The technique that assumes the puzzle is fair

The short version

A unique rectangle is four cells at the corners of a rectangle — two rows, two columns, spanning exactly two boxes. Three of them hold the same two candidates, {X, Y}, and nothing else. The fourth holds {X, Y} plus at least one more candidate.

If the fourth cell were X or Y, the four cells could be filled two different ways with no rule broken. The puzzle would have two solutions. Standard sudoku has one. So the fourth cell is neither X nor Y.

The shape

• Four cells: R1C1, R1C2, R2C1, R2C2 — a rectangle of two rows and two columns.
• Exactly two boxes spanned. Four cells in two rows always span two or three boxes; this pattern needs exactly two.
• Three cells: {X, Y} exactly.
• Fourth cell: {X, Y, ...} — contains both, plus something else.
• Eliminate X and Y from the fourth cell.

Which cell is the "extra" one does not matter. Any corner can be the odd one out.

Why it works

This is the only technique on this site that does not follow from the rules of sudoku alone. It follows from the rules plus the assumption that the puzzle is well-formed.

Here is the argument. Suppose the fourth cell takes X. Then all four corners hold only X and Y, arranged so that each row and each column of the rectangle has one X and one Y. That arrangement satisfies every constraint — and the other arrangement, with X and Y swapped, satisfies them just as well.

Two valid solutions. Published sudoku has one. So the assumption is violated, which means the fourth cell is not X.

The same argument with Y removes the other candidate. What is left is the cell's true value, and it is usually a single candidate.

This is worth sitting with, because it is a different kind of reasoning from everything else here. Every earlier technique said: the grid forces this. This one says: the puzzle would be unfair otherwise. Both are valid; only one is about the grid.

A real board

The four corners are R8C2, R8C8, R9C2 and R9C8 — two rows, two columns, spanning two boxes.

• R8C2 holds {3, 6}.
• R8C8 holds {3, 6}.
• R9C8 holds {3, 6}.
• R9C2 holds {1, 3, 4, 5, 6} — the odd one out.

Three corners are exactly {3, 6}. If R9C2 were 3 or 6, all four corners would be {3, 6}, and the rectangle could be filled two ways: 3-6 on the top row and 6-3 on the bottom, or the reverse. Both arrangements satisfy every row, column and box.

That would mean two solutions. It does not, so R9C2 loses both 3 and 6, leaving {1, 4, 5}.

Note that this does not solve the cell. It removes two candidates, which is a real gain but not a placement — and on a board where nothing else is working, it is often the move that breaks the stall.

How to scan for one

Unique rectangles are found by looking for rectangles, which is a different kind of scan from everything else on this site.

1. Find candidate pairs. Every {X, Y} cell is a potential corner. Two cells in the same row or column with identical {X, Y} lists are the start of a rectangle.
2. Close the rectangle. Look for a matching pair on the parallel line. Two rows each with {X, Y} cells in the same two columns is the shape.
3. Check the box count. The four cells must span exactly two boxes. If a corner sits in a third box, the pattern does not apply.
4. Find the odd corner. Count how many of the four have exactly {X, Y}. Three is the standard form. If all four do, the puzzle is broken — that is a two-solution puzzle, not a technique.
5. Delete both digits from the odd corner. Not one, both. The argument rules out each separately.

There are many variants with extra candidates distributed differently across the corners — Type 2, Type 3, and so on. They all rest on the same two-solution argument. The three-corners-pure form is by far the most common, and it is where to start.

The traps

Rounding up to two boxes. If the rectangle spans three boxes, the swap argument fails and the elimination is invalid. This is the most common mistake, because a rectangle in rows 1 and 2, columns 3 and 7 spans three boxes and looks perfectly innocent.

All four corners pure. If all four cells hold exactly {X, Y}, you have not found a technique — you have found a broken puzzle with two solutions. Stop and check for a transcription error.

Applying it to hand-made puzzles. Puzzle books and apps that generate puzzles sometimes produce non-unique grids. On those, the unique rectangle is unsound. This is a genuine limitation, not a theoretical quibble.

Confusing it with an X-Wing. They share the word "rectangle" and nothing else. An X-Wing is one digit confined to four cells; a unique rectangle is two digits across four cells with a uniqueness assumption.

Where to use it

Unique rectangles appear in hard and expert puzzles, and they are unusually high-value because the elimination is often two candidates from one cell — enough to turn it into a single.

Use them after the rule-based techniques have failed. Not because they are weaker, but because they depend on an assumption, and it is better to have exhausted the techniques that do not. If a unique rectangle and a skyscraper are both available, take the skyscraper first: it needs no faith.