Alternating Inference Chains (AIC)

Alternating Inference Chains (AIC) unify and generalise various chain techniques into one powerful framework. While X-Cycles work with a single digit and Simple Colouring uses conjugate pairs, AICs extend these concepts to multiple digits — linking different candidates through alternating strong and weak inferences.

Note: This technique uses chain colouring to display candidates. Square (▢) and Diamond (◇) shapes represent alternating ON/OFF states in the chain.

Understanding Links

AICs are built from two fundamental link types that must alternate throughout the chain.

Strong Links (Bidirectional)

A strong link exists when one candidate being OFF forces another ON. Two ways to create strong links:

Bi-location (Conjugate Pair): A digit appears in exactly 2 cells within a unit.

If R1C2 ≠ 5 → R1C7 = 5 (and vice versa)

Bi-value Cell: A cell contains exactly 2 candidates.

If R3C4 ≠ 3 → R3C4 = 7 (and vice versa)

Strong links work both directions — they're bidirectional implications.

Weak Links (Unidirectional)

A weak link exists when one candidate being ON forces another OFF:

Same Unit: Two cells sharing a row, column, or box.

If R1C2 = 5 → R1C7 ≠ 5 (but NOT vice versa)

Same Cell: Two different candidates in the same cell.

If R3C4 = 3 → R3C4 ≠ 7 (but NOT vice versa)

Weak links work one direction only — they're unidirectional implications.

The Key Difference

Link Type	Direction	Meaning
Strong	Bidirectional ↔	"If A is OFF, B must be ON"
Weak	Unidirectional →	"If A is ON, B must be OFF"

Building Mixed Chains

AICs combine both link types, alternating between them. The standard Eureka notation uses:

= for strong links
- for weak links

(5)r1c3=(8)r1c3-(8)r1c6=(8)r4c6-(4)r4c6=(4)r5c6-(4)r5c3=(5)r5c3

Reading this chain:

(5)r1c3 — Start with candidate 5 in R1C3
=(8)r1c3 — Strong link (bi-value): connects to 8 in same cell
-(8)r1c6 — Weak link (same row): 8 sees 8 in R1C6
=(8)r4c6 — Strong link (bi-location): conjugate pair in column 6
And so on...

The Three Rules

AICs produce eliminations through three distinct patterns.

Rule 1: Nice Loop (Continuous)

When a chain forms a continuous loop with consistent alternation, we get "off-chain" eliminations.

Pattern: The chain returns to its starting point, alternating perfectly.

Logic: Within any unit that contains both an ON node and an OFF node of the same digit, all OTHER instances of that digit can be eliminated.

puzzle: S9B1V8J4I6A046B83030B020C074J0Z09060D4J0Z7V4Q02030683074J090R0C4J0F4Q020Z071V3F162B0B2I0H0I0C082B020C0I2R0Z0F040C050A0907020D0H0F0G0H06040Z0Z030B090D020I0F080C0G0Z0Z
mode: guided
technique: AIC
settings:
  showCandidates: true
  showControls: true
  showDescription: true
  navigation: numbered
  coordinateFormat: rncn
steps:
  - text: >
      A Nice Loop is a continuous AIC that returns to its starting point. This puzzle
      contains a loop involving digits 4, 5, and 8.
    hint: subtle
    technique: AIC

  - text: >
      Identify the bi-value cells. R1C3 has {5,8}, R5C3 has {4,5}, and R4C6 has {4,8}.
      These cells allow digit-crossing strong links.
    hint: subtle
    technique: AIC
    state:
      selection:
        cells: [R1C3, R5C3, R4C6]

  - text: >
      Build the chain starting from R1C3. The bi-value link connects 5 and 8 in the same cell.
      Written as: (5)r1c3=(8)r1c3
    hint: subtle
    technique: AIC
    state:
      selection:
        cells: [R1C3]
      annotations:
        - cells: [R1C3]
          label: "{5,8}"
          style: pattern

  - text: >
      Continue with a weak link. R1C3 and R1C6 share Row 1 — they see each other.
      Written as: (8)r1c3-(8)r1c6
    hint: subtle
    technique: AIC
    state:
      selection:
        cells: [R1C3, R1C6]

  - text: >
      R1C6 and R4C6 form a conjugate pair for digit 8 in Column 6. This is a strong link.
      Written as: (8)r1c6=(8)r4c6
    hint: obvious
    technique: AIC
    state:
      selection:
        cells: [R1C6, R4C6]

  - text: >
      R4C6 is bi-value {4,8}. The bi-value strong link connects 8 and 4 in the same cell.
      Written as: (8)r4c6=(4)r4c6
    hint: obvious
    technique: AIC
    state:
      selection:
        cells: [R4C6]
      annotations:
        - cells: [R4C6]
          label: "{4,8}"
          style: pattern

  - text: >
      R4C6 and R5C6 form a conjugate pair for digit 4 in Box 5. Strong link.
      Written as: (4)r4c6=(4)r5c6
    hint: obvious
    technique: AIC
    state:
      selection:
        cells: [R4C6, R5C6]

  - text: >
      R5C6 and R5C3 share Row 5. Weak link — they see each other.
      Written as: (4)r5c6-(4)r5c3
    hint: obvious
    technique: AIC
    state:
      selection:
        cells: [R5C6, R5C3]

  - text: >
      R5C3 is bi-value {4,5}. Strong link connects 4 and 5 in the same cell.
      Written as: (4)r5c3=(5)r5c3
    hint: obvious
    technique: AIC
    state:
      selection:
        cells: [R5C3]
      annotations:
        - cells: [R5C3]
          label: "{4,5}"
          style: pattern

  - text: >
      R5C3 and R1C3 share Column 3. Weak link back to start — the loop closes!
      Written as: (5)r5c3-(5)r1c3
    hint: detailed
    technique: AIC
    state:
      selection:
        cells: [R1C3, R5C3]

  - text: >
      Complete chain: (5)r1c3=(8)r1c3-(8)r1c6=(8)r4c6=(4)r4c6=(4)r5c6-(4)r5c3=(5)r5c3-(5)r1c3.
      This is a Nice Loop — it returns to the start with consistent alternation.
    hint: detailed
    technique: AIC
    state:
      selection:
        cells: [R1C3, R1C6, R4C6, R5C3, R5C6]

  - text: >
      Nice Loop eliminations: In any unit containing both ON and OFF nodes of the same digit,
      eliminate that digit from other cells. Result: R1C4~8, R3C3~5, R4C6~5, R5C2~4.
    hint: detailed
    technique: AIC
    state:
      selection:
        cells: [R1C4, R3C3, R4C6, R5C2]

Rule 2: Strong Junction

When a chain has a discontinuity where two strong links meet, the candidate at the junction must be TRUE.

Pattern: Both ends of the chain force the same candidate ON.

Logic: If following the chain from either direction leads to the same candidate being ON, that candidate must be placed.

puzzle: S9B0504037N0H7N0B0G060BDUDU0C2216014Q82B60AC2072202BE4Q0C074O4I11090Z4Y52047Y86010616080G7Q0B06B8BE0S030G05B6017UAAAY080B038Q0A82464602820A968Q220G018Y968A0G96030208
mode: guided
technique: AIC
settings:
  showCandidates: true
  showControls: true
  showDescription: true
  navigation: numbered
  coordinateFormat: rncn
steps:
  - text: >
      A Strong Junction occurs when both ends of a chain force the same candidate ON.
      This puzzle demonstrates Rule 2 with digits 5, 8, and 9.
    hint: subtle
    technique: AIC

  - text: >
      Identify key cells. R4C3 has {5,8}, R3C7 has {8,9}, and R2C9 has {5,9}.
      These bi-value cells enable digit-crossing links.
    hint: subtle
    technique: AIC
    state:
      selection:
        cells: [R4C3, R3C7, R2C9]

  - text: >
      Start building the chain from R4C3. The chain will traverse through multiple
      cells and digits, eventually reaching R2C9 from two directions.
    hint: subtle
    technique: AIC
    state:
      selection:
        cells: [R4C3]
      annotations:
        - cells: [R4C3]
          label: "Start"
          style: pattern

  - text: >
      One path: R4C3{5,8} links via 8 to R4C7, then to R3C7{8,9}, arriving at 9.
      R3C7 connects to R2C9{5,9} via strong link on 9.
    hint: obvious
    technique: AIC
    state:
      selection:
        cells: [R4C3, R4C7, R3C7, R2C9]

  - text: >
      Another path goes through R3C5, R3C8, R5C5, R5C2, back to R4C3.
      Both paths converge at R2C9 with strong links to candidate 9.
    hint: obvious
    technique: AIC
    state:
      selection:
        cells: [R3C5, R3C8, R5C2, R5C5]

  - text: >
      Strong Junction at R2C9! Both incoming links are strong, both pointing to 9 being ON.
      This means R2C9 MUST be 9.
    hint: detailed
    technique: AIC
    state:
      selection:
        cells: [R2C9]
      annotations:
        - cells: [R2C9]
          label: "=9"
          style: pattern

  - text: >
      Result: Since R2C9 must be 9, eliminate the other candidate.
      Elimination: R2C9~5.
    hint: detailed
    technique: AIC
    state:
      selection:
        cells: [R2C9]

Rule 3: Weak Junction

When a chain has a discontinuity where two weak links meet, the candidate at the junction must be FALSE.

Pattern: Both ends of the chain force the same candidate OFF.

Logic: If assuming the candidate is ON leads to it being OFF (contradiction), then it cannot be ON.

puzzle: S9BBU4Q060A039M2Q02CYB6030A9E0E020604CY070B8A8I0H8Q0C820102067U050G01087Q0U832Q080D0F0C029F9E0R2I03020I082J0605034Q18CY0L9U2Z2R06220907030L1U17080S5E01106Q043M092U0O
mode: guided
technique: AIC
settings:
  showCandidates: true
  showControls: true
  showDescription: true
  navigation: numbered
  coordinateFormat: rncn
steps:
  - text: >
      A Weak Junction occurs when both ends of a chain force the same candidate OFF.
      This creates a contradiction that eliminates the candidate.
    hint: subtle
    technique: AIC

  - text: >
      This puzzle has a chain involving digits 7, 8, and 9 that creates a Weak Junction
      at R9C4 for digit 7.
    hint: subtle
    technique: AIC
    state:
      selection:
        cells: [R9C4]
      annotations:
        - cells: [R9C4]
          label: "Target"
          style: pattern

  - text: >
      The key cells are R2C1, R2C4, R9C1, and R9C4. These form a rectangular pattern
      with strong and weak links for digits 7, 8, and 9.
    hint: subtle
    technique: AIC
    state:
      selection:
        cells: [R2C1, R2C4, R9C1, R9C4]

  - text: >
      Build the chain: Start with +7[R9C4]. Following strong and weak links through
      R2C4, R2C1, R9C1, we eventually return to R9C4.
    hint: obvious
    technique: AIC
    state:
      selection:
        cells: [R2C1, R2C4, R9C1, R9C4]

  - text: >
      Chain notation: (7)r9c4-(7)r2c4=(7)r2c1=(8)r2c1-(8)r9c1=(8)r9c4=(7)r9c4.
      The chain starts and ends at the same candidate — forming a loop!
    hint: obvious
    technique: AIC
    state:
      selection:
        cells: [R9C4]

  - text: >
      Actually, the discontinuity is at weak links: the chain shows that if 7 is ON in R9C4,
      following the logic forces 7 to be OFF in R9C4. Contradiction!
    hint: detailed
    technique: AIC
    state:
      selection:
        cells: [R9C4]
      annotations:
        - cells: [R9C4]
          label: "≠7"
          style: deadly

  - text: >
      Weak Junction result: R9C4 cannot be 7. Elimination: R9C4~7.
    hint: detailed
    technique: AIC
    state:
      selection:
        cells: [R9C4]

Chain Notation (Eureka)

AICs use Eureka notation — the standard format across most Sudoku resources:

Symbol	Meaning
`=`	Strong link (bidirectional inference)
`-`	Weak link (unidirectional inference)
`(digit)cell`	Candidate and position
`=>`	Conclusion/elimination

Example: (5)r1c3=(8)r1c3-(8)r1c6=(8)r4c6

Reading: "5 in R1C3 strongly links to 8 in R1C3, weakly links to 8 in R1C6, strongly links to 8 in R4C6"

Strong links (=) connect candidates that cannot both be false. Weak links (-) connect candidates that cannot both be true.

Tips

Master the prerequisites — Understand X-Cycles and Simple Colouring first
Look for bi-value cells — These are your bridges between different digits
Check conjugate pairs — Strong links within units enable chain extension
Verify alternation — Chains MUST alternate strong-weak-strong-weak
Use solver hints — AICs are extremely difficult to find manually

When to Use

AICs are typically useful when:

Simpler chain techniques (X-Cycles, XY-Chains) don't find eliminations
Multiple bi-value cells exist across the grid
You need to link different digits together
The puzzle requires expert-level logic

Summary Table

Rule	Pattern	Result
Nice Loop	Continuous alternating cycle	Off-chain eliminations in shared units
Strong Junction	Two strong links meet	Candidate MUST be true → place value
Weak Junction	Two weak links meet	Candidate MUST be false → eliminate

More Puzzles

Related Techniques

X-Cycles — Single-digit chains (AIC subset)
Simple Colouring — Conjugate pair colouring
3D Medusa — Multi-digit colouring extension
XY-Chains — Bi-value cell chains (AIC subset)