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.