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Alternating Inference Chain (AIC)

A powerful chain technique that alternates strong and weak links to prove which candidates can be eliminated.

An Alternating Inference Chain (AIC) is one of the most powerful logical techniques in Sudoku. It connects candidates across the grid using a chain of alternating strong and weak links to prove that certain candidates can be eliminated.

If you already understand strategies like XY-Chain or X-Cycle, AIC is a natural next step — it generalizes both of these patterns into a single, unified framework.

Interaktives Beispiel

1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
8
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
6
1 2 3 4 5 6 7 8 9
3
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
3
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
2
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1
1 2 3 4 5 6 7 8 9
9
1 2 3 4 5 6 7 8 9
7
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
3
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
7
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
4
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
7
1 2 3 4 5 6 7 8 9
4
1 2 3 4 5 6 7 8 9
9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
5
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
2
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
3
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
5
1 2 3 4 5 6 7 8 9
9
1 2 3 4 5 6 7 8 9
1
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
5
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
4
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
3
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
3
1 2 3 4 5 6 7 8 9
5
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
6
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9

Klicken Sie auf "Logik anwenden", um die Strategie in Aktion zu sehen.

What Are Strong and Weak Links?

Before understanding AIC, you need to know the two types of logical connections between candidates:

Strong Link (=)

A strong link means: if one is OFF, the other MUST be ON.

This happens when: - A digit appears in only 2 cells within a row, column, or box (conjugate pair). If one cell isn't that digit, the other must be. - A cell has only 2 candidates (bivalue cell). If one candidate is wrong, the other must be right.

Weak Link (-)

A weak link means: if one is ON, the other MUST be OFF.

This happens when: - Two cells share a unit (row, column, or box) and both contain the same candidate. If one cell IS that digit, the other cannot be. - Two candidates are in the same cell. If one is correct, the other is wrong.

Real Example Explanation

In the example above, our solver found this chain:

Chain: R1C2=2(OFF) = R1C2=4(ON) - R1C9=4(OFF) = R1C9=7(ON) - R1C6=7(OFF) = R1C6=5(ON) - R1C1=5(OFF) = R2C1=5(ON) - R2C7=5(OFF) = R2C7=8(ON) - R3C8=8(OFF) = R3C8=2(ON)

Let's read it step by step:

  1. Start: Assume R1C2 is not 2 (OFF).
  2. Strong link (=): R1C2 has candidates {2, 4}. If it's not 2, it must be 4 (ON).
  3. Weak link (-): R1C9 also has candidate 4 in the same row. If R1C2 is 4, then R1C9 cannot be 4 (OFF).
  4. Strong link (=): R1C9 has candidates {4, 7}. If it's not 4, it must be 7 (ON).
  5. Continue alternating through R1C6, R1C1, R2C1, R2C7, R3C8...
  6. Final strong link (=): R3C8 has candidates {2, 8}. If it's not 8, it must be 2 (ON).

The Key Insight: The chain starts with "R1C2 is not 2" and after 10 links concludes "R3C8 must be 2." But we don't actually know if R1C2 is 2 or not! Here's the trick:

  • If R1C2 IS 2 → R1C2 = 2 is established.
  • If R1C2 is NOT 2 → the chain forces R3C8 = 2.

Either way, at least one of R1C2 = 2 or R3C8 = 2 must be true. So any cell that sees both R1C2 and R3C8 and has candidate 2 can have that 2 eliminated.

Result: R1C3 shares Row 1 with R1C2 and Box 1 with R3C8, and has candidate 2. Since one of those cells must be 2, we eliminate 2 from R1C3.

How AIC Works (The General Rule)

An AIC chain always alternates between strong and weak links:

OFF =strong= ON -weak- OFF =strong= ON -weak- OFF =strong= ON

The two ON endpoints at the start and end of the chain are the critical elements:

At least one of the two ON endpoints must be true.

This is because: - If the first endpoint is false (OFF), the strong link forces its partner ON, and the chain propagates all the way to the final endpoint being ON. - If the first endpoint is true (ON), then it's already established.

What Can You Eliminate?

Same digit at both endpoints: If both ON endpoints have the same digit (e.g., both are "9 ON"), eliminate that digit from any cell that sees both endpoints.

Different digits in the same cell: If the two ON endpoints are in the same cell but different digits, the cell must be one of those digits — eliminate all other candidates.

How to Spot an AIC

  1. Look for strong links: Find conjugate pairs (a digit with only 2 positions in a unit) and bivalue cells (cells with exactly 2 candidates).
  2. Build a chain: Start from a candidate assumed OFF. Follow a strong link to get ON. Then follow a weak link to get OFF. Alternate until you reach another ON endpoint.
  3. Check for eliminations: See if any cell with the target digit can see both ON endpoints.

AIC vs. Other Chain Strategies

Strategy Link Types Chain Shape Scope
X-Cycle X-links only (same digit) Closed loop Single digit
XY-Chain Y-links only (bivalue cells) Open chain Multiple digits
AIC Both X and Y links Open chain Multiple digits
3D Medusa Both X and Y links Coloring network Multiple digits
  • X-Cycle: An AIC restricted to a single digit.
  • XY-Chain: An AIC restricted to bivalue cell links.
  • AIC: Combines both — the most flexible chain technique.
  • 3D Medusa: Explores the full coloring graph simultaneously, while AIC finds individual chains.