A hidden pair is two numbers whose only remaining homes in a row, column or box are the same two squares. Those two squares have to take those two numbers between them, so every other candidate pencilled into either of them can be rubbed out.
That's the definition. Here's the part that matters more, and that almost nobody leads with: a hidden pair doesn't look like anything.
A naked pair announces itself — two squares sitting there with two pencil marks each, the same two, and you can spot it across the room. A hidden pair is buried inside squares carrying four, five, six candidates. Nothing about them is unusual. Nothing draws your eye. The pattern isn't in the squares at all. It's in where two numbers can't go, and that isn't written down anywhere on your grid.
So this is the technique people skip, and it isn't because the logic is hard. It's the same logic as a hidden single, which you already own, run at count two. It's because there's nothing to notice.
Two numbers with nowhere else to go
Every region has to end up holding all nine numbers, exactly once each. That's the only fact this technique uses.
Pick a region and take two of the numbers it's still missing. Work out where each of them could go. If both come down to the same two squares, those two squares are spoken for: one takes the first number, the other takes the second, and it doesn't matter which way round. There's no room left for a third occupant, because there is no third square.
And if both squares are spoken for, everything else pencilled into them is dead. Whatever was in there — a 3, a 7, a 4 — has nowhere to sit.
Notice what you did and didn't learn. You still don't know which square takes the 1 and which takes the 9. You've placed nothing. What you've done is empty two squares of everything else, and that is very often worth more.
Let's do one.
That's the seventh row down, with its candidates filled in. Six empty squares, and nothing about them looks special. The third one along is carrying six candidates. If you were hunting for squares that had run out of options you would walk the length of this row and find nothing.
So don't hunt squares. Take a number the row is missing, and ask where it can go.
Start with the 1. Rows count from the top, columns from the left.
- Columns 4, 5 and 6. All three sit in the bottom-middle box, and that box already holds a 1 — row 8, column 5. All three are out in one stroke.
- Column 9. Its column already has a 1, up at row 5. Out.
- Columns 1 and 3. Neither their columns nor their box has a 1. Both survive.
Two homes for the 1, and no way to choose between them. On its own that's a dead end — this is where most people stop.
Now do the 9, and do it before you move on, because one number with two homes tells you nothing and two numbers with the same two homes tells you everything.
- Columns 4, 5 and 6. Same box, and it holds a 9 as well — row 8, column 6. All three out again.
- Column 9. Its column has a 9, down at the bottom. Out.
- Columns 1 and 3. Both survive.
The same two squares. The same four eliminations did it, because the box and the column that blocked the 1 happened to be carrying a 9 too.
That's the hidden pair. The 1 and the 9 have to go in columns 1 and 3, one each.
What it buys you
Look again at what those two squares were holding. The one in column 1 was carrying 1, 3 and 9. The one in column 3 was carrying 1, 3, 4, 5, 7 and 9.
Both of them are now spoken for — one takes the 1, the other takes the 9. So the 3 in the first square is dead. And in the second square, the 3, the 4, the 5 and the 7 are all dead.
Five candidates gone, and one square went from six down to two.
Nothing was placed. If you've read the earlier pieces in this series you'll recognise the feeling by now, and know it isn't a disappointment.
You didn't find out what either square is. You found out what neither of them can be, and that emptied four squares you weren't even looking at.
Because look what those two squares have just become. They both read 1 and 9 and nothing else, they're both in the same box — the one in the bottom-left corner — and that is a naked pair. The hidden pair made one.
The 1 and the 9 are used up inside the top row of that box, so every other square in the box loses them. Three squares on the bottom row were each carrying a 1. All three just lost it.
One hidden pair, five candidates cleared directly, three more cleared by the naked pair it created. That's the technique doing its job, and it's why the two techniques are worth learning together.
The jargon, deflated
Hidden pair. The name is doing its job for once: a pair, hidden. It's a pair because two numbers own two squares. It's hidden because the squares don't show it — they're carrying other candidates, so nothing about them looks like a pair until you've asked the question about the numbers.
Now the distinction that costs people the most time, because the names collide and the techniques run in opposite directions:
- A naked pair is two squares holding the same two candidates and nothing else. It clears those two numbers from every other square in the region.
- A hidden pair is two numbers whose only homes are the same two squares. It clears every other number from those two squares.
Read those twice. One works outward and one works inward. A naked pair strips the pair's digits from the rest of the region; a hidden pair strips the rest of the digits from the pair's squares. Same shape, opposite direction.
There's a practical test that gets you there most of the time. Count the candidates in the two squares. If they hold only the pair, you can already see it and there's nothing hidden to find. If they hold the pair plus other things, those other things are your payoff. It's a test for whether there's anything to gain, which is usually the question you actually have.
It scales the way you'd expect. Three numbers whose only homes in a region are the same three squares is a hidden triple, and it clears everything else out of those three. Same caveat as the naked version, and it's the bit most guides get wrong: each of the three numbers doesn't have to be available in all three squares. Two of the three is enough, as long as none of the three numbers can go anywhere else in the region.
When to reach for it
Not first. Naked and hidden singles both place numbers, and a technique that places a number is always worth trying before one that doesn't.
Reach for this one when a region looks crowded — when you've written candidates into it, nothing has one option, no number has one home, and several squares are carrying four or five marks each. That crowding is the signal. A hidden pair can only exist where squares are holding more than they need to.
The search itself is cheap if you do it the right way round. Don't read the squares. Take the region's missing numbers one at a time and mark where each could go. You're looking for two numbers with an identical, two-square answer. Numbers with three or four homes are noise; numbers with one home are hidden singles and you should place them and start again.
Worth being straight about, since this whole series has run on one board: this puzzle doesn't need any of it. It's an easy grid and singles alone will finish it — you can see two of them sitting in that seventh row right now, a lone 7 and a lone 4, and if you were actually solving rather than reading you should place them. The hidden pair above is completely genuine. But you're seeing the technique on a board that won't punish you for missing it.
And a second thing worth being straight about, because we went and counted. Hidden pairs are rare. Of the 300 puzzles bundled inside our app, exactly one has a hidden pair anywhere in its solve path, and it's an Expert. You could play this app for a long time and never need this technique.
So why learn it? Because it's the other half of naked pairs, and understanding the pair of them together is what makes either one stick — the contrast above is the actual lesson. And because rare isn't the same as never: the day a board does hinge on one, nothing else on this ladder will move it.
Now find one yourself
Same grid, back at its starting position.
The box to work on is the one in the middle of the grid — rows four to six, columns four to six. It holds a 6, an 8, a 3 and a 2, and five squares are empty.
Write the candidates into those five squares first — you cannot do this one in your head, and pretending otherwise is how people conclude the technique doesn't work.
Then take the numbers that box is missing, one at a time, and find every square each could occupy. Two of them share the same two squares.
A warning that will save you a wrong answer: one of the five squares in that box has only one candidate, and two more are already down to two. None of them is what you're looking for. The first is a naked single and you could simply place it. The other two are squares that happen to be small, which is not the same thing as a pair — and they don't match each other, so they're not a naked pair either. Ignore all three and keep asking the question about numbers.
The answer — open when you've had a go
Rows count from the top, columns from the left.
| Number | Its only homes in the box |
|---|---|
| 1 | row 4 column 6, row 6 column 6 |
| 4 | row 4 column 6, row 6 column 6 |
Two numbers, the same two squares, and the identical second column is the whole answer. That's the hidden pair.
What it clears. Row 4, column 6 was carrying 1, 4 and 7. The 7 is dead — that square has to take the 1 or the 4. Row 6, column 6 was already down to 1 and 4, so it loses nothing.
Why the 1 is confined. The box has three empty squares outside its right-hand column: two in column 4 and one in the centre. Column 4 already holds a 1, up at row 2, which rules it out of both of those. The centre square is blocked by the 1 sitting in row 5 over at column 9 — and it's a naked single anyway. What survives is the right-hand column, rows 4 and 6.
Why the 4 is confined. Same three squares to eliminate. Column 4 holds a 4 down at row 8, which rules it out of both squares in that column. The centre square is blocked by the 4 at the far left of row 5. Same two survivors.
Yes, this one is thinner than the worked example, and it's worth seeing why. Only one candidate got cleared, because the second square was already down to the pair. A hidden pair pays out in proportion to how much junk the two squares were carrying — find one where both squares are holding five things and it clears eight candidates at once. Find one where they're already small and it barely moves the board.
That's not a failed hidden pair. It's a dry one, and knowing the difference before you spend two minutes on it is most of what getting quick means.
Where you are now
Five 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.
- Naked pair — two squares hold the same two numbers, so those numbers clear from the rest of the region. Clears, outward.
- Hidden pair — two numbers hold the same two squares, so everything else clears out of those squares. Clears, inward.
The bottom two are the same idea seen from opposite ends, and the pair of them is the real step up from singles. Everything above this point on the ladder — starting with X-wing — is a longer-range version of the same move: find something that has run out of room, and take the possibilities away from wherever it isn't.
Run singles until they stall. Write your candidates down. Then read a crowded region twice: once looking at squares, once looking at numbers. The second reading is the one most people never take.
Earlier in this series: naked singles, hidden singles, pointing pairs and naked pairs. The whole ladder is laid out in Sudoku techniques, in the order worth learning them.