[Chain Reasoning③] Applications: Pattern Classification and Advanced Structures
In the previous two articles, we learned about the concepts of strong links and weak links and chain building and propagation rules. This article will systematically introduce various application patterns of chain reasoning and demonstrate how to understand specific techniques using a unified chain framework.
Classification by Form: Open Chains vs Closed Chains
Based on whether the chain's endpoints connect, chains can be classified as open chains and closed chains (loops).
Open Chain
- The chain has a clear start and end point
- The endpoints do not connect
- Conclusions are based on the relationship between endpoints
Open chains are the most common chain structure. When the two ends of a chain have a weak link relationship (can see each other), candidate elimination can be performed.
A ═ B - C ═ D - E ═ FIf A and F can see each other (weak link exists), then one of A or F must be true, allowing elimination of other same-digit candidates that can see both A and F.
Closed Chain / Loop
- The chain's endpoint connects back to the starting point, forming a loop
- Can be used to directly determine the truth value of certain candidates
- The parity of the loop determines the conclusion type
Closed chains can be divided into continuous loops (Nice Loop) and discontinuous loops based on their structure.
All nodes in the loop can be divided into two color groups: same color means same truth value, different colors mean opposite.
The candidate at the contradiction point can be determined to be true or false.
Classification by Content: Single-digit Chains vs Bi-value Chains
Based on the type of candidates in the chain, chains can be classified as single-digit chains and bi-value chains.
Single-digit Chain
All nodes in the chain are candidates of the same digit. Links come from conjugate pairs (only two positions in a unit have this digit).
- Tracks only one digit's relationships across different positions
- Strong links come from conjugate pairs
- Weak links come from other positions in the same unit
- Representative techniques: X-Wing, Skyscraper, X-Chain
Bi-value Chain (XY-Chain)
All nodes in the chain come from bi-value cells (cells with only two candidates). Links transition between different digits.
- All nodes come from bi-value cells
- The two candidates within a cell form a strong link
- Adjacent cells sharing a candidate form a weak link
- Representative techniques: XY-Wing, XY-Chain, Remote Pairs
XY-Chain is an alternating chain composed purely of bi-value cells. For example:
R1C1{3,5}(5) - R1C4{5,7}(7) - R3C4{7,9}(9) - R3C8{4,9}(4)Start is 3, end is 4; candidates 3 and 4 that can see both endpoints can be eliminated.
Mixed Chain (AIC)
The chain contains both single-digit chain nodes and bi-value chain nodes. This is the most versatile chain structure.
- Flexibly combines various link sources
- Can freely transition between single-digit and bi-value nodes
- Most expressive power, can discover more eliminations
- Representative technique: AIC (Alternating Inference Chain)
Grouped Links
Grouped links treat multiple candidates as a single entity in chain reasoning. This greatly expands the application range of chain techniques.
When all candidate positions of a digit within one unit (row/column/box) are concentrated in the intersection area of another unit, these positions can be treated as a "group".
Example: Digit 5 in box 1 appears only in three positions in row 1; these three positions can participate in chains as a group.
Grouped Strong Links
When a group and another candidate/group satisfy the relationship "exactly one is true", a grouped strong link exists.
In row 1's other positions (box 2 and box 3), digit 5 appears only in position R1C8, as single point B.
A strong link exists between group A and B: row 1 must have one 5, either in group A (box 1) or in B (R1C8).
Grouped Weak Links
When a group and another candidate/group are in the same unit, a grouped weak link exists between them.
Discontinuous Loop
A discontinuous loop is a special type of closed chain where a "discontinuity" occurs at some node—the two adjacent links at that node are the same type (both strong links or both weak links).
- Type 1 (Two consecutive strong links):The candidate at the discontinuous point must be false
- Type 2 (Two consecutive weak links):The candidate at the discontinuous point must be true
Type 1: Two Consecutive Strong Links
A ═ B - C ═ D - ... ═ A (returns to start with strong link)Assume A is false:
→ through loop propagation → A is true (contradiction!)
Assume A is true:
→ the other end of the last strong link (let's call it X) can be true or false → no contradiction
However, if we track "false" starting from X:
X false → A true (strong link) → ... → X true
This means X cannot be false, so X is true, therefore A is false.
Conclusion: The discontinuous point A must be false.
Type 2: Two Consecutive Weak Links
A - B ═ C - D ═ ... - A (returns to start with weak link)Assume A is true:
→ through loop propagation → A is false (contradiction!)
Conclusion: The discontinuous point A must be false... wait, that doesn't seem right?
Actually, for Type 2, we need to analyze more carefully. The correct conclusion is:
If tracking "true" from A eventually returns to A requiring A to be false, this creates a contradiction.
Conclusion: The discontinuous point A must be true.
Understanding Common Techniques Through Chain Reasoning
Many seemingly different Sudoku techniques can be understood uniformly through the chain reasoning framework.
| Technique Name | Chain Description | Chain Characteristics |
|---|---|---|
| X-Wing | 4-node single-digit chain loop | Conjugate pairs in 2 rows and 2 columns forming rectangle |
| Skyscraper | 4-node single-digit open chain | Two conjugate pairs sharing one end |
| 2-String Kite | 4-node single-digit open chain | Row and column conjugate pairs connected through box |
| XY-Wing | 3-node bi-value chain | Pivot connecting two wings |
| XY-Chain | Multi-node bi-value chain | Pure bi-value cell chain |
| Remote Pairs | Even-node bi-value chain | Bi-value cell chain with same candidates |
| W-Wing | Mixed chain | Bi-value cells connected through conjugate pair |
| AIC | General mixed chain | Alternating chain with any combination |
Chain Technique Selection Strategies
In actual solving, how do you choose the appropriate chain technique? Here are some suggestions:
Start with simple techniques like conjugate pair reasoning and Skyscraper, then try complex AICs.
Bi-value cells are excellent materials for building chains. When there are many bi-value cells, prioritize XY-Wing and XY-Chain.
For a digit that's difficult to eliminate, check if it forms conjugate pairs in various units—you might discover single-digit chains.
If you want to eliminate a specific candidate, try building a chain where both ends can "see" that candidate.
The Value of Chain Reasoning
The value of learning chain reasoning theory lies not only in being able to use more advanced techniques, but also in:
- Unified Understanding:Understand numerous specific techniques with one framework
- Flexible Application:Not bound by fixed patterns, flexibly build chains according to the situation
- Discover New Chains:Don't rely on memorizing specific patterns, but discover them by understanding principles
- Deep Understanding of Sudoku:Understand the relationships between candidates from a logical essence
Summary
Through these three articles, we have systematically learned the theoretical foundation of chain reasoning:
- First Article:Definitions, sources, and properties of strong links and weak links
- Second Article:Chain building rules, propagation logic, and coloring concepts
- Third Article:Chain classification, application patterns, and unified understanding of common techniques
After mastering these theories, you possess the ability to understand and discover various chain techniques. Through continuous application and consolidation in practice, chain reasoning will become a powerful tool for solving complex Sudoku puzzles.
Start a Sudoku game, and try analyzing candidate relationships with chain thinking! When you encounter difficulties, think:
- Where are the bi-value cells? Can they form chains?
- In which units does a certain digit form conjugate pairs?
- Can I find a chain where both ends see the candidate I want to eliminate?