Alternating Inference Chains (AIC)

Alternating Inference Chains, or AICs, are the master framework that every other chain is a special case of. An X-chain is an AIC on one digit; an XY-chain is an AIC through bi-value cells. Learn the AIC and you hold the key to them all.

The idea

An AIC is a sequence of candidates joined by links that alternate in strength. A strong link says "if this is false, that is true" — the two ends cannot both be false. A weak link says "if this is true, that is false" — they cannot both be true. Begin and end the chain on strong links, and the two endpoints obey a simple truth: at least one of them is true. From that, two eliminations follow. If both endpoints are the same digit, any cell seeing both loses that digit. If the endpoints are two candidates in the same cell, that cell must be one of them, so its other candidates go.

A worked example

Follow the digit 7 through five alternating links: 7r1c1 =strong= 7r4c1 (column 1 holds 7 in only those two cells) −weak− 7r4c5 (row 4) =strong= 7r7c5 (column 5) −weak− 7r7c8 (row 7) =strong= 7r1c8 (column 8). The chain begins and ends on a strong link, so at least one endpoint — 7r1c1 or 7r1c8 — must be true. Both are 7s in row 1, so any cell that sees both, such as r1c4, loses its 7. Thread the same idea through bi-value cells instead of one digit and you have an XY-chain; the AIC framework covers every case at once.

An AIC on 7: strong links (solid) alternate with weak links (dashed). One of the two endpoints — r1c1 or r1c8 — must be 7, so r1c4, seeing both along row 1, loses its 7.

Step by step

Start on a strong link from a candidate you want to test.
Alternate links — strong, weak, strong, weak — through cells and units.
End on a strong link so both endpoints cannot be false together.
Apply the endpoint rule — clear the shared digit from common peers, or resolve the shared cell.

Common mistakes

Two weak links in a row. The chain must alternate; consecutive weak links prove nothing.
Ending weak. An AIC used for eliminations should start and end on strong links.
Reading the wrong conclusion. Same-digit ends eliminate on shared peers; same-cell ends resolve that cell — do not mix them up.

Which Sudoku types is this best for?

The same logic applies across variants, but it pays off more in some than others.

Great fit Classic Sudoku The unifying chain technique — X-chains, XY-chains and coloring are all AICs in disguise. → Works Killer Sudoku Useful once cage arithmetic stalls, linking candidates across the grid. → Works Sudoku X Diagonal strong links give chains extra rails to travel along. → Limited Samurai Sudoku Deep single-grid chains rarely pay off across five grids. →