Alternating Inference Chain (AIC)

An Alternating Inference Chain, or AIC, is the general form of the chains in Sudoku. Its items are candidates, each a digit in a cell, joined by two kinds of links. A strong link means at least one of two candidates is true: they are the only two places for a digit in a unit, or the only two candidates of a cell. A weak link means at most one of them is true: the same digit in cells that see each other, or two digits in the same cell. An AIC alternates strong and weak links and starts and ends with a strong link, so at least one of its two end candidates is true. When both ends are the same digit, every cell that sees both ends loses that digit. The X-Chain uses only one digit and the XY-Chain only two-candidate cells; an AIC can mix both. It is a Master technique in Bare Sudoku.

When to use it

When single-digit chains and chains of two-candidate cells find nothing. Mixed chains reach further, because they can switch digits inside a cell or move between cells along one digit, but they are also harder to follow. Bare Sudoku tries them after the simpler chains.

How to spot it

Write each candidate as a cell and a digit, for example r1c2 (5). Start with a strong link from a candidate, then take a weak link, then a strong link, and so on, choosing at each step any link of the right kind: along one digit within a unit, or between the two digits of a two-candidate cell. Each time you finish on a strong link, compare the two ends. If they are the same digit, cells that see both ends lose it; if they are two digits of the same cell, every other candidate of that cell can go.

Example

Bold digits are given, blue digits were placed while solving, small digits are candidates. Highlighted cells form the pattern, crossed-out candidates are removed, and a circled candidate is the digit to place.

1. Read the chain r1c1 (3=4) - r2c1 (4=1) - r4c1 (1) = r4c2 (1) - r7c2 (1=3). Each item is a digit in a cell. = is a strong link: at least one of the two items is true. - is a weak link: at most one of them is true.
2. Suppose r1c1 (3) is false. The strong link after it makes the next item true, the weak link after that makes the following item false, and so on, until r7c2 (3) is true.
3. So at least one of r1c1 (3) and r7c2 (3) is true. Both are digit 3.
4. A cell that sees both end cells cannot be 3. Remove 3 from r3c2.

Common mistakes

• Calling a link strong when the unit has a third place for the digit, or the cell a third candidate. A strong link needs exactly two possibilities.
• Drawing a conclusion from a chain whose links do not alternate. The chain must alternate strong and weak links and start and end with a strong link.
• Removing a candidate that sees only one end. The removed candidate must conflict with both ends.

In Bare Sudoku

Needed from level: Master
In the game, the hint says: AIC

Play this puzzle · Play Sudoku · Sudoku solver

Practice puzzles

In these puzzles, AIC is the hardest technique you need. Each link opens the puzzle in the game; the solver link shows every step.

1. Master 7.0 · Sudoku solver
2. Master 7.0 · Sudoku solver
3. Master 7.0 · Sudoku solver
4. Master 7.0 · Sudoku solver
5. Master 7.0 · Sudoku solver
6. Master 7.0 · Sudoku solver
7. Master 7.0 · Sudoku solver
8. Master 7.0 · Sudoku solver
9. Master 7.0 · Sudoku solver
10. Master 7.0 · Sudoku solver

Frequently asked questions

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Made by Alparslan Selçuk Develioğlu · Updated 2026-10-04