Unique Rectangle

A Unique Rectangle exploits the fact that valid Sudoku puzzles have exactly one solution. Certain candidate patterns would create multiple solutions — a deadly pattern. By preventing these patterns, we can make eliminations.

The Deadly Pattern

Imagine four cells at the corners of a rectangle:

This creates a deadly pattern because you can swap A and B between the diagonals and still have a valid solution:

puzzle: S9B9E9E06030204080A0E08050O0F090A0S0G0U0S0W0A070H0E7O8M1I857P04820C0706087O030H82820F0B0A0D077O0607040A0H030E7O987W840A0G038A8I0H9U9I860H040F8202011N0R08020509071I1Q
mode: static
settings:
  showCandidates: true
  coordinateFormat: none
initial:
  annotations:
    - cells: [R4C1]
      label: "A"
    - cells: [R4C9]
      label: "B"
    - cells: [R6C1]
      label: "B"
    - cells: [R6C9]
      label: "A"

Since valid puzzles have unique solutions, this pattern cannot occur. The different types of Unique Rectangle exploit this in various ways.

Type 1: Single Extra Corner

The simplest case. Three corners have only {A, B}, but the fourth has {A, B, C, ...}.

The Logic

Example

puzzle: S9B9E9E06030204080A0E08050O0F090A0S0G0U0S0W0A070H0E7O8M1I857P04820C0706087O030H82820F0B0A0D077O0607040A0H030E7O987W840A0G038A8I0H9U9I860H040F8202011N0R08020509071I1Q
mode: guided
settings:
  showCandidates: true
  showControls: true
  showDescription: true
  navigation: numbered
  coordinateFormat: rncn
steps:
  - text: >
      Look for a rectangle spanning two boxes. R4C1, R4C9, R6C1, and R6C9 form
      the corners of a potential Unique Rectangle.
    technique: UR1
    hint: subtle
    state:
      selection:
        cells: [R4C1, R4C9, R6C1, R6C9]
  - text: >
      Examine the floor cells. R4C9 and R6C9 each contain only {2, 9} —
      these are the bi-value "floor" cells of our rectangle.
    technique: UR1
    hint: obvious
    state:
      selection:
        cells: [R4C9, R6C9]
      annotations:
        - cells: [R4C9, R6C9]
          label: "Floor"
          style: pattern
  - text: >
      Now look at R4C1 and R6C1 — it has {1, 2, 5, 9}. This are the "roof" cells with
      extra candidates (1 and 5) beyond the deadly pair {2, 9}.
    technique: UR1
    hint: obvious
    state:
      selection:
        cells: [R4C1, R6C1]
      annotations:
        - cells: [R4C9, R6C9]
          label: "Floor"
          style: pattern
        - cells: [R4C1, R6C1]
          label: "Roof"
          style: pattern
  - text: >
      If R4C1 were reduced to just {2, 9}, we'd have four cells each containing
      only {2, 9}. This creates a deadly pattern — swapping 2s and 9s would
      give two valid solutions, which is impossible in a proper Sudoku.
    technique: UR1
    hint: detailed
    state:
      selection:
        cells: [R4C1, R4C9, R6C1, R6C9]
  - text: >
      Therefore, R4C1 cannot be 2 or 9 — it must be one of the extra candidates.
      **Eliminate 2 and 9 from R4C1**, leaving only {1, 5}.
    technique: UR1
    hint: detailed
    state:
      selection:
        cells: [R4C1]

Type 2: Two Corners with Same Extra

Two cells have the same extra candidate. Two corners have {A, B}, two corners have {A, B, C}.

The Pattern

       C4       C8
R2   {A,B}    {A,B,C}
R6   {A,B}    {A,B,C}

The Logic

Key Requirement

The two cells with the extra candidate must share a unit AND that unit must contain other cells with candidate C.

Example

puzzle: S9B0D022R0I2R2R0C0H0F1306130B4N4J070904082E090U062I020E0A070R4Z56BE037N0B0E09164I0162020F2I030B0V1J059MAI7N2I080N7R044602B6050F070608020G0Z0Z0D030I129U2Q1Q7Y8Q0H010B
mode: guided
settings:
  showCandidates: true
  showControls: true
  showDescription: true
  navigation: numbered
  coordinateFormat: rncn
steps:
  - text: >
      Find the rectangle: R1C5, R1C6, R8C5, and R8C6 form a Unique Rectangle
      spanning boxes 2 and 8.
    technique: UR2
    hint: subtle
    state:
      selection:
        cells: [R1C5, R1C6, R8C5, R8C6]
  - text: >
      The floor cells R8C5 and R8C6 each contain only {1, 5} — these are the
      deadly pair candidates.
    technique: UR2
    hint: obvious
    state:
      selection:
        cells: [R8C5, R8C6]
      annotations:
        - cells: [R8C5, R8C6]
          label: "Floor"
          style: pattern
  - text: >
      The roof cells R1C5 and R1C6 both contain {1, 5, 7}. They share the same
      extra candidate: **7**. Both roof cells are in the same box (Box 2).
    technique: UR2
    hint: obvious
    state:
      selection:
        cells: [R1C5, R1C6]
      annotations:
        - cells: [R1C5, R1C6]
          label: "Roof"
          style: pattern
  - text: >
      To avoid the deadly pattern, at least one roof cell must be 7.
      Since both roof cells are in Row 1 and Box 2, any cell that sees BOTH
      of them cannot be 7.
    technique: UR2
    hint: detailed
    state:
      selection:
        cells: [R1C5, R1C6]
  - text: >
      R1C3 sees both roof cells (same row), and R3C6 sees both (same box/column).
      **Eliminate 7 from R1C3 and R3C6**.
    technique: UR2
    hint: detailed
    state:
      selection:
        cells: [R1C3, R3C6]

Type 3: Pseudo-Naked Subset

Extra candidates form a naked pair/triple with another cell. Two corners have {A, B}, two have extra candidates that combine with a nearby cell.

The Pattern

       C4       C8
R2   {A,B}    {A,B,C}
R6   {A,B}    {A,B,D}
         + nearby cell {C,D}

The Logic

This is essentially a hidden naked pair/triple using the rectangle to identify it.

Note: Type 3 is rare in practice. The pattern requires specific conditions where the extras form a naked subset with an external cell.

Type 4: Locked Candidate (Strong Link)

A conjugate pair within the rectangle forces elimination. One of the deadly pair digits is "locked" to the roof cells.

The Pattern

       C7       C9
R2   {A,B,X}  {A,B}    ← roof cells
R7   {A,B,Y}  {A,B}    ← floor cells have only {A,B}

The Logic

  1. Look at candidate A in the roof cells (R2C7 and R7C7)
  2. If A forms a conjugate pair in column 7 (only two places for A)
  3. Then one roof cell MUST be A
  4. This prevents the deadly pattern
  5. Eliminate the OTHER digit (B) from both roof cells

Key Insight

The strong link "locks in" one of the deadly pair digits, breaking the pattern.

Example

puzzle: S9B8I8I43050B03040743052E0B082B047R067Q042E432B09064705020805070P0N09060L040302040643070509437N7N060S4A054403070208052E06019I047Q371F092M2M082H0L052B04030905022B0806
mode: guided
settings:
  showCandidates: true
  showControls: true
  showDescription: true
  navigation: numbered
  coordinateFormat: rncn
steps:
  - text: >
      Find the rectangle: R2C7, R2C9, R7C7, and R7C9 form a Unique Rectangle
      spanning boxes 3 and 9.
    technique: UR2
    hint: subtle
    state:
      selection:
        cells: [R2C7, R2C9, R7C7, R7C9]
  - text: >
      The floor cells R2C9 and R7C9 each contain only {3, 9} — the deadly pair.
    technique: UR2
    hint: obvious
    state:
      selection:
        cells: [R2C9, R7C9]
      annotations:
        - cells: [R2C9, R7C9]
          label: "Floor"
          style: pattern
  - text: >
      The roof cells R2C7 and R7C7 have extras beyond {3, 9}. Look at digit 9
      in column 7 — it appears **only** in these two roof cells.
    technique: UR2
    hint: obvious
    state:
      selection:
        cells: [R2C7, R7C7]
      annotations:
        - cells: [R2C7, R7C7]
          label: "Roof"
          style: pattern
  - text: >
      This creates a strong link: one of R2C7 or R7C7 **must** be 9.
      If one roof cell is 9, the deadly pattern is broken.
    technique: UR2
    hint: detailed
    state:
      selection:
        cells: [R2C7, R7C7]
  - text: >
      Since 9 is locked to the roof, 3 cannot also be in both roof cells
      (that would still allow the deadly pattern). **Eliminate 3 from R2C7
      and R7C7**.
    technique: UR2
    hint: detailed
    state:
      selection:
        cells: [R2C7, R7C7]

Type 5: Diagonal Exclusion

A restricted digit forces diagonal placement. One of the deadly pair digits appears only within the rectangle cells.

The Pattern

       C3       C5
R1   {A,B}    {A,B}    ← roof: A appears here AND in floor
R2   {A,B}    {A,B}    ← but B ONLY appears in rectangle

The Logic

  1. Digit B appears only in the rectangle cells (within rows 1-2 and columns 3-5)
  2. B must occupy exactly two cells — one per row and one per column
  3. This forces B into a diagonal: either (R1C3, R2C5) or (R1C5, R2C3)
  4. The other diagonal must contain A
  5. Eliminate A from the floor cells on the forced diagonal

Key Insight

When a digit is restricted to the rectangle, it must occupy a diagonal, forcing the other digit to the opposite diagonal.

Example

puzzle: S9B8I8I45050L03040743052E0L082D047R067Q042E432B09064705020805070P0N09060L040302040643070509437N7N060S4A054403070208052E06019I047Q371F092M2M082H0L052B04030905022B0806
mode: guided
settings:
  showCandidates: true
  showControls: true
  showDescription: true
  navigation: numbered
  coordinateFormat: rncn
steps:
  - text: >
      Find the rectangle: R1C3, R1C5, R2C3, and R2C5 form a Unique Rectangle
      spanning boxes 1 and 2.
    technique: UR2
    hint: subtle
    state:
      selection:
        cells: [R1C3, R1C5, R2C3, R2C5]
  - text: >
      All four cells contain candidates {1, 2}. This looks like a deadly
      pattern, but let's check if one digit is restricted.
    technique: UR2
    hint: obvious
    state:
      selection:
        cells: [R1C3, R1C5, R2C3, R2C5]
  - text: >
      Look at digit 2 in rows 1-2 and columns 3-5. Digit 2 appears **only**
      in the rectangle cells — nowhere else in these rows/columns!
    technique: UR2
    hint: obvious
    state:
      selection:
        cells: [R1C3, R1C5, R2C3, R2C5]
  - text: >
      Since 2 must appear once per row and once per column, it must occupy
      a diagonal: either (R1C3, R2C5) or (R1C5, R2C3).
    technique: UR2
    hint: detailed
    state:
      selection:
        cells: [R1C3, R2C5]
  - text: >
      The opposite diagonal will contain 1. To avoid the deadly pattern,
      **eliminate 1 from the floor cells** R1C5 and R2C3.
    technique: UR2
    hint: detailed
    state:
      selection:
        cells: [R1C5, R2C3]

Summary Table

Type Configuration Action
1 Three {A,B}, one {A,B,C...} Eliminate A,B from fourth corner
2 Two {A,B}, two {A,B,C} same extra Eliminate C from cells seeing both
3 Two {A,B}, two with extras forming subset Apply naked subset elimination
4 Roof cells have strong link for A or B Eliminate other digit from roof
5 Bi-value cell adjacent Forces placement via chain

Finding Unique Rectangles

  1. Look for rectangles — Four cells spanning exactly two boxes, two rows, two columns
  2. Check candidate pairs — Find cells with mostly the same two candidates
  3. Find the asymmetry — The "extra" candidates prevent the deadly pattern
  4. Identify the type — Match the pattern to apply the correct rule

Valid Puzzle Assumption

Unique Rectangle techniques rely on the puzzle having exactly one solution. If you're solving a puzzle that might have multiple solutions, these techniques could give incorrect results. All legitimate Sudoku puzzles have unique solutions.

More Puzzles

Related Techniques