Every rule in sudoku comes from three constraints: each row, column, and box holds the digits 1-9 exactly once. But a genuine puzzle carries a fourth, unwritten rule that solvers quietly rely on without always naming it: it has to have exactly one solution. This page is about that guarantee, and how a handful of advanced techniques turn it into an actual move.

What the guarantee rules out

If a puzzle could be finished two different ways, that would mean somewhere on the board, a group of cells could swap values between the two solutions without breaking any row, column, or box constraint. The classic shape this takes is a rectangle: four cells at the corners of two rows and two columns, each restricted to the same two candidates, arranged so that swapping all four between the two digits still satisfies every row, column, and box.

A properly constructed puzzle can never contain that shape fully formed, because if it did, there’d be no way to tell which of the two swapped arrangements was “the” solution — and we require exactly one. Our own generator checks this directly rather than hoping for it: every time a clue is removed while building a puzzle, the backtracking solver counts solutions and the removal is rejected the instant a second one appears. See how our puzzles are generated for the actual mechanics.

Turning the guarantee into a move

Advanced solvers use that fact in reverse. If you can see that a certain cell would complete a deadly rectangle — one where swapping two digits between two rows and two columns would still be valid — you can rule out the candidate that would create it, on the grounds that a genuine puzzle can’t have more than one answer. It’s a different kind of reasoning from everything else on this site: instead of deducing what a cell must be from the rules directly, you’re deducing what it can’t be from the fact that you’re holding a real, uniquely-solvable puzzle.

A concrete puzzle, one answer

7
8
4
8
6
1
2
4
2
7
1
9
3
7
1
2
4
9
1
5
9
3
2
7
8
5
4
1

This puzzle — generated the same way as every puzzle on this site — has exactly one way to fill in the rest of the grid:

6
5
1
2
9
7
8
3
4
7
9
3
8
4
6
1
2
5
4
2
8
5
3
1
9
6
7
2
1
9
3
7
4
5
8
6
3
4
6
9
8
5
7
1
2
8
7
5
6
1
2
4
9
3
1
8
7
4
6
3
2
5
9
9
3
2
7
5
8
6
4
1
5
6
4
1
2
9
3
7
8

Every clue that could be removed from a puzzle like this without creating a second valid completion already has been; that’s what “generated down to a unique solution” means in practice, not just in theory.

This isn’t something you’ll reach for as often as X-Wing or XY-Wing, but it’s worth knowing exists, especially on puzzles graded Evil where every other technique has already been tried.