Last free cell
The last free cell is the first technique anyone learns, usually without being taught it. A row, column or box has one square left; the digit it is missing goes there. It needs no notation and no scanning, only the ability to notice that eight of nine squares are full.
The idea
Every row, every column and every box holds each of the digits 1 to 9 exactly once. That is the entire rule of Sudoku, and it has an immediate consequence: if one of those units has eight digits in it, the ninth square can only be the digit that is absent.
There is no reasoning to do beyond reading. You are not weighing possibilities against each other, as every later technique asks you to — you are naming the one digit from 1 to 9 that is not on display.
A worked example
The row shows 4, 8, 3, 9, 1, 7, 5 and 6. Run through the digits in order and 2 is the one that never appears, so 2 belongs in r5c5. Nothing outside the row was consulted, and nothing needed to be.
The same argument works on a column with one gap, and on a box with one gap. The unit shape is irrelevant — all three carry the same promise about the nine digits.
When it fires
Late, and then often. A unit needs to be eight-ninths solved before this applies, which almost never happens at the start of a puzzle. Its real use is the endgame: once placements begin to cascade, each one tends to leave another unit with a single gap, and a grid can close out in a run of these without any other technique being needed.
That is also why it is not what you open with. Cross-hatching is the technique for the start of a grid, because it asks a question you can answer while the board is still mostly empty.
Common mistakes
The one real trap is miscounting — glancing at a unit, seeing it looks nearly full, and placing a digit that is already there somewhere you did not look. Eight squares is more than it is comfortable to hold at a glance, so it is worth running 1 through 9 against the unit rather than trusting the impression that something is missing.
It is also easy to confuse this with a naked single, which sounds similar and is not the same. The last free cell is a statement about a unit that is nearly complete. A naked single is a statement about one square whose row, column and box have between them excluded eight digits — and that square can be sitting in a row with six gaps left.
Common questions
What is the last free cell technique in Sudoku?
When a row, column or box has exactly one empty square, that square's digit is whichever of the nine the unit does not already contain. There is nothing to work out beyond reading the eight digits that are there and naming the one that is missing.
Is the last free cell the same as a naked single?
No, although they often agree. The last free cell is a fact about a unit: it has one gap, so the gap takes the missing digit. A naked single is a fact about a square: its row, column and box between them have ruled out eight digits. A square can be a naked single while its row still has six empty squares, which the last free cell could never tell you.
Why does this technique stop being useful?
Because it needs a unit that is already eight-ninths solved, and that is a late-stage condition. It clears up the tail end of a puzzle and almost never fires at the start, which is why cross-hatching rather than this is the technique you open with.
Where to go next
The next technique in the order is cross-hatching — Sweep one digit across the grid and let the rows and columns you already have rule out squares for you.
All nine techniques: the techniques index. New to the rules? Start here. Ready to use it? Open a board.