Subsets & Locked Candidates

Naked Pair

A Naked Pair is two cells in one row, column or box whose candidates are exactly the same two digits, such as {4,8} and {4,8}. The two cells must take those two digits between them, so both digits can be removed from every other cell of that unit — and of their box as well when the two cells also share one. It is the smallest naked subset, and the first one to look for once singles and locked candidates run dry.

What you need to see on the grid

Look for a Naked Pair as soon as singles and locked candidates stop producing moves — it is the first subset the app's hint engine tries. The signal is two cells with exactly two candidates each in the same row, column or box. Four terms carry the method:

Bivalue cell: A cell with exactly two candidates left. A Naked Pair is two of them with identical lists. A cell with three candidates can never join, even when two of its digits match — that is where a Naked Triple starts.
The shared units: one or two: The pair clears every unit that holds both cells. Two cells of a row or column in different boxes share that line only. Two cells of a line inside one box share the line and the box, and clear both. Two identical bivalue cells that share no unit lock nothing.
Which cell takes which digit: The pair proves that its two digits are in those two cells, not which one is where. You do not need the order to clear the rest of the unit; another unit will settle it later.
Naked against hidden: A Naked Pair shows only its two digits and cleans the rest of the unit. A Hidden Pair is two digits with only two places left; its cells may carry other candidates, and the cleaning happens inside those two cells. In row 9 below, 4 and 5 can go only in R9C1 and R9C6 — a Hidden Pair, which would strip those two cells down to {4,5} while the Naked Pair {6,9} clears the rest of the row.

How to apply it, step by step

One board runs through all five steps: a pair in row 9 and the cell it opens, early in an expert puzzle from the app, at the exact point where its hint needs a Naked Pair. 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. The pencil marks are complete: a digit is missing from a cell only because its row, column or box already holds it — even when that column or box is not drawn — or because an earlier step removed it.

Step 1Find two cells in one unit with the same two candidates

Pick a row, column or box and look only at its cells with exactly two candidates. If two of them show the same two digits, you have a Naked Pair. Compare both digits: a single shared digit is not enough.

Row 9 has six empty cells. Four of them carry three or four candidates; the other two, R9C2 and R9C5, both read {6,9}. That is the pair. The two cells do not need to be neighbours — these two are three columns apart.

Technique: Naked Pair

Only row 9 matters here. R9C2 and R9C5 are its only cells with two candidates, and both read {6,9}.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1
R2
R3
R4
R5
R6
R7
R8
R9			2	3					8

Digit already on the grid
Row 9, the unit being searched
A cell of the pair
Candidates 6 and 9, the digits studied
Not needed for this step

Filled in: row 9. 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 2See why the two digits are used up, and where

Try both ways. If R9C2 is 6, R9C5 must be 9; if R9C2 is 9, R9C5 must be 6. Either way, row 9 gets its 6 and its 9 from these two cells, so no other cell of row 9 can hold either digit. We still do not know which cell takes which, and we do not need to.

The argument only covers units that contain both cells. R9C2 sits in box 7 and R9C5 in box 8, so here that is row 9 alone. Box 7 holds just one of the two cells: if R9C2 turns out to be 6, box 7 still needs a 9, and it can go in R7C1 or R7C3. So the pair says nothing about either box.

Technique: Naked Pair

Row 9 with boxes 7 and 8. Each box holds only one of the two cells and still has other places for 6 and 9, so the pair lives in row 9 alone.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1
R2
R3
R4
R5
R6
R7		1			8
R8		3		7	1
R9			2	3					8

Digit already on the grid
Row 9, the only unit holding both cells
A cell of the pair
Candidates 6 and 9, the digits studied
Not needed for this step

Filled in: row 9; boxes 7 and 8. 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 3Remove both digits from the rest of the unit

Go through every other cell of row 9 and erase 6 and 9 wherever they appear: R9C1 loses 9, R9C6 loses 6 and 9, R9C7 loses 6 and R9C8 loses 6. That is five candidates. The pair itself keeps {6,9} — it is still unsolved.

Check every cell of the row, including those that list only one of the two digits: R9C1 shows 9 but not 6, and it still loses the 9.

Technique: Naked Pair

Row 9 in full. 6 and 9 leave the four other empty cells: five candidates in all.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1
R2
R3
R4
R5
R6
R7
R8
R9			2	3					8

Digit already on the grid
Row 9, the unit being searched
A cell of the pair
Candidates 6 and 9, the digits studied
Candidate removed
Not needed for this step

Filled in: row 9. The other 72 cells are left blank: they are not needed for this step. Removed: 6 from R9C6, R9C7 and R9C8; 9 from R9C1 and R9C6.

Step 4When the pair also shares a box, clear the box too

Before moving on, check the box. When both cells of a pair also lie in one box, the argument of step 2 holds there as well, and the two digits leave the rest of that box too — often the half a player forgets.

Here there is nothing more to clear: R9C2 is in box 7 and R9C5 in box 8, so the pair shares no box. Box 7 keeps the 9s of R7C1 and R7C3, and box 8 keeps the 6 of R8C6 — any of them may still be the box's digit.

Boxes 7 and 8 after step 3. Each holds one cell of the pair, so neither box loses anything.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1
R2
R3
R4
R5
R6
R7		1			8
R8		3		7	1
R9			2	3

Digit already on the grid
Boxes 7 and 8: one cell of the pair each
A cell of the pair
Candidates 6 and 9, the digits studied
Not needed for this step

Filled in: boxes 7 and 8. The other 63 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.

Step 5Collect what the clean-up unlocks

A Naked Pair never places a digit by itself, so look at the other units of the cells it just cleared. R9C7 and R9C8 are in row 9, but also in box 9, and both have just lost their 6. Box 9 had three places for 6: R7C7, R9C7 and R9C8. Now R7C7 is the only one left: a Hidden Single, R7C7 = 6 — the app's next hint in this puzzle.

Row 9 itself places nothing yet; the move appears in a box the pair never touched directly. That is why you check every unit of every cell that lost a candidate.

Technique: Hidden Single

Box 9 after the pair. R9C7 and R9C8 have lost their 6, so 6 now fits only R7C7.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1
R2
R3
R4
R5
R6
R7							6	3	5
R8							4	2	9
R9									8

Digit already on the grid
Box 9: one place left for 6
The cell that must be 6
Candidate 6, the digit studied
Digit placed
Not needed for this step

Filled in: box 9. The other 72 cells are left blank: they are not needed for this step. Placed: R7C7 = 6.

Common mistakes

A pair that only half matches

R5C2 {2,9} and R9C2 {6,9} share column 2, and both have two candidates — but only the 9 is common. Nothing forces 9 into either of them: they could take 2 and 6 and leave the 9 of the column to R3C2, so nothing can be removed. Both digits must match before you erase anything.

One cell with a third candidate

If R9C5 read {6,7,9}, row 9 would have no pair: R9C5 could be 7 and leave one of 6 and 9 for another cell of the row. Do not drop the extra digit to make the pair fit. Keep the two cells in mind and look for a third one that uses only 6, 7 and 9 — that is a Naked Triple.

Getting the box wrong

It goes wrong both ways. Stop after the line when the pair also shares a box, and you lose moves. Sweep a box the pair does not share, and you erase without proof: R9C2 sits in box 7 but R9C5 does not, so removing 9 from R7C3 would be a guess — and a wrong one, because the solution puts the 9 of box 7 in R7C3.

Recent daily puzzles that needed this technique

Learn the move, not just the digit

In a real game, when a Naked Pair is what opens the next cell, the hint in the app points to that cell and names the pair in its reasoning — the same move as step 5 above.

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