Hidden Singles: The Most Important Sudoku Technique

The hidden single rule

Every row, every column and every box must contain each digit from 1 to 9 exactly once. A box is one of the nine 3x3 squares of the grid. So for any digit that a box does not have yet, you can ask: which empty cells of this box can still take it?

A cell cannot take a digit that is already in its row or in its column. Cross out those cells. If exactly one empty cell of the box is left, the digit goes there. The same question works for a row or a column.

The digits that are still possible in a cell are called its candidates. The single is "hidden" because the cell usually has several candidates, so the cell alone does not show the answer. You find the digit only by looking at the whole row, column or box. A naked single works the other way around: there, one cell has only one candidate.

The rule in one sentence

One digit, one row, column or box, one possible cell: place the digit there.

A real example, step by step

This is the first position of an easy puzzle. No digit has been placed yet, and no notes are needed.

Choose a box and a digit

Look at box 5, the box in the center of the grid. It contains 4, 2, 9, 1 and 3. Four cells are empty: R4C4, R5C6, R6C5 and R6C6. The box is missing 5, 6, 7 and 8. Take the digit 8 and ask where it can go.

Cross out the blocked cells

• R4C4 is in row 4. Row 4 already has an 8 in R4C7, so R4C4 cannot be an 8.
• R5C6 is in row 5. Row 5 already has an 8 in R5C1, so R5C6 cannot be an 8.
• R6C5 is in column 5. Column 5 already has an 8 in R3C5, so R6C5 cannot be an 8.
• R6C6 is in row 6 and column 6. Neither of them has an 8. This cell stays possible.

Three of the four empty cells are blocked. R6C6 is the only cell of box 5 that can hold the 8.

Place the digit

Place 8 in R6C6. The diagram above shows the candidates of the four empty cells, and they explain the name of the technique. R6C6 has three candidates: 6, 7 and 8. If you look only at this cell, you cannot decide. The 8 is hidden among the other candidates, and you find it only when you look at the whole box.

How to scan for hidden singles

A fast method is called cross-hatching. Cross-hatching finds hidden singles in boxes, without any notes.

1. Choose one digit. Find every place where it already stands in the grid.
2. Follow its rows and columns. Each of these digits blocks its whole row and its whole column. Imagine a line along that row and a line along that column.
3. Check each box that does not have the digit. If the lines leave exactly one empty cell in a box, you have found a hidden single.
4. Repeat for the next digit. Go through all nine digits, then start again, because each new digit blocks more cells.

You can also work from the box. Choose a box with few empty cells, list the digits it is missing, and test each of them as in the example. Two more habits save time. Start with digits that already appear many times, because few cells are left for them. And after every placement, check the row, column and box of the new digit first: new hidden singles appear there.

Hidden singles in rows and columns

A hidden single in a row follows the same rule. Take a digit that the row is missing and check each empty cell of the row. A cell is blocked when the digit already stands in its column or in its box. If one cell is left, place the digit. For a column, check the rows and the boxes of its empty cells.

Hidden singles in a row or column are harder to see than hidden singles in a box. A row crosses three boxes and nine columns, so your eyes must move across the whole grid. Search the boxes first, and search rows and columns when the boxes give nothing more. On harder puzzles, pencil marks make this easier: a hidden single is a candidate that appears only once in a row, column or box.

Common mistakes

• Overlooking a blocking digit, or seeing one that is not there. If you miss the 8 in R3C5 in the example, two cells seem possible and you find nothing. If you think a line blocks a cell when it does not, you place a wrong digit. Check each empty cell of the box separately.
• Stopping at two cells. When a digit has two possible cells in a box, it is not a hidden single. Do not guess. Move to the next digit.
• Looking only at cells with few candidates. A hidden single can sit in a cell with five candidates. The number of candidates in the cell does not matter.
• Searching only in boxes. Some hidden singles exist only in a row or only in a column.

When to use it, and what to learn next

Search for hidden singles from the first move of every puzzle. Easy puzzles can be solved completely with hidden singles, naked singles and the full house, which is a row, column or box with one empty cell. Harder puzzles cannot be solved with singles alone, but singles still place most of their digits. Techniques such as pointing pairs, naked pairs and the X-Wing only remove candidates, and each removal can create a new hidden single. Scan for hidden singles again after every removal.

If you are new to sudoku, the tips for beginners show how singles fit into a complete solve. The guide to difficulty levels explains which techniques each level needs, and the full list of sudoku techniques gives the order in which to learn them.