Interactive Sudoku Technique
Unique Rectangles: One Danger, Six Kinds of Context
Four cells threaten to exchange the same two digits without changing any row, column, or box. The rectangle warns us what cannot be allowed. Its surroundings tell us what can be removed.
The Promise Behind the Technique
Uniqueness Is an Assumption, Not a Sudoku Rule
A standard Sudoku is expected to have exactly one solution. Unique Rectangle techniques use that promise. If a candidate arrangement would leave two interchangeable completions, the arrangement cannot be allowed to finish.
Sudoku Paradise puzzles are checked for a single solution. If you are solving a puzzle from an uncertain source, verify that it is unique before relying on this family.
The rectangle defines the danger. Context proves the elimination.
Clear the Name Once More
An Empty Rectangle and a Unique Rectangle Share Only a Word
Empty Rectangle
Follows one digit through one box and an outside Strong Link. Its proof works without any appeal to uniqueness.
Unique Rectangle
Follows two digits through four corners. Its proof prevents a second valid completion from surviving.
Need the other rectangle? Review the Empty Rectangle lesson.
Interactive Foundation
Why the Basic Rectangle Is Deadly
Consider four unsolved cells at r2c2, r2c5,
r3c2, and r3c5. They occupy exactly two
rows, two columns, and two boxes. If all four are reduced to the
same pair, (2, 7), the digits can trade places around the corners.
Completion A places 2 and 7 once in each affected row, column, and box.
Both completions obey the ordinary Sudoku house rules. A puzzle promising one solution must prevent this four-cell state before it becomes final.
The Geometry Before the Type
First Confirm the Rectangle
- Four unsolved corners sit at the intersections of two rows and two columns.
- Exactly two boxes contain those corners. A rectangle spread over four boxes is not a standard UR.
- The same two rectangle digits remain available around the four corners.
- Additional candidates prevent the deadly pair from becoming the whole story.
Floor
A clean side whose two cells contain only the rectangle pair. It gives the possible UR a firm base.
Roof
The opposite side, usually carrying additional candidates that keep the puzzle from falling into ambiguity.
Heavy Cell
A rectangle corner containing the deadly pair plus one or more extra candidates.
Floor and roof are useful reading aids, not universal laws. Type 5 can place its heavy cells diagonally, and the Hidden Rectangle can bury most of the structure beneath extra candidates.
Preview the Family
What Kind of Context Breaks the Ambiguity?
Select a pattern. The rectangle supplies the same warning each time; the source of the useful conclusion changes.
Context: One corner alone carries the extra candidates.
Result: Remove both rectangle digits from that corner.
Unique Rectangle Type 1
One Heavy Corner Must Escape the Pair
r23c25, the lone heavy corner
r3c5 loses 2 and 7, leaving 9.
Recognize it
Three corners contain only the rectangle pair, (X, Y). The fourth contains X and Y plus one or more extras.
Reason it
If the heavy corner chose X or Y, all four corners could collapse into the interchangeable pair. Therefore that corner must use one of its extra candidates.
Remove
Eliminate X and Y from the heavy corner. When only one extra remains, Naked Singles may place it immediately afterward.
Type 1: one heavy corner must leave the deadly pair.
Unique Rectangle Type 2
The Same Extra Candidate Guards Both Roof Cells
r23c25, the roofs share only the
extra 7, so r1c5 loses 7.
Recognize it
The two non-diagonal roof cells lie in one shared house. Each contains the rectangle pair plus the same single extra, Z.
Reason it
At least one roof cell must become Z. If neither did, the roof would reduce to X and Y and the deadly rectangle would remain.
Remove
Any outside cell that sees both possible homes for Z cannot itself contain Z.
Type 2: the shared extra must live in one roof cell, so common peers lose it.
Unique Rectangle Type 3
The Roof Extras Join a Naked Set
r46c25, the roof extras and
r5c5 lock (6, 8), so r5c4 loses both.
Recognize it
Two non-diagonal roof cells carry several extras rather than one matching escape candidate. Both roof cells share a house with one or more supporting cells.
Reason it
To avoid the deadly rectangle, at least one roof extra must be true. For subset counting, combine both roof cells into one container holding only their extra digits. If that container and the supporting cells lock N digits into N places, they form a Naked Pair, Triple, or Quad within the shared house.
Remove
Remove the locked subset digits from other cells in that house. The victims are outside the subset; the roof cells are part of its proof.
Type 3: count the two roof cells as one home for their extras, then finish the Naked Set.
Unique Rectangle Type 4
One Rectangle Digit Is Strongly Linked Across the Roof
r56c25, 3 is Strongly Linked across
the roof in Column 5, so both roof cells lose 6.
Recognize it
Two non-diagonal roof cells contain extras. In a house shared by those roof cells, one rectangle digit—say X—appears in exactly those two positions.
Reason it
The Strong Link guarantees that one roof cell is X. Allowing the other roof cell to become Y would complete the interchangeable X/Y pattern.
Remove
Eliminate Y, the other rectangle digit, from both roof cells. X remains strongly linked; the extras provide the necessary escape from ambiguity.
Type 4: a Strong Link on one rectangle digit removes the other digit from the roof.
Unique Rectangle Type 5
The Repeated Extra Moves Off the Roof Line
r23c25, diagonal extras at
r2c2 and r3c5 remove 6 from
r3c1.
Recognize it
The same single extra candidate, Z, accompanies the rectangle pair in two diagonal corners—or in three of the four corners.
Reason it
At least one of those Z candidates must be true. Removing Z from every heavy corner would leave nothing to stop the rectangle from becoming the deadly X/Y exchange.
Remove
An outside Z that sees every Z-bearing rectangle corner can be eliminated. With diagonal extras, seeing only one corner is not enough; with three extras, the victim must see all three.
Type 5: one repeated extra must survive somewhere in the rectangle, so a common peer cannot hold it.
Keep the Solver Roles Honest
A Unique Rectangle May Reveal a Placement Without Becoming a Guesser
The logical action of a UR is usually expressed as an elimination. Sometimes that elimination leaves one candidate in a cell. A human may naturally say, “therefore this cell is 1.” Sudoku Paradise can preserve the same reasoning while letting Naked Singles record the resulting placement.
This is especially visible in Type 1: the heavy corner cannot use either rectangle digit, so its only extra may be solved. The UR broke the ambiguity; the Single cleaned up the residue.
Uniqueness tells us what cannot be allowed. Context tells us what can be removed. Singles place what remains.
Interactive Classification
Name the Context, Not Merely the Rectangle
Three corners are exactly (2, 7). The fourth is (2, 7, 9). Which pattern removes 2 and 7 from the heavy corner?
Identify the extra piece of context that creates the elimination.
Example 1 of 6 · 0 mastered
Before You Eliminate
A Rectangle Alone Proves Nothing
- Confirm two rows, two columns, and exactly two boxes.
- Confirm that the same two deadly digits belong to the possible rectangle.
- Identify the exact additional context required by the claimed type.
- Do not remove every extra candidate merely because a rectangle is present.
- For Type 2 and Type 5, verify that the victim sees every possible home of the repeated extra.
- For Type 3, prove a genuine subset in one common house before touching a victim.
- For Type 4 and the Hidden Rectangle, verify the required Strong Links rather than trusting the picture.
- Use uniqueness only on a puzzle known to have one solution.
Common Questions
Unique Rectangles Clarified
Must all four cells contain only the same two candidates?
No. That finished state is the deadly pattern we are preventing. In a usable UR, one or more corners normally contain additional candidates. This lesson begins with the clean visible form in which both rectangle digits are still penciled into all four corners.
Does every rectangle use a floor and a roof?
No. Those names are most helpful when two clean cells face two heavy cells. Type 5 can use diagonal heavy corners, and the Hidden Rectangle may be too cluttered for a tidy roofline.
Is Type 5 the same as the Hidden Rectangle?
No. Type 5 follows one repeated extra candidate placed in two diagonal corners or three corners. A Hidden Rectangle instead uses row-and-column restrictions on one of the rectangle digits to expose an elimination beneath the clutter.
Why do some references number these patterns differently?
Sudoku terminology grew through several communities and solving programs. Names are not perfectly universal. Judge a pattern by its candidates, geometry, proof, and elimination—not by the type number alone.
Continue Practicing
Carry the Rectangle into a Real Puzzle
Scan Bivalue Cells for a repeated pair, confirm the two-box geometry, then ask what prevents the deadly pattern from closing.