A digit with only one place left in a box, row or column goes there, even if the cell it lands in still shows two other candidates.
How to spot it
Most people hunt for a cell. They read down the grid looking for a square that seems nearly decided, and there are eighty-one of those to check.
Hunt for a digit instead. Pick one and ask the same question of each box in turn: where can this digit still go? That is nine questions, and most of them answer themselves, because a box that already contains the digit is not asking anything. You are looking for the box that answers with exactly one cell.
This technique works on a bare grid. The pairs and triples further along the path cannot start until the candidates are written in. This one needs nothing written down: pick a box, then look along the rows and columns that already carry the digit.
Pick the digit before you pick the cell
Choose a digit that is nearly finished. The 4 is already on this board seven times, so only two boxes are still missing one. Every 4 already placed closes a row and a column, so the boxes that are left have few cells still able to take it. The top-left box is one of the two.
Sweep the rows that already own a 4
You do not need to know what the empty cells contain. You only need to know where a 4 already sits. Row two has one and row three has one, and a digit cannot appear twice in a row, so both rows are closed to a 4 along their whole length, including the stretch that crosses this box.
One cell survives, and it still shows three candidates
Three of the box's four empty cells sit in a closed row. The fourth does not, so the 4 belongs there.
Now ask that cell what it thinks it is. Its own pencil marks would read 124, and nothing in them singles out the 4. The claim was never about the cell. It was about the box running out of places, and the box is what answered.
Write it in, then clean up after it
A placement is worth more than an elimination because it closes three houses at once. Before moving on, take the 4 out of the pencil marks of every cell that shares a row, a column or a box with it, or the next technique you try will be reading a grid that no longer exists.
Hidden singles do not stop existing when a puzzle gets hard. They stop being obvious, because by then the grid is covered in pencil marks and the cell that matters is buried under four of them. After any elimination technique fires, check the houses it touched for a hidden single before moving further down the path.
Why it works
A house holds each of the nine digits exactly once, so for every digit the question which cell? always has an answer, and that answer is one of the house's empty cells. Close enough of them and one is left standing. That is the entire argument, and it is why the deduction is safe the moment you see it: it does not depend on how anything else in the grid resolves.
What makes the single hidden is that the cell never tells you. The cell knows only that it could be a 1, a 2 or a 4. The deduction lives one level up, in the house's count of where each digit can still go, and the box is the thing that knows it has run out of places to put a 4. A solver who only ever reads cells will pass over every hidden single on the board without noticing.
Boxes are not the whole board
Almost everyone learns this on boxes and then scans boxes forever. Rows and columns hide singles too, and they hide them better, because a row's nine cells are strung across the grid instead of sitting in a square the eye can take in at once.
There is a second hidden single on the board above, and box scanning cannot see it.
Two of the box's cells can still take an 8. Two is not one, so the box has nothing to say and a box-only scan moves on.
The same 8, read along the row instead. Column one and column six already carry an 8, and between them they close two of row three's three empty cells. The third takes it.
Common mistakes
- Looking for a cell that looks decided. The cell whose turn it is usually shows three or four candidates. Scanning for a lone pencil mark finds a naked single, which is a different technique that happens to place a digit as well.
- Stopping at the placement. Writing the digit in is half the move. The other half is taking it out of the pencil marks of every cell that shares a house with it.
- Scanning without a system. Work the digits in order, 1 through 9, and go back to 1 after every placement. It is dull, and it is what stops you walking past one.
Which digit is a hidden single in this box?
Take the digits the box is missing one at a time, and count how many cells could still hold each of them.
الخلاصة
- ابحث عن
- A digit that has run out of cells in one box, row or column.
- النطاق
- Any box, row or column, with or without pencil marks.
- النتيجة
- The digit is placed. Nothing is eliminated, but the placement closes three houses at once.
- العلامة
- A house down to two or three empty cells, or a digit that is already sitting on the grid six or seven times.
- إذا لم تنجح
- If nothing in any house is down to a single home, the grid needs an elimination before it will give up another digit. The hidden pair is the next tool on the path, and it is this same counting argument run on two digits at once.