Hidden Unique Rectangle

A Hidden Unique Rectangle extends Unique Rectangle logic to cluttered rectangles where only 1-2 cells are bi-value. By combining uniqueness constraints with strong links, we can force eliminations that prevent deadly patterns.

The Deadly Pattern

Like standard Unique Rectangle, Hidden UR prevents patterns that would create multiple solutions. The difference is that most cells are "cluttered" with extra candidates — the deadly pair is hidden among other digits.

puzzle: S9B040F032B082Q83830B0I070B060N124J04430E434302090D0C060G010O827U0W0818070F0744827V0F7O4T4J0354044Y052H2G454309480563DA2K06CZBB4B52096Y660E015U024A484963DA2K9G06BB05
mode: static
settings:
  showCandidates: true
  coordinateFormat: none
initial:
  annotations:
    - cells: [R6C1]
      label: "Anchor"
      style: pattern
    - cells: [R6C3, R8C1, R8C3]
      label: "Cluttered"

The key insight is combining:

When a strong link exists within the rectangle's rows or columns, we can prove that certain candidates would lead to the deadly pattern — and therefore must be eliminated.

Type 1: Single Anchor

One bi-value cell anchors the pattern. The anchor contains exactly {A, B}, while the other three cells have extras.

puzzle: S9B040F032B082Q83830B0I070B060N124J04430E434302090D0C060G010O827U0W0818070F0744827V0F7O4T4J0354044Y052H2G454309480563DA2K06CZBB4B52096Y660E015U024A484963DA2K9G06BB05
mode: static
settings:
  showCandidates: true
  coordinateFormat: none
initial:
  selection:
    cells: [R6C1, R6C3, R8C1, R8C3]
  annotations:
    - cells: [R6C1]
      label: "Anchor"
      style: pattern

The Logic

  1. The anchor cell (bi-value {A, B}) fixes one corner of the deadly pattern
  2. If a strong link on digit A or B connects other corners, the pattern becomes forced
  3. One of the cluttered cells must NOT be A or B to break the pattern
  4. Eliminate the offending candidate from that cell

Strong Link Requirement

For Type 1, we need strong links on one of the deadly digits connecting corners. A strong link means the digit appears in only two cells within a row or column — if one is false, the other must be true.

Example

puzzle: S9B040F032B082Q83830B0I070B060N124J04430E434302090D0C060G010O827U0W0818070F0744827V0F7O4T4J0354044Y052H2G454309480563DA2K06CZBB4B52096Y660E015U024A484963DA2K9G06BB05
mode: guided
settings:
  showCandidates: true
  showControls: true
  showDescription: true
  navigation: numbered
  coordinateFormat: rncn
steps:
  - text: >
      Find a rectangle spanning two boxes. R6C1, R6C3, R8C1, and R8C3 form
      a potential Hidden Unique Rectangle in boxes 4 and 7.
    technique: HUR
    hint: subtle
    state:
      selection:
        cells: [R6C1, R6C3, R8C1, R8C3]
  - text: >
      Identify the anchor cell. R6C1 contains exactly {6, 8} — this is our
      bi-value anchor. The other three corners have extra candidates.
    technique: HUR
    hint: obvious
    state:
      selection:
        cells: [R6C1]
      annotations:
        - cells: [R6C1]
          label: "Anchor"
          style: pattern
  - text: >
      Look for strong links. In row 6, digit 6 appears only in C1 and C3.
      In column 1, digit 6 appears only in R6 and R8. These are strong links!
    technique: HUR
    hint: obvious
    state:
      selection:
        cells: [R6C1, R6C3, R8C1]
      annotations:
        - cells: [R6C1]
          label: "Anchor"
          style: pattern
  - text: >
      Trace the logic: If R8C1 = 8, then by strong link R6C1 must be 6.
      Combined with other strong links, this could force a deadly pattern.
      To prevent this, R8C1 cannot be 8.
    technique: HUR
    hint: detailed
    state:
      selection:
        cells: [R8C1]
      annotations:
        - cells: [R6C1]
          label: "Anchor"
          style: pattern
  - text: >
      **Eliminate 8 from R8C1**. This breaks the potential deadly pattern
      by ensuring R8C1 won't participate in the {6, 8} configuration.
    technique: HUR
    hint: detailed
    state:
      selection:
        cells: [R8C1]

Type 2: Same-Axis Floor

Two bi-value cells share a row or column. These form the "floor" of the pattern, while the other two cells (the "roof") have extra candidates.

puzzle: S9B0E0A0H0D0702060C0I0C2C0F080E0I2D2D040D2C0903010F2S4K5W09040E0F022B0C5V5U0H0F0A9E0C0D9G9G05070C027N0H0E7V7V0F06052I2D0I2B082K030B0I0C2Q040H2Q0F010A082I2S0F0CA4A42C
mode: static
settings:
  showCandidates: true
  coordinateFormat: none
initial:
  annotations:
    - cells: [R7C3, R9C3]
      label: "Floor"
      style: pattern
    - cells: [R7C8, R9C8]
      label: "Roof"

The Logic

  1. Two bi-value cells {A, B} share a row or column — these anchor the pattern
  2. The opposite corners (roof) have extras but still contain {A, B}
  3. A strong link on one digit between floor and roof forces the pattern
  4. Eliminate candidates from roof cells to break the deadly pattern

Key Insight

The floor cells being bi-value means they're "locked" into the deadly pair. Any strong link that forces the roof cells to also become {A, B} would create the deadly pattern — so we eliminate to prevent this.

Example

puzzle: S9B0E0A0H0D0702060C0I0C2C0F080E0I2D2D040D2C0903010F2S4K5W09040E0F022B0C5V5U0H0F0A9E0C0D9G9G05070C027N0H0E7V7V0F06052I2D0I2B082K030B0I0C2Q040H2Q0F010A082I2S0F0CA4A42C
mode: guided
settings:
  showCandidates: true
  showControls: true
  showDescription: true
  navigation: numbered
  coordinateFormat: rncn
steps:
  - text: >
      Find a rectangle spanning two boxes. R7C3, R7C8, R9C3, and R9C8 form
      a potential Hidden Unique Rectangle in boxes 7 and 9.
    technique: HUR
    hint: subtle
    state:
      selection:
        cells: [R7C3, R7C8, R9C3, R9C8]
  - text: >
      Identify the floor cells. R7C3 and R9C3 both contain exactly {4, 7} —
      these bi-value cells share column 3, forming the "floor" of our pattern.
    technique: HUR
    hint: obvious
    state:
      selection:
        cells: [R7C3, R9C3]
      annotations:
        - cells: [R7C3, R9C3]
          label: "Floor"
          style: pattern
  - text: >
      The roof cells R7C8 and R9C8 have extras beyond {4, 7}. They contain
      the deadly pair plus additional candidates that might save us.
    technique: HUR
    hint: obvious
    state:
      selection:
        cells: [R7C8, R9C8]
      annotations:
        - cells: [R7C3, R9C3]
          label: "Floor"
          style: pattern
        - cells: [R7C8, R9C8]
          label: "Roof"
  - text: >
      Look for a strong link. In column 3, digit 4 appears only in R7C3 and
      R9C3 (the floor cells). This strong link connects to the roof through
      the rows.
    technique: HUR
    hint: detailed
    state:
      selection:
        cells: [R7C3, R9C3]
      annotations:
        - cells: [R7C3, R9C3]
          label: "Floor"
          style: pattern
  - text: >
      The strong link forces: if R9C8 = 7, tracing through the pattern creates
      a deadly configuration. To prevent this, **eliminate 7 from R9C8**.
    technique: HUR
    hint: detailed
    state:
      selection:
        cells: [R9C8]

Summary Table

Type Configuration Floor Cells Roof Cells Elimination
1 Single anchor 1 bi-value 3 with extras From cell forced by strong links
2 Same-axis floor 2 bi-value (same row/col) 2 with extras From roof cells

Complexity

Hidden Unique Rectangle is an expert-level technique because:

Tips

  1. Find bi-value cells first — These are your anchor points
  2. Look for conjugate pairs — Strong links on the deadly digits within the rectangle's rows/columns
  3. Check both row and column — Strong links can work in either direction
  4. Trust the hint system — This technique is difficult to spot manually; let the solver help

More Puzzles

Related Techniques