Chains: X-Chain, XY-Chain and AIC

Extreme

Chains are how the hardest puzzles are solved. A chain links notes one after another, and proves that one of its two ends must be true. Anything that clashes with both ends goes. Extreme puzzles in Sudoku Knockout need them, and never anything harder: no puzzle needs a guess.

Two kinds of link

  • Strong link (=): if one note is false, the other is true. A digit with only two places in a row, column or box, or a cell with only two notes.
  • Weak link (–): if one note is true, the other is false. The same digit in two cells that see each other, or two notes in the same cell.

A chain alternates strong and weak links, and starts and ends with a strong one. Follow it from one end assuming that end is false: each strong link makes the next note true, each weak link makes the next one false, and the far end comes out true. So one of the two ends is true, whatever happens.

In the pictures, solid lines are strong links and dashed ones weak; the ends are orange. Below each, the chain is written out: C2(8) is the 8 in C2.

X-Chain: one digit

An X-Chain follows a single digit through lines where it has two places. Skyscrapers and kites are the shortest X-Chains; longer ones reach across the board.

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The chain: C2(8) = C5(8) – F5(8) = D4(8) – D1(8) = G1(8). One of its ends, C2(8) or G1(8), is true, and the 8 in I2 clashes with both, so it goes.

XY-Chain: cells with two notes

An XY-Chain hops between cells with two notes, each seeing the next and sharing a digit with it. Inside each cell the link is strong (it’s one or the other), and between cells it’s weak. An XY-Wing is the shortest one.

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The chain: A6(2) = A6(5) – A3(5) = A3(9) – A2(9) = A2(3) – F2(3) = F2(2). One of its ends, A6(2) or F2(2), is true, and the 2 in F6 clashes with both, so it goes.

AIC: both kinds

An alternating inference chain mixes both: digits with two places in a line, and cells with two notes. It’s the most general chain, and it can clear notes in a cell that’s part of the chain, as in this one.

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The chain: I3(5) = B3(5) – B5(5) = B5(6) – B4(6) = I4(6). One of its ends, I3(5) or I4(6), is true, and the 5 in I4 clashes with both, so it goes.

How to find them

  • Start from a note you’d like to remove and look for strong links around it on both sides: a chain is a way to connect them.
  • Keep chains short. The game’s solver looks for the shortest one that works, and so should you: the longer the chain, the easier it is to slip.

Try it on a real puzzle

Extreme puzzles in Sudoku Knockout need techniques like this one, and never a guess. When you’re stuck, a hint shows you the step.

Play an Extreme puzzle