Techniques / Chains

Chains, loops and forcing nets

Alternating implications, followed until both ends say something.

18 techniques in this family, introduced at tier 7. How to read a sudoku grid explains the notation used below.

What these have in common

A chain is a sequence of links where truth alternates. Follow it far enough and you learn something about the two ends without ever learning which of them is true.

Why the logic holds

Two kinds of link. A strong link says at most one of these is false, so if A is false B is true. A weak link says at most one is true, so if A is true B is false. Alternate them along a chain and the two ends cannot both be false. Anything that sees both ends is therefore eliminated. Close the chain into a loop and the constraint tightens further, because now every weak link in the loop is exclusive.

When to reach for it

When everything cheaper has failed. Chains will solve almost anything, which is precisely why they are not the first thing to try: the search space is large and the reward for finding a short chain quickly is much better than for grinding out a long one.

How the members differ

X-Chains use a single digit, XY-Chains run through bivalue cells, and the Alternating Inference Chain is the general form that subsumes both. Grouped AICs let a node be several cells at once, and AICs with Almost Locked Set or Almost Hidden Set nodes let a node be a whole structure. Forcing chains work differently: instead of one alternating path they test every candidate of a cell, house or digit in turn and keep whatever consequence all branches share. Nishio assumes a single candidate and looks for a contradiction, and the dynamic and Kraken variants push that further.

The mistake to avoid

Losing the distinction between finding a proof and doing a search. A forcing net that branches repeatedly is closer to guess and check than to deduction, and a puzzle that genuinely needs one is a puzzle whose setter did not test it. Proven Sudoku never publishes a board that needs anything in this family.

Every technique in this family

• AIC with AHS NodesThe same as above but using sets of numbers that are hiding rather than sets of squares.
• AIC with ALS NodesA chain that can step through a small set of squares holding one more number than it has room for.
• Alternating Inference ChainA chain that switches between two kinds of step.
• Bowman's BingoPick a digit, assume it, and write out everything that follows until you either finish or hit a contradiction.
• Cell Forcing ChainTake one square and try each number it could hold.
• Continuous Nice LoopA loop where the reasoning closes back on itself with no weak spot.
• Digit Forcing ChainTake one number in one square.
• Discontinuous Nice LoopA loop that almost closes but has one weak spot.
• Dynamic Forcing Chain with ContradictionAssume a digit is right.
• Dynamic Forcing Chain with Double ImplicationTry both options for a square and follow each one out fully.
• Forcing Chain (general form)Pick something and assume it.
• Grouped AICA chain where a step can use a whole group of squares at once instead of a single square.
• KrakenTake a pattern that nearly works.
• NishioAssume a number goes in a square, then check whether the rest of that number can still fit everywhere it needs to.
• Nishio Forcing NetThe same check as Nishio.
• Unit Forcing ChainTake one number in one row, column or box and try each home it could have.
• X-ChainA chain about one number only.
• XY-ChainA chain of squares that each hold exactly two numbers.