Alternating Inference Chains

Alternating Inference Chains are the general form of every chain technique in Sudoku. Instead of following one digit, you hop between digits and cells, alternating two kinds of link, and read off what must be true at the ends. Once you can build them, X-Wings, wings and single-digit cycles all become special cases.

What the pattern is

A chain is a sequence of candidates — each one a digit in a specific cell. Two link types connect them:

An Alternating Inference Chain starts with a strong link, ends with a strong link, and alternates strong, weak, strong, weak all the way through. That structure guarantees one thing: at least one of the two end candidates is true. You never learn which — you do not need to.

When to look for it

Reach for chains once singles, locked sets, fish and wings are exhausted and the grid is full of bivalue cells. Chains need complete, trustworthy pencil marks; a single stale candidate invents links that do not exist. Start from a bivalue cell or from a digit that has exactly two places in some unit, because those are the strong links that seed everything.

How to apply it, step by step

1. Pick a starting candidate and a strong link leaving it.
2. From the candidate you land on, take a weak link: same cell, different digit, or same digit in the same unit.
3. From there take another strong link. Keep alternating.
4. Stop when you reach a candidate of the same digit as your start.
5. Eliminate that digit from every cell that sees both ends of the chain.

If instead the chain closes on itself with the alternation intact, you have a continuous loop: every weak link in it becomes exclusive, so all other candidates of that digit in the shared unit go, and any extra candidate in a looped cell goes too. If the alternation breaks at one node, the chain is discontinuous: a node with two weak links pointing at it is false, and a node with two strong links pointing at it is true.

A worked example

Three strong links, two weak links, both ends holding digit 5.

box 1: 5 only in r1c2, r3c1        r3c1 = {5,8}
col 1: 8 only in r3c1, r9c1        r9c1 = {5,8}
box 7: 5 only in r9c1, r7c2

r1c2#5 = r3c1#5 - r3c1#8 = r9c1#8 - r9c1#5 = r7c2#5

read as: r1c2 not 5 -> r3c1=5 -> r3c1 not 8 -> r9c1=8
                     -> r9c1 not 5 -> r7c2=5

So r1c2 or r7c2 holds the 5. Both sit in column 2, so every other cell of column 2 loses its 5 — here r4c2, r5c2 and r6c2. Note that the eliminations must see both ends; a cell seeing only one end proves nothing.

Common mistakes

• Using a weak link where the pattern only supports "not both true" and then reading it as a strong link. A cell with three candidates gives weak links, not strong ones.
• Ending on a weak link. The final link must be strong or the last node is not forced.
• Eliminating on cells that see just one end.
• Chaining from pencil marks you have not refreshed since your last placement.

Where you meet it

Alternating Inference Chains take over where X-Cycles stop, because they are free to change digit at every cell. Expect to need them in Master puzzles and to lean on them constantly at Legendary. Build short ones first: a five-node chain solves far more positions than most solvers expect.