Singles

Naked Single

A Naked Single is an empty cell whose row, column and box already hold the other eight digits between them, so only one candidate is left and that digit goes there. To find one, pick a cell where a well-filled row, column and box cross, strike every digit those three units contain, and write in the one digit that survives. It reads the grid cell by cell — Hidden Single, the next lesson, asks the opposite question, digit by digit.

What you need to see on the grid

Look for Naked Singles as soon as you start writing candidates, and again after every digit you place: each placement takes a candidate away from up to 20 cells at once. On a pencilled grid the signal is a cell showing a single mark — but only if your marks are up to date. Four terms carry the whole method:

The 20 peers: Every cell shares a unit with exactly 20 others: 8 in its row, 8 in its column, and the 4 remaining cells of its box. Those 20 are the only cells on the grid that can take a candidate away from it.
A crowded crossing: The cells worth trying first sit where a well-filled row, column and box meet: the more of its peers already hold a digit, the fewer candidates a cell can have left. The extreme case is a unit with one empty cell — the Last Digit from the scanning guide — whose answer you can read off without any pencil marks.
Different digits, not filled cells: A Naked Single needs eight different digits among the peers, not eight filled peers. A digit held by both the row and the column counts once: in the example below, R5C3 has 11 filled peers, and they hold exactly eight different digits.
Cell view, not digit view: A Naked Single is found by looking at one cell and counting its surviving candidates down to 1. A Hidden Single is found by looking at one digit inside one unit and counting its possible cells down to 1 — same word, different question, and each finds moves the other misses.

How to apply it, step by step

One example runs through all five steps: the cell R5C3, early in a puzzle. Every diagram shows the whole grid; the cells a step needs are filled in, the others are left blank and hatched, so you know you can ignore them. Steps 1 to 3 draw only the digits already placed and the candidates of R5C3 as you work them out; steps 4 and 5 add every pencil mark.

Step 1Choose a cell where a busy row, column and box cross

A Naked Single hides in a cell that can see many digits, so start where a well-filled row, column and box meet. Row 5 and column 3 hold five digits each, and box 4 adds one more filled cell, R4C2 (its other two digits are already in row 5 and column 3). R5C3 sits where all three cross: 11 of its 20 peers are filled — more than for any other empty cell on this board. That makes it the first cell worth checking.

You need no pencil marks yet, only the three units to read. A cell in a nearly empty row or column is rarely worth the effort: it seldom sees enough digits to lose eight candidates.

Row 5, column 3 and box 4 in full, digits only. R5C3 is where they cross; 11 of its 20 peers are filled.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1			5
R2
R3			6
R4		7
R5	5			9	3	2			8
R6			2
R7			9
R8
R9			4

Digit already on the grid
Row 5, column 3 and box 4: what R5C3 can see
The cell being solved
Not needed for this step

Filled in: row 5; column 3; box 4. The other 60 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.

Step 2Strike the digits its row already holds

Start from all nine digits and cross off every digit row 5 already contains: 5, 9, 3, 2 and 8. A row holds each digit only once, so none of them can go in R5C3. Four candidates survive: 1, 4, 6 and 7.

Read the units in a fixed order — row, then column, then box — so you never have to wonder which ones you have already checked.

Only row 5 is drawn. Its five digits leave R5C3 four candidates: 1, 4, 6 and 7.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1
R2
R3
R4
R5	5			9	3	2			8
R6
R7
R8
R9

Digit already on the grid
Row 5, being read
The cell being solved
Not needed for this step

Filled in: row 5. The other 72 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.

Step 3Then strike the digits its column holds

Column 3 holds 5, 6, 2, 9 and 4. Three of them — 5, 2 and 9 — are already gone, because row 5 has them too: a digit is crossed off once, however many units contain it. Only 6 and 4 are new, and they take R5C3 from four candidates down to two: 1 and 7.

That is why counting filled peers is only a first filter. What decides is how many different digits they hold.

Row 5 and column 3. The column adds only 6 and 4, so R5C3 keeps 1 and 7.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1			5
R2
R3			6
R4
R5	5			9	3	2			8
R6			2
R7			9
R8
R9			4

Digit already on the grid
Row 5 and column 3, read so far
The cell being solved
Not needed for this step

Filled in: row 5; column 3. The other 64 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.

Step 4Finish with the box — one candidate left, place it

Box 4 holds 7, 5 and 2. The 5 and the 2 are already crossed off, but the 7 in R4C2 lies in neither row 5 nor column 3: only the box could see it. It removes the 7, and R5C3 is left with a single candidate, 1. Its 11 filled peers hold exactly eight different digits, 2 to 9, so the ninth must go here: write R5C3 = 1.

Notice what the digit 1 does around it: it is still a candidate in R5C2 and R5C8 in the row, in R4C3 and R8C3 in the column, and in five other cells of the box. No unit has only one place left for 1, so this is not a Hidden Single — reading digit by digit finds nothing here; only reading the cell does.

Technique: Naked Single

Row 5, column 3 and box 4 with every candidate. R5C3 lists only 1; every other empty cell here still has two or more.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1			5
R2
R3			6
R4		7
R5	5		1	9	3	2			8
R6			2
R7			9
R8
R9			4

Digit already on the grid
The cell with one candidate left
Candidate 1, the digit studied
Digit placed
Not needed for this step

Filled in: row 5; column 3; box 4. The other 60 cells are left blank: they are not needed for this step. Placed: R5C3 = 1.

Step 5Erase the digit from its peers and follow the chain

A placed digit leaves the pencil marks of all its peers at once. R4C3, in the same column and box, had two candidates, 1 and 8; with the 1 gone it has only 8 — another Naked Single. Write R4C3 = 8.

Keep going and box 4 falls cell by cell: the 8 leaves R4C1 with 9 alone, and that 9 leaves R6C1 with 3. Each placement creates the next single, but only if you erase as you go. Skip the update and R4C3 still shows 1 and 8, and the chain never starts.

Technique: Naked Single

Row 4, column 3 and box 4 after R5C3 = 1. The 1 has left every peer, and R4C3 keeps only 8.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1			5
R2
R3			6
R4		7	8						3
R5	5		1
R6			2
R7			9
R8
R9			4

Digit already on the grid
Down to one candidate after R5C3 = 1
Candidate 8, the digit studied
Digit placed
Not needed for this step

Filled in: row 4; column 3; box 4. The other 60 cells are left blank: they are not needed for this step. Placed: R4C3 = 8.

Common mistakes

Placing on a hunch instead of on eight eliminations

"Only a 1 seems to fit here" is not a Naked Single. Name the unit that removes each of the other eight digits, as steps 2 to 4 do; if you cannot name one for even a single digit, the cell still has two candidates and you have found nothing.

Trusting stale pencil marks

Every digit you place has to disappear from the pencil marks of all its peers. A mark that should have gone keeps a cell at two candidates when it really has one — R4C3 in step 5 still showing 1 and 8 — so the very singles your placement created stay hidden. When in doubt, re-derive the cell from its row, column and box instead of trusting the marks.

Checking two units instead of three

A cell is constrained by its row, its column and its box together, and the box adds four peers the two lines do not reach. Stop after the column in the example and R5C3 still looks like 1 or 7: the 7 that settles it, in R4C2, is in box 4 but in neither row 5 nor column 3.

Recent daily puzzles that needed this technique

Learn the move, not just the digit

In a real game, when the hint in the app places a Naked Single, its explanation follows steps 2 to 4: the digits the cell's row already has, then the new ones from its column, then from its box, until one digit is left.

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