Unique Rectangles and the Deadly Pattern

The deadly pattern

Picture four cells that form a rectangle, sitting in two rows, two columns, and spanning exactly two boxes. Now imagine all four of them hold only the same two candidates, say three and seven. If the finished puzzle really ended that way, you could place three, seven, seven, three around the rectangle, or swap to seven, three, three, seven, and both arrangements would satisfy every row, column and box rule equally well. That means the puzzle would have at least two valid solutions, which a proper Sudoku never does. Because such a configuration would be fatal to the promise of a single answer, it is known as the deadly pattern, and recognising it is the whole foundation of the technique. What makes the Unique Rectangle unusual among solving methods is that it does not reason about the numbers on their own; it reasons about the puzzle guarantee of uniqueness. You are effectively using the setter promise, that exactly one solution exists, as an extra rule of the game, and that borrowed rule turns out to be surprisingly powerful.

Turning the pattern into a move

Since the deadly pattern cannot survive in a puzzle that has a single solution, something on the board must break it, and that simple fact is what licenses you to remove candidates. The friendliest case to learn is the Type 1 Unique Rectangle. Here three of the four corners hold just the bare pair, three and seven, while the fourth corner holds that same pair plus one or more extra candidates. Those extra candidates are the only thing standing between the puzzle and a fatal second solution, so they cannot be a coincidence: the pair itself must be false in that fourth cell, because if it were not, the deadly four-corner pattern would form and uniqueness would collapse. So you erase both pair values from the corner that carries the extras, leaving only the extras behind. In one stroke you have used the guarantee of a unique solution to prune two candidates that ordinary logic would have left sitting there untouched.

A worked example

Suppose r2c1, r2c2 and r5c1 all hold exactly the pair three and seven, and r5c2 holds three, seven and also five. Check the geometry first: these four cells sit in rows two and five and in columns one and two, and they cover just two boxes, box one and box four. That two-box span is essential, and here it holds. Now apply the logic. If r5c2 were three or seven, all four corners would be the bare pair, the deadly pattern would form, and the board would have two solutions. Since that is forbidden in a proper puzzle, r5c2 cannot be three and cannot be seven; it must be the extra candidate. You therefore remove three and seven from r5c2, which immediately solves it as five. Notice how the extra value, five, is what rescues the puzzle from ambiguity, and how the deletion often collapses the cell to a single answer, which is why the Unique Rectangle feels especially decisive when it appears.

How to spot one

1. Look for two rows and two columns whose four intersection cells all involve the same two candidates.
2. Confirm the four cells span only two boxes, never three or four, because the argument depends on it.
3. If three corners are exactly the bare pair and one corner carries extra candidates, you have a Type 1.
4. Delete both pair values from the corner that holds the extras, and take any resulting single as solved.

Other types, and using it with care

That two-box requirement is not a detail you can safely skip. If the four cells crossed three or four boxes instead, the swap that would create a second solution would also disturb a box constraint, so the deadly logic would no longer hold and the elimination would be unsound. Always check the boxes before you trust the pattern; in practice it means the two columns must share a box-stack, or the two rows must share a box-band, so that each side of the rectangle lives inside a single box. Type 1 is only the beginning, too. The deadly pattern spawns a whole family: in a Type 2, for instance, two corners share the same single extra candidate, and that extra can be eliminated from any cell seeing both, while other types handle extras split across corners or combine the rectangle with a conjugate pair. They all spring from the same root idea, that the bare four-corner pattern must never be allowed to form. One final caution: Unique Rectangles only work when the puzzle is genuinely guaranteed to have one solution. Every board in Shapedoku is validated for exactly that before it is served, so the logic is completely safe here, from Easy up to a fearsome Extreme; but on a puzzle from an untrusted source that secretly has two answers, a uniqueness trick can misfire, so reserve it for boards you trust.