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 exactly two boxes (only two — 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}, the configuration would let 3 and 7 swap freely 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. The "deadly pattern" never actually appears in a real puzzle; what appears are configurations one move away from it, where the move that would create the deadly pattern is precisely the one we can rule out. Six standard sub-types catalogue how that ruling-out plays out, depending on where the extra candidates sit and what they let us prove.

Type 1 — three corners locked

The simplest form, and the one most solvers meet first. Three corners of the rectangle have only {a, b} as candidates; the fourth corner has {a, b, X} — the same pair plus one extra X. The fourth cell must be X. Placing a or b would close the deadly rectangle, and the puzzle's uniqueness forbids it. Once you've internalised Type 1, the rest of the type catalogue is variations on "what if the extras are arranged differently."

Type 2 — a locked extra

Two corners on the same edge of the rectangle have only {a, b}. The other two corners — the opposite edge — both have {a, b, c}, with the same extra c in each. At least one of those two extras-corners must be c; otherwise both fall back to {a, b} pairs and the deadly pattern completes. So c can be eliminated from every cell that sees both extras-corners — meaning every cell in their shared row, column, or box. It's the UR's quietest form: the elimination doesn't happen inside the rectangle at all.

Type 3 — naked subset inside the rectangle

Same opening: two corners on one edge are {a, b}, two corners on the other edge carry extras. This time each extras-corner has a different extra — say {a, b, c} and {a, b, d}. Treat the two extras-corners as if they were a virtual cell-pair holding {c, d}, and look for a naked subset in the unit they share with other cells. If you can find cells whose candidates combine with {c, d} into a naked pair, triple, or quad, the standard naked-subset elimination fires on the rest of that unit. The UR's role is to anchor the subset — the two extras-corners must together resolve to {c, d}, which is what makes the subset valid.

Type 4 — strong link on the extras edge

Two corners on one edge are {a, b}; the other two have extras of any kind. Look at the unit shared by the two extras-corners — usually their row or their column. If one of {a, b} — say a — appears in that unit only at those two extras-corners (a strong link on a between exactly those two cells), then a must be placed in one of them. That, in turn, means b can be eliminated from both extras-corners. The deadly-pattern argument is doing the work of a strong-link argument that you might otherwise have made elsewhere; it's specifically forcing b out, because keeping b in either extras-corner would re-open the deadly rectangle.

Type 5 — extras on the diagonal

The two corners on a diagonal — not adjacent ones — share an extra. Two diagonally-opposite cells read {a, b, c}; the other two read {a, b}. By the same logic as Type 2, at least one of those diagonal corners must be c. So c can be eliminated from every cell that sees both diagonal corners. The visible reach is usually narrower than Type 2's — diagonal pairs share fewer common neighbours than line pairs do — but when an eliminating cell exists, the move is decisive.

Type 6 — X-wing inside the rectangle

The most exotic of the six, and the one most often confused with a plain hidden X-wing. Within the rectangle's two rows and its two columns simultaneously, one of {a, b} — say a — appears only at the four rectangle corners. That's a hidden X-wing on a, restricted to the rectangle. The X-wing places a on one of the two diagonals. The deadly-pattern argument rules out the diagonal that would complete the rectangle, forcing a to the other one. The two cells on the deadly diagonal can have a removed entirely.

In practice, Type 6 fires rarely, and even strong solvers tend to spot the simpler types first. When you do find it, it usually appears late in a hard-or-harder puzzle where every more conventional move has already been exhausted.

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 — Sudoku Mountain's generator guarantees it. But the technique sits one rung above the no-repeats logic in conceptual cleanliness, and there's a small minority who refuse to use it on principle. Same family of objection that some solvers raise about BUG and BUG+1, which lean on uniqueness in the same way.

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. Type 1 is the version that shows up most; Types 4 and 6 are the rarest.

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