If you have ever solved an easy Sudoku, you have already used Naked Singles and Hidden Singles, probably without knowing their names. These two techniques are the foundation of every solving method. They are enough to finish most easy puzzles and a good share of medium ones, and even in the hardest puzzles they do most of the actual placing of digits. Learning to spot them deliberately, instead of by luck, is the fastest way to become a quicker and calmer solver.
Candidates and pencil marks
A candidate is a digit that could still go into an empty cell without breaking the rules: it does not already appear in the same row, the same column or the same 3×3 box. Pencil marks (also called notes) are the small digits you write in a cell to keep track of its candidates.
To describe positions, solvers use a simple notation: r1c5 means row 1, column 5. Rows are counted from top to bottom, columns from left to right, and boxes are numbered 1 to 9 like reading a page, so box 1 is top left and box 2 is top middle.
You do not have to fill in every candidate from the start. On easy puzzles, many players solve entirely by scanning. Still, understanding candidates is what makes both singles easy to explain, and full pencil marks become essential once a puzzle gets harder.
What is a Naked Single?
A Naked Single is a cell that has only one candidate left. Every other digit is already present in its row, its column or its box, so the remaining digit must go there. It is called "naked" because the answer is visible just by looking at that one cell.
Example: finding a Naked Single in r1c5
Suppose r1c5 is empty, and you check its three units:
| Unit | Digits already placed |
|---|---|
| Row 1 | 3, 5, 7, 9 |
| Column 5 | 1, 4, 8 |
| Box 2 (top middle) | 2, 9 |
| All seen digits together | 1, 2, 3, 4, 5, 7, 8, 9 |
Eight different digits are already seen by r1c5. The only digit missing from the list is 6, so r1c5 = 6. Notice that the 9 appears in both row 1 and box 2; it only counts once. The useful habit is to count different digits, not entries.
What is a Hidden Single?
A Hidden Single looks at the puzzle from the opposite direction. Instead of asking "which digits can go in this cell?", you ask "where can this digit go in this unit?". If a digit has only one possible cell left in a row, a column or a box, it must go there, even if that cell still has several other candidates. That is why it is "hidden": the cell itself may look undecided until you focus on a single digit.
Example: a Hidden Single for the digit 4 in box 1
Here is box 1 of a different puzzle. Dots are empty cells:
| Column 1 | Column 2 | Column 3 | |
|---|---|---|---|
| Row 1 | 8 | . | . |
| Row 2 | . | 3 | . |
| Row 3 | . | . | 5 |
Box 1 does not contain a 4 yet. Outside the box, there is a 4 in r1c7, a 4 in r2c5 and a 4 in r7c2, and there is no 4 in row 3, column 1 or column 3. Now check the six empty cells one by one:
- r1c2 and r1c3 cannot hold 4, because row 1 already has a 4 in r1c7.
- r2c1 and r2c3 cannot hold 4, because row 2 already has a 4 in r2c5.
- r3c2 cannot hold 4, because column 2 already has a 4 in r7c2.
- Only r3c1 remains. Box 1 must contain a 4 somewhere, so r3c1 = 4.
We do not know (or need to know) what else r3c1 could contain. It might still have three or four candidates. The digit 4 simply has nowhere else to go.
How to scan for singles efficiently
Random searching is slow. Good solvers use a few repeatable scanning patterns and switch between them.
Scan by number (cross-hatching)
Pick one digit, say 1, and look at every 1 already in the grid. Mentally draw lines through their rows and columns. Then visit each box that has no 1 yet and see how many cells are left uncovered. If exactly one is left, you have a Hidden Single. Go through the digits 1 to 9 in order; digits that already appear many times are the most productive, because they cover the most cells.
Scan by box
Choose a box that is nearly full, for example one with six or seven givens. List the missing digits and test each one against the rows and columns passing through the box. Crowded boxes produce singles quickly, and if only one cell is empty, the last digit is found immediately.
Scan by row and column
Rows and columns with only two or three empty cells deserve the same treatment. For each missing digit, check which of the empty cells can take it. Hidden Singles in rows and columns are a little harder to see than in boxes, because the blocking digits are spread across the whole grid, so many players check them after the box scan.
A step-by-step routine
- Start with cross-hatching for each digit from 1 to 9 and place every Hidden Single in a box.
- Check nearly complete rows, columns and boxes for missing digits.
- When scanning stops producing results, write full pencil marks for the remaining cells.
- Look for cells with a single candidate (Naked Singles) and digits with a single position in a unit (Hidden Singles).
- After every placement, remove that digit from the candidates in the same row, column and box. One placement often creates the next single.
Common mistakes
- Forgetting to update pencil marks. If you place a 4 and do not erase 4 from its row, column and box, your notes start lying to you and you may "find" a single that is not real.
- Incomplete candidate lists. Notes written in a hurry often miss a digit. A cell that looks like a Naked Single might really have two candidates.
- Checking only two of the three units. For a Naked Single you must check the row, the column and the box. Skipping the box is the most frequent slip.
- Confusing the two singles. A Hidden Single cell can still show several pencil marks. Do not dismiss it just because it does not look solved.
- Guessing. A well-made Sudoku has exactly one solution and can be solved by logic. If you are stuck, re-scan; a guess that seems to work can hide an error until much later.
When singles are not enough
Sometimes every unit has been scanned, the pencil marks are correct, and no single appears. That is the sign that the puzzle needs the next layer of techniques. The usual next steps are locked candidates (pointing and claiming), where a digit in a box is confined to one row or column; naked and hidden pairs, where two cells share the same two candidates; and later patterns such as the X-Wing. Each of these does not place a digit directly. Instead, it removes candidates, and the removal almost always reveals a new single.
The Sudoku Explainer (SE) rating reflects this. In that system a Hidden Single in a box rates 1.2, a Hidden Single in a row or column rates 1.5, and a Naked Single rates 2.3. A puzzle that can be finished with singles alone therefore has an SE rating of at most 2.3. If a puzzle is rated higher, you will need at least one technique beyond singles.
Practice makes the patterns automatic
Naked and Hidden Singles are simple to understand, but speed comes from repetition. Try an easy or medium puzzle in our online Sudoku and make a point of using cross-hatching before you write any notes. The daily Sudoku is a good way to practise a little every day. After a couple of weeks you will notice that you no longer search for singles; you simply see them.