You've been finding squares that ran out of options. Then the board goes quiet. You check square after square and every one of them still has three or four numbers standing, and the puzzle sits there looking finished with itself.
Nothing is wrong. You've just run out of the first technique, and the second one is waiting about ten seconds away.
Here it is, in one line: stop asking about squares and start asking about numbers.
Turn the question around
The first technique picks a square and asks what can go in it. That question runs dry, because most squares genuinely do have several numbers available.
So ask the other question. Pick a region — a row, a column, or a 3×3 box — and pick a number. Then ask: where in this region can that number go?
Every region has to end up holding all nine numbers, exactly once each. So if you can rule out eight of the nine squares, the number goes in the ninth. It doesn't matter how many other numbers that square could also have taken. The region has nowhere else to put this one.
That's the whole idea. Let's watch it work.
Same puzzle as last time, back at its starting position — so if you placed a 5 in the middle last time, this grid hasn't got it yet.
One thing worth saying plainly, because you might notice it and wonder: this particular puzzle never actually forces you to use hidden singles. It's an easy grid, and a patient solver could finish it on naked singles alone. It's here because you already know it, which makes the technique easier to see. The hidden single below is completely genuine — that square has four candidates and the first technique cannot find it — but you're being shown the tool on a familiar board rather than on one that would beat you without it.
Look at the box across the top middle. It already holds 7, 1, 9 and 5, which leaves five empty squares and five numbers still to place.
Now pick one of those numbers. We'll take the 8, and ask the only question that matters: where in this box can an 8 go?
Go square by square, and remember that a number is blocked if its row or its column already has one. Positions below are given inside the box — "the box's top row" means the box's own top row, not the grid's.
- The box's top row, left-hand square. Its column already has an 8, four rows down. Out.
- The box's bottom row, all three squares. That whole row of the grid already has an 8, over on the left. All three are out in one stroke.
- The box's top row, right-hand square. Its row has no 8. Its column has no 8. It survives.
Five squares, four of them eliminated by two numbers that were already on the board. One square left, and the 8 goes there.
Why that square, when the square itself doesn't know
Here's the part worth slowing down for. Write in the candidates for those five squares — what each one could still legally hold — and look at what you get.
Five squares, and the 8 appears in exactly one of them.
Now notice what that square is holding: 2, 4, 6 and 8. Four candidates. If you'd gone hunting for a square that had run out of options — the whole of the first technique — you'd have walked straight past it. It has four options. It isn't close to being finished.
But every other square in the box has already lost its 8. So the box has one place left to put an 8, and that settles it.
The square didn't know it was an 8. The box knew.
The jargon, deflated
This one's called a hidden single. Same deal as last time — the name is worse than the idea, and it's at least descriptive. It's a single, because one number has one home. It's hidden, because the square it lands in is still carrying other candidates, so it doesn't announce itself the way a naked single does.
Naked single: this square has one number left. Hidden single: this number has one square left.
That's the whole difference. Two questions, opposite directions, same crossing-off underneath. Most people find hidden singles harder to spot, and it isn't because the logic is harder — it's because you have to pick a number to hunt for before you start looking, and there are nine of them.
The trick for finding them
Don't sweep all nine numbers through all nine regions. That's eighty-one searches and you'll give up around the fourth.
Look for a number that's already well represented on the board. If there are six 8s placed, every one of them is casting a line across the grid, and those lines do your eliminating for free. In the box above, two 8s did all four eliminations between them.
So: pick the number the board already has a lot of. Find a region where it's missing. Follow its lines and see how many squares they knock out. Numbers with only one or two placed are poor hunting — they don't block enough to corner anything.
Which raises a fair question: how are you supposed to know which number the board has a lot of, short of counting all nine across eighty-one squares? On paper, you count. In our app you don't — every key on the number pad shows how many of that digit are still unplaced, so the number with the smallest count is the one worth hunting, and it's already on screen. That counter exists for exactly this, and it's the difference between the technique taking ten seconds and taking two minutes.
This is also the technique that keeps paying after you place something. Every number you write in is a new line across the grid, which is why a puzzle that felt stuck often produces three or four moves in a row once the first one lands.
Now find one yourself
Same board, with the 8 placed. From here on I'll use numbers for squares as well as words: rows count from the top, columns from the left, so row 5, column 3 is the third square along on the fifth row down.
The box to work on is the one on the left-hand edge, level with the middle of the grid — rows four to six, columns one to three. It holds 8, 4 and 7 down its left-hand column, and six squares are empty.
There are two hidden singles in that box, and either one counts. Pick a number that's well represented elsewhere on the grid, find where its lines fall, and see if the box has only one square left for it.
Expect to try more than one number. The box is missing six and only two of them are hidden singles — try the 9 and you'll find three homes and no answer. That's the search working, not failing: you've ruled out a number, which is progress of exactly the kind this whole series is about.
A hint on where to start, if you want one: both answers land in the same column of the box — the right-hand one. That's not a coincidence. The box's left-hand column is already full, and its middle column is cut off by a line running down column 2, so the right-hand column is where the survivors end up.
Don't check whether the square you land on has other candidates. It will, and that's the point of the technique.
Both answers — open when you've had a go
| Number | Only home | That square's other candidates |
|---|---|---|
| 6 | row 5, column 3 | 2, 5, 9 |
| 3 | row 6, column 3 | 1, 5, 9 |
Why the 6. Three lines do it, and the box has six empty squares to clear.
- Row 4 has a 6, which takes out both empty squares on the box's top row.
- Row 6 has a 6, which takes out both empty squares on the box's bottom row.
- Column 2 has a 6, down at row 7, which takes out the middle column.
That's five squares gone. The survivor is row 5, column 3.
Why the 3. Same shape, different lines.
- Row 4 has a 3 and row 5 has a 3, which between them clear the box's top two rows — four squares.
- Column 2 has a 3, up at the very top of the grid, which clears the middle column.
Five squares gone again. The survivor is row 6, column 3.
Both squares are carrying four candidates each. Neither is a naked single, and neither ever will be until something else on the board moves.
What to do with this
You now have both halves of the same idea, and between them they'll finish almost every Easy puzzle in our app — 205 of the 207 that ship inside it. Medium is a different story: the two singles alone finish 5 of 57. That gap is not you getting worse. It's the point where a puzzle starts asking for a third technique.
When the board is busy, hunt squares — look for one that's cornered by a crowded row, column and box at once. When that dries up, switch questions and hunt numbers: take the digit the board already has most of, and go looking for a region with one place left to put it.
Alternate between them. The two techniques feed each other, because every placement you make tightens a row, a column and a box for both kinds of search.
If a puzzle stops answering to either, it isn't that you've hit a wall. It's that the puzzle has a harder technique in it, and that one has a name too. (It's called a pointing pair, and it's next.)
Missed the first one? Naked singles: the square that can only be one number covers naked singles, which this piece assumes. The rest of the path is laid out in Sudoku techniques, in the order worth learning them.