Direct Hidden Single

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A Direct Hidden Single is a single technique that determines a placement by ruling out impossible positions using only the digits already placed on the board (givens and answers), without writing candidates.

How to spot it

• For each digit x from 1 to 9, examine all cells where x has already been placed. Rule out every unfilled cell that shares a row, column, or box with any of them.
• Examine each house (row, column, or box) in turn. If only one unfilled cell has not been ruled out, x must be placed in that cell.

How it works

• The rules of Sudoku require each digit from 1 to 9 to appear exactly once in every house. Once a cell contains x, no other cell in the same row, column, or box can contain x, so those positions can be ruled out.
• After ruling out these positions, consider a house where x has not yet been placed. If only one unfilled cell can contain x, x must be placed in that cell; otherwise, the house would be missing digit x, violating the Sudoku rules.

Examples

On the board above, focus on digit 1. For every cell that already contains a 1, whether it is a given or an answer, we cross out all unfilled cells in the same row, column, and box. That tells us that only the unfilled cells not crossed out can still take digit 1.

Looking at row 4, only cell R4C2 remains available for digit 1. Therefore, 1 must be placed in R4C2.

This pattern of crossed-out cells resembles the cross-hatched shading used in drawing, so it is called cross-hatching.

If we inspect the peer cells of R4C2, we find that only digits 3, 5, 6, 7, 8, and 9 have appeared. In other words, 1, 2, and 4 could all still go in R4C2; these are its candidates. That observation alone is not enough to determine the digit. Only by using cross-hatching can we see that, in this house, digit 1 has exactly one possible location. That is why this technique is called a Hidden Single.

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Direct Hidden Singles can also appear in a column or a box. For example, on the board below, if we focus on digit 2, we find that in column 3, only R7C3 is not crossed out. Therefore, 2 must be placed in R7C3.

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If we focus on digit 2 on the board below, we find that in box 3, only R3C8 is not crossed out. Therefore, 2 must be placed in R3C8.