Unique Rectangle: Using the "One Solution" Rule

Every technique so far has reasoned purely from candidate placement — where a digit can and can't go. The Unique Rectangle is different: it leans on the fact that a real sudoku has exactly one solution, and uses that to rule out configurations that would create a second one.

What is a Unique Rectangle?

Four cells at the corners of a rectangle spanning two boxes, all holding only the same two digits, could be filled two different ways by swapping them — but a valid puzzle has exactly one solution, so that situation can't exist. This shows up in a few patterns:

• Type 1: three corners are just those two digits. The fourth corner can't be either of them.
• Type 2: two corners are just those two digits, and the other two share one extra candidate between them. That extra candidate must be real in one of them.
• Type 4: two corners are just those two digits, and a shared row or column locks one of the two digits to the other two corners. The other digit can be eliminated from both.

A worked example (Type 1)

Here's the exact board the app uses for its own Unique Rectangle lesson:

Singles clear everything except one corner of the bottom band:

That's where the rectangle shows up. Three of its four corners hold only {2, 5} — the fourth, R9C3, holds {2, 5, 9}, the odd one out:

Blue & violet: the cells the logic runs on · Rose: where the elimination lands

If R9C3 also held only {2, 5}, the four corners could swap 2s and 5s between rows 7 and 9 and still satisfy every row, column, and box — a second solution, which can't happen. So R9C3 can't be 2 or 5. Erase both, and it's left with a single candidate: R9C3 is a 9.

How to spot one

1. Find four cells forming a rectangle across exactly two boxes where every corner contains the same two candidates.
2. Check whether three corners contain only those two — if so, the fourth loses both.
3. Otherwise, look at the two corners with extra candidates and check for a shared extra digit (Type 2) or a shared unit that locks one digit (Type 4).

Rarer configurations of the same idea (types 3, 5, and 6) exist, but the three above cover the large majority of Unique Rectangles you'll actually run into.