AIC

An alternating inference chain proves one of its two ends is true.

An Alternating Inference Chain is the master pattern behind almost every advanced chaining move in Sudoku. It strings together (cell, digit) nodes using links that strictly alternate: a strong link says "if this one is false, that one must be true", while a weak link says "if this one is true, that one must be false". When both ends of the chain assert the same digit, any cell that sees both ends can lose that digit. Learning the AIC (alternating inference chain) sudoku technique means X-Chains and XY-Chains come along for free — they are simply special cases of the same idea.

Worked example

An alternating inference chain over column 1: through the strong (within-cell) and weak (column) links of R1C1{1,2} and R4C1{1,2}, one end must be 1 — so R7C1, seeing both, loses 1.121219
An alternating inference chain over column 1: through the strong (within-cell) and weak (column) links of R1C1{1,2} and R4C1{1,2}, one end must be 1 — so R7C1, seeing both, loses 1.

On this grid, column 1 holds two bivalue cells: R1C1 and R4C1 are both {1,2}. Follow the chain from R1C1=1. Inside R1C1 a strong link forces R1C1=2 to be the alternative, then a weak link along column 1 pushes across to R4C1=2, and inside R4C1 a strong link swings back to R4C1=1. Both ends now assert a 1, so at least one of R1C1 and R4C1 must be a 1. R7C1 sees both of them down column 1, so it cannot also be a 1 — the 1 is eliminated from R7C1.

When to use it

Reach for an AIC once the named patterns — wings, fish, coloring — have stalled but the grid is still dense with bivalue cells and conjugate (strong-link) pairs. Scan those cells and try to build a chain that alternates strong and weak links, switching digits through bivalue cells where it helps. If the chain starts and ends on the same digit assertion, look for any cell that sees both ends: that cell loses the shared digit.

How to use it

  1. Build an alternating chain. Link candidates by alternating strong links (if one is off the other is on) and weak links (they cannot both be on), starting and ending on a strong link. Links may switch digit through a cell.
  2. Read the two ends. A chain that starts and ends on a strong link proves at least one of its two end candidates is true — you do not know which, but one of them holds.
  3. Eliminate what sees both ends. Any candidate that is weakly linked to BOTH ends would force both ends off, which is impossible — so that candidate can be removed.
Practice on a real puzzle

Frequently asked questions

What is an AIC in Sudoku?

An AIC, or Alternating Inference Chain, is a sequence of (cell, digit) nodes joined by links that alternate between strong ('at least one of the pair is true') and weak ('at most one of the pair is true'). When both ends of the chain claim the same digit, every cell that can see both ends must drop that digit. In the example above, the chain across R1C1 and R4C1 both asserts a 1, so R7C1 loses its candidate 1.

How do I spot an AIC on the grid?

Start hunting once the obvious wings and fish have dried up but bivalue cells and conjugate pairs are still plentiful. Pick a node and try to extend outward, alternating strong and weak links and switching digits through bivalue cells as needed. A useful AIC begins and ends on the same digit assertion — then look for a cell, like R7C1 here, that sees both ends and can be pruned.

What is the difference between an AIC and an X-Chain or XY-Chain?

They are the same tool at different levels of specialization. An AIC is the general form; an X-Chain is an AIC that never leaves a single digit, and an XY-Chain is an AIC that travels through bivalue cells, swapping digits as it goes. The column-1 example above stays on the digits 1 and 2 through two bivalue cells, so it reads like a short XY-style AIC.