Extended Unique Rectangle

An Extended Unique Rectangle (EUR) expands the standard Unique Rectangle from a 2×2 to a 2×3 (or 3×2) configuration. Instead of four cells with two digits, we look for six cells with three digits forming a deadly pattern.

The Deadly Pattern

The deadly pattern in EUR involves:

Six cells forming a 2×3 or 3×2 rectangle
Three candidate digits {A, B, C} — the "EUR triple"
Pattern spans exactly 3 boxes

puzzle: S9B091F07430204034Y050804020306050901071V030Z094307045044116B042Q4709064K48135V0N0204062B4I091W3N092Q47432B0448130Z13040902080706070906430543020304040208060703050901
mode: static
settings:
  showCandidates: true
  coordinateFormat: none
initial:
  annotations:
    - cells: [R3C1]
      label: "A"
    - cells: [R3C3]
      label: "B"
    - cells: [R5C1]
      label: "B"
    - cells: [R5C3]
      label: "A"
    - cells: [R7C1]
      label: "A"
    - cells: [R7C3]
      label: "B"

If all six cells contained only the same three candidates, the puzzle would have multiple solutions — you could swap A↔B on the diagonals and still satisfy all constraints. Since valid puzzles have unique solutions, this pattern cannot occur.

Floor and Roof

Like standard Unique Rectangle, EUR uses floor/roof terminology:

Floor cells: Contain only the EUR triple {A, B, C}
Roof cells: Contain the EUR triple plus extra candidates

The different EUR types exploit these extra candidates in various ways.

Type 1: Odd Cell

The simplest case. Five cells contain only {A, B, C}, but the sixth — the "odd cell" — has extra candidates.

puzzle: S9B091F07430204034Y050804020306050901071V030Z094307045044116B042Q4709064K48135V0N0204062B4I091W3N092Q47432B0448130Z13040902080706070906430543020304040208060703050901
mode: static
settings:
  showCandidates: true
  coordinateFormat: none
initial:
  selection:
    cells: [R3C3, R5C1, R5C3, R7C1, R7C3]
  annotations:
    - cells: [R3C1]
      label: "Odd Cell"
      style: pattern

The Logic

If the sixth cell (the "odd" cell) contained only {A, B, C}, we'd have a deadly pattern
The extra candidates prevent this
Therefore, the odd cell MUST be one of the extra candidates
Eliminate A, B, and C from the odd cell

Example

puzzle: S9B091F07430204034Y050804020306050901071V030Z094307045044116B042Q4709064K48135V0N0204062B4I091W3N092Q47432B0448130Z13040902080706070906430543020304040208060703050901
mode: guided
settings:
  showCandidates: true
  showControls: true
  showDescription: true
  navigation: numbered
  coordinateFormat: rncn
steps:
  - text: >
      Look for a 3×2 rectangle spanning 3 boxes. R3C1, R3C3, R5C1, R5C3, R7C1,
      and R7C3 form the corners of a potential Extended Unique Rectangle.
    technique: EUR
    hint: subtle
    state:
      selection:
        cells: [R3C1, R3C3, R5C1, R5C3, R7C1, R7C3]
  - text: >
      Examine the candidates. Five cells (R3C3, R5C1, R5C3, R7C1, R7C3) contain
      only {1, 3, 5} — these are the "floor" cells with just the EUR triple.
    technique: EUR
    hint: obvious
    state:
      selection:
        cells: [R3C3, R5C1, R5C3, R7C1, R7C3]
  - text: >
      Now look at R3C1 — it has {1, 3, 5, 6}. This is the "odd cell" with
      an extra candidate (6) beyond the EUR triple {1, 3, 5}.
    technique: EUR
    hint: obvious
    state:
      selection:
        cells: [R3C1]
      annotations:
        - cells: [R3C1]
          label: "Odd Cell"
          style: pattern
  - text: >
      If R3C1 were reduced to just {1, 3, 5}, all six cells would contain
      only the EUR triple. This creates a deadly pattern — rearranging
      1s, 3s, and 5s would give multiple valid solutions.
    technique: EUR
    hint: detailed
    state:
      selection:
        cells: [R3C1, R3C3, R5C1, R5C3, R7C1, R7C3]
  - text: >
      Therefore, R3C1 cannot be 1, 3, or 5 — it must be 6 (the extra candidate).
      **Eliminate 1 and 5 from R3C1**, leaving only {3, 6}. (Note: 3 was
      already eliminated by other means in this puzzle state.)
    technique: EUR
    hint: detailed
    state:
      selection:
        cells: [R3C1]

Type 2: Shared Extra

Two roof cells share the same extra candidate. Four floor cells have only {A, B, C}, while two roof cells have {A, B, C, X} — the same extra candidate X.

puzzle: S9B010H167U070F7O84030W0W0G8E138206087V091Q26020N082B2R0R4Y054Y0G0B04037N7N0U0U02080901050F0G0G0I010612124404441S1S5A014Y07BG7S055C0709124Y140R0P44050A467Q047S5W2G06
mode: static
settings:
  showCandidates: true
  coordinateFormat: none
initial:
  annotations:
    - cells: [R2C1, R2C2, R5C1, R5C2]
      label: "Floor"
      style: pattern
    - cells: [R7C1, R7C2]
      label: "Roof"
      style: pattern

The Logic

To avoid the deadly pattern, at least one roof cell must NOT be A, B, or C
Both roof cells have the same extra candidate X
Therefore, X MUST appear in one of the roof cells
Eliminate X from any cell that sees BOTH roof cells

Key Requirement

The two roof cells must share a unit (row, column, or box) AND that unit must contain other cells with candidate X for eliminations to occur.

Example

puzzle: S9B010H167U070F7O84030W0W0G8E138206087V091Q26020N082B2R0R4Y054Y0G0B04037N7N0U0U02080901050F0G0G0I010612124404441S1S5A014Y07BG7S055C0709124Y140R0P44050A467Q047S5W2G06
mode: guided
settings:
  showCandidates: true
  showControls: true
  showDescription: true
  navigation: numbered
  coordinateFormat: rncn
steps:
  - text: >
      Find the rectangle: R2C1, R2C2, R5C1, R5C2, R7C1, and R7C2 form an
      Extended Unique Rectangle spanning boxes 1, 4, and 7.
    technique: EUR
    hint: subtle
    state:
      selection:
        cells: [R2C1, R2C2, R5C1, R5C2, R7C1, R7C2]
  - text: >
      The floor cells R2C1, R2C2, R5C1, and R5C2 each contain only {2, 3, 4} —
      these are the EUR triple digits.
    technique: EUR
    hint: obvious
    state:
      selection:
        cells: [R2C1, R2C2, R5C1, R5C2]
      annotations:
        - cells: [R2C1, R2C2, R5C1, R5C2]
          label: "Floor"
          style: pattern
  - text: >
      The roof cells R7C1 and R7C2 both contain {2, 3, 4, 6}. They share the
      same extra candidate: **6**. Both roof cells are in Row 7.
    technique: EUR
    hint: obvious
    state:
      selection:
        cells: [R7C1, R7C2]
      annotations:
        - cells: [R2C1, R2C2, R5C1, R5C2]
          label: "Floor"
          style: pattern
        - cells: [R7C1, R7C2]
          label: "Roof"
          style: pattern
  - text: >
      To avoid the deadly pattern, at least one roof cell must be 6.
      Since both roof cells are in Row 7 and Box 7, any cell that sees BOTH
      of them cannot be 6.
    technique: EUR
    hint: detailed
    state:
      selection:
        cells: [R7C1, R7C2]
      annotations:
        - cells: [R7C1, R7C2]
          label: "Roof"
          style: pattern
  - text: >
      R7C3 and R7C5 see both roof cells (same row). R8C1 sees both (same box).
      **Eliminate 6 from R7C3, R7C5, and R8C1**.
    technique: EUR
    hint: detailed
    state:
      selection:
        cells: [R7C3, R7C5, R8C1]

Type 4: Locked Pair

An EUR digit is locked to the roof cells. Four floor cells have only {A, B, C}, while two roof cells have extras. Crucially, one of the EUR digits (say A) appears only in the roof cells within their shared unit.

puzzle: S9B7Q01074Q06B602164M0204B601BM034I07060546064A0207BA7U0107090U060R0213081201050246BAB607060408060U070R057R027Q0407BM4M4M06B6010206024I094I010403077Q46010207040682BM
mode: static
settings:
  showCandidates: true
  coordinateFormat: none
initial:
  annotations:
    - cells: [R4C3, R4C5, R6C3, R6C5]
      label: "Floor"
      style: pattern
    - cells: [R4C7, R6C7]
      label: "Roof"
      style: pattern

The Logic

Look at candidate A (or B or C) in the shared unit of the roof cells
If A forms a conjugate pair in that unit (only appears in the two roof cells)
Then one roof cell MUST be A
This prevents the deadly pattern
Eliminate the OTHER EUR digits (B and C) from both roof cells

Key Insight

The "locked" digit guarantees at least one roof cell isn't part of the deadly triple, breaking the pattern.

Example

puzzle: S9B7Q01074Q06B602164M0204B601BM034I07060546064A0207BA7U0107090U060R0213081201050246BAB607060408060U070R057R027Q0407BM4M4M06B6010206024I094I010403077Q46010207040682BM
mode: guided
settings:
  showCandidates: true
  showControls: true
  showDescription: true
  navigation: numbered
  coordinateFormat: rncn
steps:
  - text: >
      Find the rectangle: R4C3, R4C5, R4C7, R6C3, R6C5, and R6C7 form an
      Extended Unique Rectangle spanning boxes 4, 5, and 6.
    technique: EUR
    hint: subtle
    state:
      selection:
        cells: [R4C3, R4C5, R4C7, R6C3, R6C5, R6C7]
  - text: >
      The floor cells R4C3, R4C5, R6C3, and R6C5 each contain only {1, 3, 4} —
      the EUR triple.
    technique: EUR
    hint: obvious
    state:
      selection:
        cells: [R4C3, R4C5, R6C3, R6C5]
      annotations:
        - cells: [R4C3, R4C5, R6C3, R6C5]
          label: "Floor"
          style: pattern
  - text: >
      The roof cells R4C7 and R6C7 have extras beyond {1, 3, 4}. Look at digit 4
      in column 7 — it appears **only** in these two roof cells.
    technique: EUR
    hint: obvious
    state:
      selection:
        cells: [R4C7, R6C7]
      annotations:
        - cells: [R4C3, R4C5, R6C3, R6C5]
          label: "Floor"
          style: pattern
        - cells: [R4C7, R6C7]
          label: "Roof"
          style: pattern
  - text: >
      This creates a strong link: one of R4C7 or R6C7 **must** be 4.
      If one roof cell is 4, the deadly pattern is broken since that cell
      won't participate in the EUR triple arrangement.
    technique: EUR
    hint: detailed
    state:
      selection:
        cells: [R4C7, R6C7]
      annotations:
        - cells: [R4C7, R6C7]
          label: "Roof"
          style: pattern
  - text: >
      Since 4 is locked to the roof, the other EUR digits (1 and 3) cannot
      BOTH be in roof cells (that would still allow the deadly pattern).
      **Eliminate 3 from R4C7 and R6C7**.
    technique: EUR
    hint: detailed
    state:
      selection:
        cells: [R4C7, R6C7]

Summary Table

Type	Configuration	Action
1	Five {A,B,C}, one {A,B,C,X...}	Eliminate A,B,C from odd cell
2	Four {A,B,C}, two {A,B,C,X} same extra	Eliminate X from cells seeing both roof
4	Four {A,B,C}, two roof with locked digit	Eliminate non-locked EUR digits from roof

Comparison to Standard UR

Aspect	Unique Rectangle	Extended Unique Rectangle
Size	2×2 (4 cells)	2×3 or 3×2 (6 cells)
Digits	2 candidates	3 candidates
Boxes	Exactly 2	Exactly 3
Rarity	Common	Rare
Difficulty	Advanced	Expert

Finding Extended Unique Rectangles

Look for aligned triples — Find cells in a row/column with the same three candidates
Check for 2×3 or 3×2 geometry — Six cells, two in one dimension, three in the other
Verify box coverage — Pattern must span exactly 3 boxes
Identify floor vs roof — Which cells have only the triple? Which have extras?
Match the type — Apply the appropriate elimination rule

Complexity

Extended Unique Rectangle is an expert-level technique because:

Must identify 2×3 or 3×2 patterns across the grid
Three digits to track instead of two
More cells to verify (typically five floor, one or two roof)
Crosses more box boundaries
Multiple type variations to recognise

More Puzzles

Related Techniques

Unique Rectangle — Standard 2×2 version
Hidden Unique Rectangle — Uses strong links
Avoidable Rectangle — Works on solved cells