The Uniqueness Argument
Guide4 / 4Why 'this puzzle has one solution' is itself a solving weapon.
Every proper sudoku has exactly one solution: publishers guarantee it, and our generator verifies it by exhaustive search. Uniqueness techniques turn that guarantee around: any candidate arrangement that would allow two solutions must be false.
The deadly pattern
Picture four cells on the corners of a rectangle, lying in exactly two boxes, each holding the same two candidates {a,b}. If that ever became the final state, the solved grid could swap a and b around the rectangle and stay perfectly valid: two solutions from one puzzle.
The two-box condition is essential: spread over four boxes, the swap would break box constraints, and no paradox arises. In two boxes, the swap is invisible to every row, column and box involved, which is what makes the pattern deadly.
From threat to technique
Since the puzzle is unique, the deadly pattern can never complete. Whatever would finish it is therefore impossible, and that single sentence spawns a whole family of techniques: the five Unique Rectangle types deny the last free corner, exploit shared extra candidates, or lock a pair digit onto the roof; BUG+1 applies the same idea to the whole board at once.
These are strange techniques the first time you meet them: they never argue about what digits can go somewhere, only about what the puzzle as a whole is allowed to be. The eliminations are just as binding.
Is that even fair?
Purists note that uniqueness techniques assume the puzzle-maker did their job: on a broken puzzle with two solutions, a Unique Rectangle deduction could 'eliminate' a digit that one of the solutions actually uses. That's true, and irrelevant in practice: every puzzle on this site is generated with a machine-verified unique solution, so the assumption is a fact.
If you'd rather not rely on it, every campaign puzzle stays solvable: uniqueness techniques give you shortcuts, not lifelines. Learn them in the Unique Rectangle and BUG+1 lessons, and enjoy one of sudoku's most elegant ideas.