Sudoku Chaining Techniques: A Roadmap From Colouring to AIC

What Sudoku chaining techniques do

Sudoku chaining techniques trace a sequence of linked candidates and follow the consequences of one assumption until they force a conclusion. Two link types do all the work. A strong link says if this candidate is false then that one must be true, which happens when a digit has only two possible cells in a house. A weak link says if this candidate is true then that one must be false, which happens when two candidates share a house and cannot both hold. Chains alternate these links. Because the endpoints of a well-built chain leave you with a guaranteed truth somewhere along it, you can eliminate the chained digit from any cell that sees both ends. Everything from basic colouring to advanced forcing chains is just a longer or fancier version of this alternation.

Simple colouring and X-chains

The gentlest entry is simple colouring on one digit. Pick a candidate such as the green circle for 4, find houses where 4 has exactly two spots, and paint one blue and its partner yellow, following every conjugate pair so the cluster is two-toned. One colour is the truth. Two eliminations follow: if two cells of the same colour share a house, that colour is impossible, so the other colour is all true; and any uncoloured cell seeing both a blue and a yellow candidate cannot be 4. For instance, if r2c3 is blue, r2c9 is yellow, and r5c3 sees both, then 4 leaves r5c3. An X-chain stretches this into a single path on one digit, mixing strong and weak links. Imagine 7 running r1c2 strong to r1c8, weak to r6c8, strong to r6c4. The ends r1c2 and r6c4 cannot both be false, so any cell seeing both loses 7. A short X-chain like this is often the same thing a book calls a skyscraper or a turbot fish.

XY-chains across bi-value cells

XY-chains move through bi-value cells and can carry different digits from link to link. Each cell in the chain has two candidates, so you enter on one and leave on the other, using the shared digit between neighbours as the weak link. If the chain begins and ends on the same digit, say it starts forcing 5 to be false at one end and true at the other, then any cell seeing both ends drops 5. XY-chains are powerful because bi-value cells are common in the mid-game once pairs have thinned the grid. They are the natural next step after the XY-Wing, which is really just the shortest possible XY-chain of three cells, so learning wings first makes chains feel familiar rather than foreign.

AICs and forcing chains

Alternating inference chains, or AICs, are the general form that contains all of the above. An AIC strictly alternates strong and weak links and may pass through whole houses as grouped nodes, letting a chain treat several candidates in one box as a single unit. Forcing chains take a different stance: you assume a cell is a particular digit and follow the fallout down two or more branches, and if every branch reaches the same conclusion, that conclusion is true regardless of the assumption. Forcing chains and nets can solve almost anything, but they are slow and error-prone by hand, so most solvers reserve them for the final locked door on an Extreme puzzle. Because Shapedoku renders each digit as its own shape, tracing a chain is kinder on the eyes, and its Notes view with smart cleanup keeps your candidate map honest; when a chain defeats you, the free solver at shapedoku.com can show the shortest one available.

1. Learn simple colouring on one digit until both eliminations are automatic.
2. Extend to X-chains, which quietly teaches skyscrapers and turbot fish.
3. Add XY-chains through bi-value cells once naked pairs feel comfortable.
4. Study grouped AICs so a whole box can act as a single chain node.
5. Keep forcing chains as a last resort for the very hardest puzzles.