Uniqueness
Unique Rectangle
A unique rectangle is the one technique that uses a fact about the puzzle rather than about the grid: a proper sudoku has exactly one solution. Four cells forming a rectangle across two rows, two columns and exactly two boxes must not all end up holding the same two digits, because swapping them would produce a second, equally valid solution. So at least one of the four cells has to be something else, and that removes candidates.
- Level
- Advanced
The pattern
What it does.
Line the four cells up: two in one row, two in the same columns in another row, spanning two boxes. If all four could hold only the digits X and Y, the puzzle would have two solutions, which a published sudoku can't.
In the simplest form, one of the four cells carries a third candidate. That cell must not be X or Y, so it becomes a single candidate, or at least loses both.
The technique is the only one on this site that depends on the puzzle being well formed. That's a fair assumption for a published sudoku and worth stating: on a grid with several solutions, the eliminations don't hold.
Example
A worked example.
Find four cells in a rectangle
Look for two rows and two columns where all four crossing cells are unsolved, and where the four span exactly two boxes rather than four.
Check they share two digits
Each of the four cells must be able to take both X and Y. If one of them can't take both, there's no rectangle to begin with.
Look for the extra candidate
In the basic form, exactly one of the four cells holds a third candidate Z and the other three hold only X and Y. That cell is the one the rectangle acts on.
Remove the two digits from it
Delete X and Y from that cell, leaving Z. If it becomes a single candidate, place it; the removal is certain because the alternative would make the puzzle ambiguous.
Spotting it
How to spot it.
- Scan for pairs of cells holding the same two candidates in two rows, exactly as you would for a naked pair, but look for the second row rather than the rest of the unit.
- Check the boxes: the four cells must cover two boxes, not four. A rectangle spread across four boxes proves nothing.
- Look for a cell among the four with more than two candidates. The extra candidates are the ones the rectangle deletes, which is why the pattern is worth hunting in grids thick with marks.
- Remember the orientation: two rows and two columns, or the same shape turned ninety degrees. Both are the same pattern.
Where it stops working
The eliminations rest on the puzzle having one solution, so they are only as good as the puzzle. It also removes candidates from the four cells rather than from the surrounding units, and it won't fire at all unless a rectangle of two-candidate cells exists. Where a puzzle is known to be ambiguous, don't use it.
Keep going
Related topics.
New to sudoku? The rules take two minutes to read, and the first-puzzle walkthrough covers the basics that come before any of these.
Beyond the lessons.
The game's lessons stop at X-Wing, so there's no Unique Rectangle lesson. The eight it does teach will get you through almost any puzzle; this page is here for when you run into the pattern anyway.