candidates technique first, there are no ways to determine which of the four empty cells in block B7 should be the correct position for the digit 3.
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After applying the Locked Candidates technique first, followed by the Hidden Single technique, we can determine that the cell (R8,C2) should be filled with the digit 3.
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After applying the Locked Candidates technique to eliminate some possibilities in some cells, we may reach a situation where we can apply the Naked Single technique for a cell. In this example, the orange cell (R5,C1) in Figure 6 is empty. We are going to check how many possible digits can be filled in this cell. First, we highlight the related area (cells in the same row, in the same column, or in the same 3x3 block) of the orange cell with light brown as shown in Figure 6. After that, we count how many distinct digits appear in this area. In this case, seven digits, which are 1, 2, 4, 5, 6, 8, and 9, are in the related area already. As a result, they cannot be candidates for the orange cell. Now only digits 3 and 7 are possible for the orange cell. Without applying the Locked Candidate technique, we cannot determine which of these two digits is the correct digit for the cell. However, the Locked Candidate technique rules out the possibility of the digit 3. So, the digit 7 turns out to be the only possible digit for the orange cell.
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After applying the Locked Candidates technique first, followed by the Naked Single technique, we can determine that the cell (R5,C1) should be filled with the digit 7.
In the example above, we can apply the two basic techniques right after the Locked Candidates technique. However, in many real cases, we have to apply advanced techniques several times to eliminate some possibilities before we can apply the two basic techniques. It becomes very difficult to remember all eliminations. For this reason, many players would like to make some notes in the cells with pencils. For difficult puzzles, players may fill in all obvious candidate digits (those digits that are not in the same row, in the same column, or in the same block) for all empty cells first, and then remove the candidates from the cells if the advanced techniques rule them out so that we do not need to remember all the eliminations with our heads. We just need to concentrate on the techniques and logic when solving the puzzles.
In order to demonstrate the advanced techniques more clearly, the examples for the advanced techniques will have candidate digits filled in all empty cells so that we can show how to spot the patterns of the advanced techniques and which possibilities could be eliminated from the cells as the consequences of the advanced techniques.
Next, we will use real Sudoku puzzles as examples to demonstrate this technique from different scenarios.
List of Sudoku Solving Techniques
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