Alternating Inference Chains in Sudoku Explained

Read each link as an inference

An Alternating Inference Chain, often shortened to AIC, is a sequence of candidates connected by alternating strong and weak links. A weak link says that two candidates cannot both be true. If one is true, the other is false. A strong link says that at least one of two candidates must be true. If one is false, the other is true. The most familiar strong link is a conjugate pair, where one shape has exactly two positions in a row, column, or box. A bivalue cell also supplies a strong inference between its two candidates. The chain works by switching direction at every step: true crosses a weak link and produces false, then false crosses a strong link and produces true. In Shapedoku, the candidates may be hearts, crescents, crystals, or other shapes, but the logic is identical to a numbered grid. Complete Notes are important because a supposed conjugate pair stops being strong if a third legal position was overlooked.

Use endpoints to prove an elimination

A useful AIC usually begins and ends with candidates that create a shared consequence. Suppose the chain starts with a heart candidate in one cell and ends with a heart candidate elsewhere. If assuming the first heart false forces the last heart true, then at least one endpoint must be true. Any outside heart candidate that sees both endpoints can be removed, because it would conflict whichever endpoint becomes true. Another form begins with one candidate assumed true and ends by forcing a target candidate false. If every link is valid, the implication itself justifies the elimination. AICs can move through several symbols and several kinds of units, which makes them more general than an X-Chain or XY-Chain. The endpoints provide the result, while the internal candidates simply carry the logic. Do not remove candidates that merely sit near the chain. A target must follow from the exact endpoint relationship, not from the chain looking impressively complicated across the board.

Build short chains before attempting long ones

Start with a bivalue cell or conjugate pair near another strong relationship. Write the candidates as letters if the glowing symbols are hard to track in a long argument. Move from a strong link to a weak link, then back to a strong link, checking each shared row, column, box, or cell. Stop when you reach a useful endpoint rather than extending the chain for decoration. Four or five links are enough for many real eliminations. On an Extreme Shapedoku puzzle, use Notes with smart cleanup and recheck the candidate map after every final placement. If a chain fails to produce a target, leave it. A valid chain can still be useless in the current position. Return to singles, locked candidates, subsets, and fish before searching elsewhere. The best advanced deduction is not the longest one. It is the shortest proof that removes a candidate without asking you to remember an entire small novel.

1. Choose a candidate with a verified strong link.
2. Cross the strong link by reading false implies true.
3. Use a weak link from the true candidate to force the next candidate false.
4. Continue alternating strong and weak links without skipping a step.
5. Stop when the endpoints prove a placement or elimination.
6. Trace the chain backward before changing the board.