A naked pair is two squares in the same row, column or box holding the same two candidates and nothing else. Between them those two squares use up both numbers, so both numbers can be crossed off every other square in that region.
Like pointing pairs, it places nothing. You've already made that adjustment once, so this time it needs one line rather than a section: the payoff is squares getting smaller, and squares getting smaller is what produces the next number you write in.
What's new here is where you look. The first three techniques all read the board through regions and numbers. This one reads two squares against each other, and it's the first technique that is completely invisible unless you've written your candidates down.
Two squares, two numbers, no room for anyone else
Here's why it works, and the whole argument is one sentence long.
If a square can only be a 1 or a 5, and another square in the same region can also only be a 1 or a 5, then between them those two squares take the 1 and the 5. You don't know which way round. You don't need to. Either the first takes the 1 and the second takes the 5, or the other way about — and in both cases, both numbers are used up.
So no other square in that region can be a 1 or a 5. Not because you've worked out what they are, but because the two numbers they might have wanted are already spoken for.
That's the whole technique. The hard part isn't the logic; it's that you can't see it unless your candidates are written down. A board with no pencil marks on it is a board where this technique doesn't exist yet.
The board
Same puzzle the rest of this series runs on, three squares further along.
The three highlighted squares are ones a solver working this board would already have placed: the 5 in the centre and the 8 near the top are the two worked examples from naked singles and hidden singles, and the 2 on the fifth row is the first answer in the naked singles exercise. If you've never seen any of that, it doesn't matter — the board above is a complete, legal position and everything below works from it as it stands.
Now look down the second column, and write its candidates in.
Five empty squares in that column. Read them and one thing should stop you.
The fourth square down reads 1 and 5. The sixth square down reads 1 and 5. Not similar sets — the same set, exactly, with nothing else in either.
That's a naked pair, and unlike almost everything else in this series you really can spot it by looking. Two small squares, matching.
What it buys you
Those two squares are going to take the 1 and the 5 between them. So every other square in the column loses both numbers.
Go and look at what that hits. The second square down is 4 and 7 — no 1, no 5, nothing to take. The eighth is 7 and 8 — nothing again. And the last one, at the bottom of the column, reads 1, 4 and 5.
Two of its three candidates just died.
It's a 4. One candidate left, which makes it a naked single, which means you can write it in.
That's worth sitting with for a second, because it's the answer to the complaint people have about every technique after the singles. You didn't place a number. You cleared two candidates off one square — and clearing them placed a number anyway, one square over.
A technique that places nothing and a technique that places nothing useful are different things. This one manufactures singles.
The jargon, deflated
Naked pair. The name is honest, for once. It's a pair because two squares hold two numbers; it's naked because both squares are showing you exactly what they've got, with nothing else in the way.
Which is the whole difference from the one you'll meet next. A naked pair is visible — two small squares, matching. A hidden pair is the same idea buried inside squares carrying four or five candidates each, where nothing looks unusual at all. Same logic, opposite visibility.
And it is not a pointing pair, despite the names. The two collide badly and it's worth being precise, because they do different jobs:
- A pointing pair is one number with two homes, both on the same line inside a box. It clears that number from the rest of the line.
- A naked pair is two squares with the same two numbers. It clears both numbers from the rest of the region.
One number, two squares versus two numbers, two squares. If you can say which you're looking at, you know what you're allowed to cross off.
It scales, too. Three squares sharing three candidates between them is a naked triple; four sharing four is a naked quad. The logic is identical — three squares, three numbers, no room for a fourth — and the clearing works the same way.
There's one detail there that most explanations get wrong, and it's worth having right. The three squares in a triple don't each need all three candidates. Squares reading 1 2, 2 3 and 1 3 are a perfectly good naked triple on 1, 2 and 3: no square has all three, but between them the three squares use nothing else, and that's the only condition that matters. If you go hunting for three identical squares you'll miss most of the triples on the board.
When to reach for it
After the singles have stopped paying, and only once your candidates are written down.
That second part isn't a style note. A naked pair is defined by what two squares are carrying, so a board without pencil marks contains no findable naked pairs at all. If you take one thing from this piece, take that: this is the technique that pays you back for the bookkeeping.
The search is cheap once you're set up. You're looking for squares with exactly two candidates — no other size matters for a pair — and then checking whether any two of them in the same row, column or box match. Two-candidate squares are rare enough to scan for and common enough to find, which is why this is usually the fourth technique people learn rather than the tenth.
One caution that saves wasted effort, the same one that applies to pointing pairs. A naked pair only pays if the rest of the region actually wanted those numbers. Find a genuine pair whose region has nothing else carrying a 1 or a 5 and you've found something real that buys you nothing. That's a dry one. Note it and move on.
And worth being straight about, since the whole series has run on this one board: this puzzle doesn't require any of this. It's an easy grid and singles alone will finish it. The pair above is completely genuine and it really does hand you that 4 — but you're meeting the technique on a board that won't punish you for missing it. On a Hard puzzle it looks exactly the same and the board stops moving without it.
Now find one yourself
No new board, and no new pair. The pair you've already found does its work twice.
Look again at where those two squares sit. They're both in the second column — that's the region you just cleared. But they're also both inside the same 3×3 box, the one on the left-hand edge, level with the middle of the grid — rows four to six, columns one to three.
A region is a region. The same argument applies there, unchanged: those two squares take the 1 and the 5 between them, so nothing else in that box can have either.
Write the candidates into that box's empty squares and find out what it clears.
There are three empty squares in that box besides the pair itself. Work out which of them were carrying a 1 or a 5, take those away, and see what's left.
The answer — open when you've had a go
Rows count from the top, columns from the left. The pair is at row 4 column 2 and row 6 column 2, and it clears the box's third column.
| Square | Was carrying | After the pair clears |
|---|---|---|
| row 4, column 3 | 1, 5, 9 | 9 — a naked single |
| row 5, column 3 | 6, 9 | 6, 9 — untouched, it had neither |
| row 6, column 3 | 1, 3, 5, 9 | 3, 9 |
So the same pair produced a second single. Row 4 column 3 had three candidates, lost the 1 and the 5, and came down to a lone 9 you can write straight in.
That's two numbers placed — the 4 at the bottom of column 2, and this 9 — from one pair of squares that never got solved themselves. Both squares of the pair are still sitting there reading 1 and 5, exactly as they were.
Why it fired twice. Nothing clever: the two squares happen to share both a column and a box, so both regions are subject to the same argument. Any naked pair sitting in one line and one box does this. It's worth checking for every time, and it's free — you've already done the hard part by finding the pair.
Where you are now
Four techniques, and they stack:
- Naked single — this square has one number left. Places a number.
- Hidden single — this number has one square left in a region. Places a number.
- Pointing pair — this number's squares in a box share a line, so it clears from the rest of that line. Clears one number.
- Naked pair — these two squares hold the same two numbers, so both clear from the rest of the region. Clears two numbers, and often hands you a single.
The pattern in that list is the thing to carry forward. Each technique finds something that has run out of room — a square, a number, a line, now a pair of squares — and turns it into eliminations somewhere else. The names get longer further up the ladder. The move doesn't change.
Write your candidates down. Run both singles until they stall. Then scan for squares with exactly two marks, and check whether any two of them match.
Earlier in this series: naked singles, hidden singles and pointing pairs. Next: hidden pairs, which is this one turned inside out. The whole ladder is laid out in Sudoku techniques, in the order worth learning them.