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[Chain Reasoning②] Building: Alternating Rules & State Transfer

2025-06-06 · 15 min read

In the previous article, we learned about the two basic building blocks of chain reasoning: strong links and weak links. This article will further explore how to combine these links to build complete inference chains and derive valid conclusions from them.

Chain Reasoning Series (2/3)

← ① Basics ② Building (Current) ③ Applications →

This article continues from the basics, please make sure you've read article ①

Chain Building: Strong and weak links alternately connect to form complete inference paths

Basic Structure of Chains

A chain is a sequence of candidate nodes and links. Each node represents a candidate (a specific digit in a specific cell), and adjacent nodes are connected by strong links or weak links.

         Formal Representation of Chains:

        A ═ B - C ═ D - E ═ F

        Where:

        • A, B, C, D, E, F are candidate nodes

        • ═ represents a strong link

        • - represents a weak link

        • The entire chain describes a logical inference path from A to F

Representation of Candidate Nodes

In chain reasoning, we typically represent candidate nodes in the following ways:

Position + Digit: e.g., R3C5(4) means "candidate 4 in row 3, column 5"
Abbreviated form: e.g., r3c5=4 or (3,5)4

Each node represents an assertion: the candidate is true (the cell is filled with that digit) or false (the candidate is eliminated).

Alternating Rule of Links

The core rule for building valid chains is: strong links and weak links must alternate. This rule ensures the validity of logical inference.

Why Alternation is Necessary?

Strong link: Transfers "false → true", cannot transfer "true → true"
Weak link: Transfers "true → false", cannot transfer "false → false"

If two strong links are used consecutively (false → true → ?), the second strong link cannot continue the transfer;
If two weak links are used consecutively (true → false → ?), the second weak link cannot continue the transfer.
Only by alternating can we form a continuous inference chain.

Special Case: Consecutive Strong Links
When multiple strong links appear consecutively (e.g., A ═ B ═ C ═ D), it may seem to violate the alternating rule, but this is actually valid.

Reason: A strong link's condition is "exactly one true, one false", while a weak link's condition is "at most one true". Since "exactly one" necessarily satisfies "at most one", every strong link is also a weak link.

Interpretation:
A ═ B ═ C ═ D
Can be understood as:
A ═ B - C ═ D (the middle strong link acts as a weak link)

Therefore, consecutive strong links are not an error in notation; rather, the middle strong link implicitly assumes the role of a weak link.

Alternating rule of strong and weak links: Only alternation can form valid inference chains

Valid Chain Patterns

According to the alternating rule, valid chains must be in one of the following forms:

1 Starting with strong link, ending with strong link:
A ═ B - C ═ D - E ═ F
Chain length is odd number of links (strong-weak-strong-weak-strong)

2 Starting with weak link, ending with weak link:
A - B ═ C - D ═ E - F
Chain length is odd number of links (weak-strong-weak-strong-weak)

3 Starting with strong link, ending with weak link (or vice versa):
A ═ B - C ═ D - E
Chain length is even number of links

Coloring Concept

Coloring is a powerful mental tool for understanding chain reasoning. We alternately assign two "colors" to nodes on the chain, representing two possible true/false states.

         Coloring Rules:

        Assign color A (e.g., blue) to the starting point of the chain
The next node connected by a strong link is assigned the opposite color B (e.g., green)
The next node connected by a weak link is assigned the same color
Continue alternating until the end of the chain

    

Coloring concept: Strong links flip colors, weak links maintain colors

Logical Explanation of Coloring

Strong Strong links flip colors:
Both ends of a strong link are "exactly one true, one false". If one end is false, the other must be true; if one end is true, the other must be false.
Therefore, the colors at both ends of a strong link are opposite, representing opposite true/false states.

Weak Weak links maintain colors:
Both ends of a weak link are "at most one true". If we assume one end is true (color A = true), the other end must be false.
But if one end is false, the state of the other end is uncertain. Therefore, when coloring, we focus on the scenario "if the previous node is true", so the node after a weak link has the same "true/false assumption" as the previous node.
(Note: "maintaining color" here refers to behavior when tracking "true" state transfer)

Core Meaning of Coloring:
Same-color nodes: Either all true or all false
Different-color nodes: Opposite true/false states

Through coloring, we can quickly determine the true/false relationship between any two nodes on the chain.

Two Perspectives on State Transfer

Understanding chain reasoning has two complementary perspectives: tracking "true" state and tracking "false" state.

Perspective One: Tracking "True" State Transfer

Assume the starting point of the chain is true, observe how this "true" state transfers along the chain:

A ═ B - C ═ D - E ═ F

Assume A = true
→ A-B is a strong link, when A is true B can be true or false, state is uncertain

(Tracking "true" cannot effectively transfer on pure strong links)

A - B ═ C - D ═ E - F

Assume A = true
→ A-B is a weak link, A true → B must be false
→ B-C is a strong link, B false → C must be true
→ C-D is a weak link, C true → D must be false
→ D-E is a strong link, D false → E must be true
→ E-F is a weak link, E true → F must be false

Conclusion: A true → F false

Perspective Two: Tracking "False" State Transfer

Assume the starting point of the chain is false, observe how this "false" state transfers along the chain:

A ═ B - C ═ D - E ═ F

Assume A = false
→ A-B is a strong link, A false → B must be true
→ B-C is a weak link, B true → C must be false
→ C-D is a strong link, C false → D must be true
→ D-E is a weak link, D true → E must be false
→ E-F is a strong link, E false → F must be true

Conclusion: A false → F true

         Key Observation:

        For chains starting and ending with strong links:

        • Start false → end true (by tracking "false" state)

        • Start and end have opposite colors

        For chains starting and ending with weak links:

        • Start true → end false (by tracking "true" state)

        • Start and end have the same color

Deriving Conclusions from Chains

After building a valid chain, how do we derive conclusions that can be used for elimination? This depends on the structure of the chain and the relationship between the endpoints.

Conclusion Type One: Weak Link Between Endpoints

1 Scenario: The two endpoints A and F can "see" each other (weak link exists)
Chain: A ═ B - C ═ D - E ═ F, and A and F are in the same row/column/box or same cell
Analysis:
• If A false → F true (chain transfer)
• If A true → F false (weak link between A and F)
Conclusion: Regardless of A's truth value, at least one of A and F is true (if A is false then F is true, if A is true then A itself is true).
Application: Other same-digit candidates that can see both A and F can be eliminated!

Conclusion Type Two: Same Candidate at Both Ends

2 Scenario: The endpoints are the same candidate in the same cell (forming a loop)
Chain: A ═ B - C ═ D - E ═ A (returns to start)
Analysis:
• If A false → ... → A true (contradiction!)
Conclusion: A cannot be false, so A must be true.

Conclusion Type Three: Coloring Conflict

3 Scenario: Two same-color nodes on the chain have a weak link between them (they can see each other)
Analysis:
• Same color means they have the same true/false state
• Weak link means they cannot both be true
Conclusion: These two nodes must both be false. All same-color nodes are false, all opposite-color nodes are true.

Three main ways to derive conclusions from chains

Alternating Inference Chain (AIC)

Alternating Inference Chain (AIC) is the standard form of chain reasoning. Its characteristics are:

Strong links and weak links strictly alternate
Starts with a strong link, ends with a strong link
A weak link exists between the chain endpoints

         Standard Form of AIC:

        A ═ B - C ═ D - ... - Y ═ Z

        Where a weak link exists between A and Z (they can see each other).

        Conclusion: At least one of A and Z is true, therefore other candidates that can see both A and Z can be eliminated.

AIC is a powerful framework, and many specific techniques can be viewed as special forms of AIC:

X-Wing, Swordfish: Can be described using AIC
Skyscraper: A simple AIC
XY-Wing: A three-node AIC
XY-Chain: An AIC composed of bivalue cells

Practical Tips for Building Chains

In actual puzzle solving, building effective chains requires some skills and experience:

1 Start from bivalue cells:
Bivalue cells provide both strong links (two digits within the cell) and easily discoverable weak links (same-digit candidates in the same unit). They are ideal starting points for building chains.

2 Look for conjugate pairs:
In rows, columns, and boxes, look for digits that appear only twice. The conjugate pairs they form are important sources of strong links.

3 Pay attention to link type identification:
The same pair of candidates may have both a strong link and a weak link simultaneously (such as bivalue cells or conjugate pairs). When building chains, be clear about which type of link you are using.

4 Goal-oriented approach:
If you want to eliminate a certain candidate X, try to build a chain such that both endpoints can "see" X.

Common Mistakes:

Using two weak links consecutively (cannot transfer state)
Misjudging a weak link as a strong link (leads to incorrect conclusions)
Forgetting to verify the relationship between chain endpoints (cannot derive conclusions)

Note: Consecutive strong links are allowed, because strong links can be used as weak links (see "Special Case" above)

Next Steps

This article introduced how to build chains and derive conclusions from them. In the next article, we will discuss:

Various application patterns of chains (open chains, closed chains, loops)
Unified understanding of common chain techniques
Grouped links and complex chain structures
Discontinuous loops and advanced reasoning

Related Reading:

Chain Reasoning Basics - Review the concepts of strong and weak links
XY-Chain Technique - Specific application of chain reasoning
Skyscraper Technique - Example of simple AIC

Chain Reasoning Basics Chain Application Patterns Techniques Index