Unique rectangle

The unique rectangle is the only basic technique that leans on a meta-property of Sudoku rather than the no-repeats rule directly: well-formed Sudoku puzzles have exactly one solution. If a candidate configuration would produce two solutions, that configuration cannot be allowed to complete — the puzzle's uniqueness rules out the deadlock.

The deadly pattern

Four cells sitting at the corners of a rectangle, spanning two rows, two columns, and two boxes (only two boxes, not four — the rectangle must be confined to a 2×2 box-pair). All four cells have the same two candidates, say the pair {3, 7}. If three of those cells were "naked pairs" with exactly {3, 7} and the fourth was also {3, 7}, you'd be looking at a configuration where 3 and 7 could be swapped between the rectangle's diagonals — two valid solutions. Sudoku doesn't allow that, so the configuration is forbidden.

The technique exploits the deadlock in advance. If three cells of the rectangle have only {3, 7} as candidates, and the fourth cell has {3, 7, X} for some additional candidate X, then the fourth cell must be X — because if it were 3 or 7, the deadly rectangle would close.

Why some solvers find it controversial

The unique rectangle is the technique that "uses the puzzle's uniqueness against itself." Some purists object: in principle a Sudoku could be ill-formed (multi-solution) by accident, and the unique-rectangle move would give a wrong answer. In practice every published Sudoku has been checked for uniqueness, so the move is safe — and Sudoku Mountain's generator guarantees uniqueness. But the technique sits one rung above the no-repeats logic in conceptual cleanliness.

When you'll see it

Unique rectangles appear on expert and master puzzles where the standard techniques have already fired. They're not common — most stuck states get unblocked by simpler moves first — but when they do appear, they often produce a placement that a Y-wing or swordfish couldn't have reached.

For a survey of when to reach for the more exotic intermediate techniques, see Which technique is this puzzle asking for.