SuuudokuuuAIC Sudoku Technique
Updated
An Alternating Inference Chain, or AIC, follows a path of alternating strong and weak links across candidates and cells, generalising X-Chains, XY-Chains and simple coloring into one technique that is not limited to a single digit or to bivalue cells.
When an AIC applies
Every chain technique earlier on this list is a restricted case of an AIC. An X-Chain sticks to one digit. An XY-Chain sticks to bivalue cells. Simple coloring sticks to strong links only. An AIC drops all three restrictions: a link can be a strong or weak link between two candidates of the same cell, or between two cells that share a unit and a digit, and consecutive links can switch digits entirely as long as the chain keeps alternating strong and weak.
Two link types feed an AIC. A cell link connects two different candidates in the same cell — always at least weak, since a cell holds only one value, and strong as well when the cell is bivalue. A unit link connects the same candidate in two cells that share a row, column or box — weak if both cells merely allow the digit, strong if the unit has no other cell that can hold it.
An AIC can close in two ways. If the two ends land on different cells that share a peer, the shared candidate can be erased from that peer, just as in an X-Chain or XY-Chain. If the chain instead loops back to a second candidate of its own starting cell, or to a cell in the same unit as its start, forcing a contradiction, the starting candidate itself is eliminated.
Worked example
Column 7’s only two candidates for 1 are r4c7 and r6c7, a strong link. r9c7 is bivalue at 1 and 5. Column 7’s only two candidates for 5 are r2c7 and r9c7, another strong link. r2c7 is bivalue at 5 and 9.
Suppose r4c7 is 9. Then r4c7 is not 1, so the strong link on 1 forces r6c7 to be 1. r6c7 and r9c7 share column 7, so r9c7 cannot be 1 either; since r9c7 only has 1 and 5 as candidates, it must be 5. r9c7 being 5 means r2c7, its strong-link partner on 5, is no longer needed to hold 5, so r2c7 is left with its other candidate and must be 9. But r2c7 and r4c7 also share column 7, and a column cannot hold the same digit twice — r4c7 was already assumed to be 9. The chain contradicts its own starting assumption, so r4c7 cannot be 9 after all. The solver removes it and leaves 1 and 4 behind.
How to spot an AIC
- List strong and weak links freelyFor any digit and any cell, note bivalue cell links and unit strong links the same way as for simpler chains, without restricting to one digit.
- Alternate strong and weak links across digitsBuild a path where the link type alternates, allowing the active digit to change from one link to the next as the chain moves through a cell.
- Check for a shared peer or a loopSee whether the chain’s two ends share a peer for the elimination to land on, or whether the chain loops back to contradict its own start.
- Apply the matching elimination ruleErase the shared candidate from a common peer, or eliminate the starting candidate outright if the chain contradicts itself.
Common mistakes
- Breaking the strong-weak alternation. Two strong links or two weak links in a row do not carry the same guarantee as a proper alternation.
- Forgetting that a cell link is only strong when the cell is bivalue. A three-candidate cell only gives a weak link between any two of its candidates.
- Missing the self-contradiction ending. A chain that loops back to the start does not need a separate target cell; it eliminates its own first candidate.
- Assuming an AIC always tracks one digit. Switching digits mid-chain, through bivalue cell links, is exactly what separates an AIC from an X-Chain.
AIC FAQ
Is an X-Chain a special case of an AIC?
Yes. An X-Chain is an AIC restricted to unit links on a single digit, with no cell links switching the active digit along the way.
Is simple coloring a special case of an AIC too?
Yes. Coloring is an AIC restricted to strong links only, which is why an AIC can find eliminations a coloring network on its own would miss.
Why does the worked example eliminate a candidate from a cell inside the chain?
Because the chain loops back to contradict its own starting assumption rather than reaching an outside peer. That self-contradiction is enough to rule the starting candidate out directly.
Is an AIC the hardest technique on this list?
It is the most general. Because it subsumes the fish, wing, and simpler chain patterns, an AIC search can find eliminations that no single one of the earlier named techniques can reach on its own, which is why it sits last here.