Every empty cell sits in a row, a column and a box. When those three houses have used eight of the nine digits between them, one digit is left over and the cell must take it.
How to spot it
Pick an empty cell and read the three houses it belongs to: its row, its column, and the box it sits in. Every digit already written in any of them is a digit this cell cannot hold. If the three readings between them account for eight of the nine digits, the ninth is the only one left and the cell is settled.
The work is a scan, not a comparison. You are not weighing this cell against its neighbours and you do not need a single pencil mark. You look along a row, down a column and around a box, and you keep count of what you have seen.
The cells to try first are where two well-filled lines cross a well-filled box. Eight cells on the grid below can be settled this way, and not one of its twenty-seven houses is down to a single empty square.
Read the row
Row five holds a 1, a 5, a 7 and a 9. Those four are out for every empty cell along it, the marked one included.
Five digits are still possible, so the row on its own settles nothing. Five of its nine cells are still empty, which is a long way from a single gap.
Read the column and the box
Column six holds a 3 and a 6 that the row did not, and box five holds a 2 and a 4 that neither the row nor the column did. Both of them repeat the 1 and the 7 already seen.
Together the three houses have used 1, 2, 3, 4, 5, 6, 7 and 9. That is eight digits, the 8 is the only one unaccounted for, and so the cell takes it.
All three houses were needed. Read only two of them and three digits are still possible, whichever two you pick.
Write it in, then re-read the neighbours
The 8 goes in. One placement adds a digit to three houses at once, so every empty cell in all three has one more digit ruled out.
Two of them were a single digit short. The cell three rows up in the same column had a 2 and an 8 left to it, and the cell one step right along row five had the same pair. Both have lost the 8, so both are now a 2.
No house has come down to one empty cell. Row five, column six and box five each still hold four gaps. This chain runs from cell to cell, not from house to house.
A digit you write in belongs to three houses, so it narrows every empty cell in all three. The ones worth re-reading are those that were down to two digits, and they are usually nearest the cell you have just filled. Both cells settled above sat in a house the 8 had joined.
Why it works
A row, a column and a box each hold the digits 1 to 9 once. Every cell belongs to one of each, so a digit written anywhere in those three houses is a digit that cell can never hold. Count the distinct digits across all three, subtract from nine, and what remains is exactly the set the cell can take. When that set has one member there is nothing further to establish and nothing elsewhere on the grid can overturn it.
This is the conclusion a naked single reaches, and the difference is the instrument rather than the logic. A naked single is read off pencil marks that have already been narrowed to one. Here the narrowing is done by eye, from the board, on a cell that may have nothing pencilled near it. That is why this sits earlier on the path: it needs no preparation.
Asking where a digit goes is a different question, and a hidden single is the technique that asks it. On the cell above it returns no answer, which is worth seeing side by side.
What can this cell be? Its row, its column and its box have used eight digits between them, so one digit is left and the cell takes it. One reading, one answer, and the pencil mark below is the result rather than the input.
Where can the 8 go? Three cells along row five could still take one, and asking the same of column six or box five gives three cells each time. Three houses, three times no answer, and the cell was decided all the same.
The cell marked below sits in row four and column seven. Which digit can it take?
Read its row, then its column, then its box, and keep a list of the digits you have already seen.
Recap
- Look for
- An empty cell whose row, column and box have used eight different digits between them.
- Scope
- Any cell, on a bare grid or a pencilled one. Nothing earlier is needed.
- Result
- One digit is placed. Nothing is eliminated directly, though every empty cell in those three houses loses a candidate.
- Tell-tale
- Two well-filled lines crossing a well-filled box. The grid above held eight such cells and not one house with a single gap.
- If it fails
- If no cell comes down to one digit, start pencilling. Marking the candidates turns the same scan into a naked single hunt, and the pairs and triples that follow need those marks anyway.