How Chains Work

Outcome

You can trace an alternating inference chain, determine what its endpoints prove, and reject a valid-looking chain that produces no useful conclusion.

Intro copy

A chain is not useful because it is long.

It is useful because it proves something.

Your job is:

1. propagate the states correctly;
2. identify the endpoint conclusion;
3. make the resulting placement or elimination.

Step 1 — Abstract four-node chain

A = B - C = D

Start:

A = false

Learner traces:

B = true
C = false
D = true

Every transition asks:

Which link rule did you use?

This prevents state clicking without understanding.

Step 2 — Reverse the chain

Start:

D = false

Trace backward.

Conclusion:

A false forces D true, and D false forces A true.

Therefore at least one endpoint is true.

Step 3 — Add a target

Place an external candidate T that conflicts with both endpoints.

Prompt:

If one endpoint is always true, can T survive?

No.

Remove T.

This is the first full chain deduction.

Step 4 — Put the same logic on Sudoku

Use one digit.

Solid = Strong.

Dashed = Weak.

Learner:

1. identifies link type at each step;
2. traces endpoint A false;
3. traces endpoint B false in reverse;
4. proves one endpoint true;
5. removes the same digit from a common peer.

Step 5 — Valid chain, no useful move

Show an alternating chain with correct endpoints but no candidate seeing both and no contradiction.

Prompt:

Is the chain logically valid?

Yes.

Does it solve anything right now?

No.

Feedback:

Chains are tools for deductions, not collectibles.

Step 6 — Find the broken chain

Show a proposed path containing one invalid Strong Link: the source unit actually has a third candidate.

Learner identifies the broken edge.

This trains proof verification instead of trusting rendered lines.

Step 7 — Recognize familiar named patterns

Show:

• Skyscraper;
• Two-String Kite;
• W-Wing preview.

Overlay the link structure.

Prompt:

Which parts of the named pattern are doing the logical work?

The goal is to collapse “many techniques” into one inference system.

Practice

Include:

• abstract state propagation;
• one-digit chain;
• broken Strong Link;
• valid-but-useless chain;
• named-pattern-to-chain conversion.

Hint ladder

1. “Start from one endpoint false.”
2. “Strong: false → true.”
3. “Weak: true → false.”
4. “Continue alternating.”
5. “What does the opposite endpoint become?”
6. “Which candidates conflict with both endpoints?”

Completion

• three correct chain traces;
• two endpoint eliminations;
• one broken-chain diagnosis;
• one valid-but-unproductive classification.

Next

W-Wing

Use the same inference language in a compact named pattern.

Reference

Read Sudoku Chains for X-Chain, XY-Chain, notation, and the later path into AICs.