Lesson 6 of 7 · Pair technique
Naked pairs: two squares that share two digits
The guide and all sudoku lessons
Every lesson so far has ended with a digit placed. A naked pair does something different: it removes candidates. Sometimes that is exactly what a stuck square needs.
If two squares in the same row, column or box both show exactly the same two marks, say 4 and 5, and nothing else, then between them they must take the 4 and the 5. You do not know which goes where yet, but you know no other square in that unit can hold either digit. Clear the 4 and the 5 from the unit's other marks.
This is the first technique that reasons about squares together instead of one at a time. It is what you reach for when no single is left, which happens more often as the givens thin out.
Example 1: In a box
Step 1 of 4
No single is left on this board. Read the pencil marks in the bottom-right box.
This board has no single left anywhere: every empty square shows at least two marks, and no digit has a single home in any row, column or box. Read the bottom-right box. Two of its squares, in row 7, carry exactly the same two marks, 1 and 6. Between them they take both digits, so no other square in the box can hold a 1 or a 6.
The ringed square, at row 8, column 9, was showing 1 and 9. Drop the 1 and only the 9 is left. Nothing about that square on its own told you that. The pair did.
Example 2: In a row
Step 1 of 4
Read the pencil marks in row 6. It has five empty squares, and no single is left on the board.
Row 6 has five empty squares. The two in columns 1 and 2 both show exactly 2 and 7. That pair owns the 2 and the 7 for the whole row.
The square in column 7 was showing 1 and 2. Clear the 2 and it can only be a 1. No single on this board would have found that 1: the row, the column and the box all still have other room for it. The pair is what decides it.
Try it yourself
A new board, and the same idea. Get it wrong and the board tells you why.
The marks in column 7 are filled in, and no single is left on the board. What goes in the ringed square?
Pick the digit for the ringed square.
How to spot it faster
- Look for squares with exactly two marks. A pair needs two of them in the same row, column or box with the same two digits.
- Check every unit the two squares share. Two squares in the same row and the same box clear the digits from both.
- After clearing, look at the squares you edited: the one that lost a mark may now be a single.
- The idea grows: three squares whose marks together use only three digits form a triple, and the same clearing applies.
Common mistakes
A third mark spoils the pair
Both squares must show the same two digits and nothing else. A square marked 4, 5 and 6 beside one marked 4 and 5 is not a pair.
Clearing outside the unit
The pair only owns its digits inside the units it shares. Two squares in the same row say nothing about a 4 in a different row.
Clearing the pair itself
Remove the digits from the other squares of the unit, never from the two squares that form the pair.
Questions
Try it
On a hard board, once singles run out, scan your marks for two squares in a unit that show the same two digits, clear them from the rest of the unit, and look again for singles.
Play solo on a hard 9×9