[Chain Reasoning ①] Basics: Strong and Weak Links
Chain Reasoning is the core theoretical framework for advanced Sudoku techniques. Almost all advanced elimination techniques—from simple X-Wing to complex AICs—can be understood and described through chain reasoning. This article explores the two most fundamental concepts of chain reasoning: Strong Links and Weak Links.
What is a Chain?
In Sudoku, a Chain is a sequence of connections formed between candidates through certain logical relationships. Imagine: if we can establish reasoning relationships like "if A is true, then B is true/false" between candidates and link these relationships together, we form a chain.
The essence of a chain is logical propagation: starting from one point, through a series of logical deductions, reaching a conclusion. This conclusion is typically used to:
- Determine that a candidate must be true (confirm placement)
- Determine that a candidate must be false (eliminate candidate)
To understand chains, we must first understand the basic units that form chains: Links. Links describe the logical relationship between two candidates, divided into strong links and weak links based on the strength of the relationship.
Strong Link
A strong link exists between two candidates A and B if and only if: exactly one of A and B is true, and the other is false.
In other words, if A is false then B must be true, and if A is true then B must be false (mutually exclusive and complete).
Notation: A = B or A ═══ B (double line)
Sources of Strong Links
Strong links can come from the following situations:
1. Strong Link within a Bi-value Cell
When a cell has only two candidates, a strong link exists between these two candidates.
Logic: If 4 is false, the cell must be 7; if 7 is false, the cell must be 4.
Bi-value cells are the most common source of strong links because they are intuitive: the cell is either this number or that number.
2. Strong Link from Conjugate Pair
When a digit appears in only two positions within a unit (row, column, or box), a strong link exists between that candidate at these two positions. This relationship is called a Conjugate Pair.
Logic: Row 5 must have a 3. If R5C2 is not 3, R5C8 must be 3; and vice versa.
The two ends of a conjugate pair strong link are the same digit at different positions, not different digits at the same position. This is fundamentally different from strong links within bi-value cells.
3. Grouped Strong Link
More broadly, when a group of candidates and another group of candidates satisfy the "exactly one group is true" relationship, a strong link exists. This will be covered in advanced techniques and discussed in detail in the third article of this series.
Core Properties of Strong Links
- Exactly one true: Exactly one end of a strong link is true, the other is false
- False propagates true: If one end is false, the other must be true
- True propagates false: If one end is true, the other must be false
Weak Link
A weak link exists between two candidates A and B if and only if: if A is true, then B must be false.
In other words, at most one of A and B is true (both can be false, but both cannot be true).
Notation: A - B or A ─── B (single line)
Sources of Weak Links
Weak links also have multiple sources:
1. Weak Link between Different Candidates in the Same Cell
Within the same cell, a weak link exists between any two different candidates.
Logic: A cell can only hold one number. If 1 is placed, it cannot be 5.
2. Weak Link between Same Candidates in the Same Unit
Within the same unit (row, column, or box), weak links exist pairwise between all positions of the same candidate.
Logic: A digit can only appear once in a box. If R1C7 is 6, then R2C8 and R3C9 cannot be 6.
Compared to strong links, weak links are more ubiquitous. In fact, the basic rules of Sudoku (no repeated digits in row, column, or box; one digit per cell) essentially define a large number of weak link relationships.
Core Properties of Weak Links
- At most one true: At most one end of a weak link is true
- True propagates false: If one end is true, the other must be false
- Can be both false: Both ends can be false simultaneously (different from strong links!)
Comparison of Strong and Weak Links
Understanding the difference between strong and weak links is key to mastering chain reasoning. Let's summarize with a comparison table:
| Property | Strong Link | Weak Link |
|---|---|---|
| Core property | Exactly one true, one false | At most one true |
| Logic propagation | False → True, True → False | True → False |
| Can both be true | ✗ No | ✗ No |
| Can both be false | ✗ No | ✓ Yes |
| Notation | ═══ (double line) or = | ─── (single line) or - |
| Common sources | Bi-value cells, Conjugate pairs | Same cell different digits, Same unit same digit |
Special Case: Strong Links are Also Weak Links
Here's an important concept to understand: Strong links are often also weak links.
Strong link perspective: If 4 is false, 7 must be true → Strong link exists
Weak link perspective: If 4 is true, 7 must be false → Weak link also exists
Conclusion: These two candidates have both a strong link and a weak link!
Strong link perspective: If 3 at R5C2 is false, 3 at R5C8 must be true → Strong link exists
Weak link perspective: If 3 at R5C2 is true, 3 at R5C8 must be false (same row can't have two 3s) → Weak link also exists
Conclusion: Conjugate pairs also satisfy both strong and weak link conditions!
When two candidates satisfy the "exactly one true, one false" relationship (neither can both be true, nor can both be false), they have both a strong link and a weak link. This is the "strongest" link relationship and is very useful in chain construction.
Memory tip: Bi-value cells and conjugate pairs always have both strong and weak links.
The Concept of "Seeing"
In chain reasoning, the concept of "seeing" (see) is frequently used. Understanding "seeing" is crucial for identifying link relationships.
Candidate A "sees" candidate B means there is a weak link between A and B.
If A is true, then B must be false—A can "eliminate" B.
"Seeing" relationships exist between:
- Different candidates in the same cell
- Same candidate in the same row
- Same candidate in the same column
- Same candidate in the same box
This concept will be frequently used when discussing chain applications, such as "candidates that can be seen by both ends can be eliminated."
Why is Distinguishing Strong and Weak Links So Important?
The distinction between strong and weak links is the cornerstone of chain reasoning. Their differences determine:
Strong links allow inferring "true" from "false"; weak links allow inferring "false" from "true". Chain reasoning uses these two different propagation directions to construct complex logical deductions.
When building chains, you must correctly identify whether each step is a strong link or weak link to ensure correct reasoning. Incorrectly treating a weak link as a strong link will lead to wrong conclusions.
Many seemingly different techniques (like X-Wing, Skyscraper, XY-Wing, etc.) are essentially chains with specific patterns. Understanding strong and weak links allows you to comprehend these techniques within a unified framework.
Next Steps
This article introduced the two most fundamental concepts of chain reasoning: strong links and weak links. With these concepts understood, we can begin learning how to combine them to construct complete chains.
In the next article, we will discuss:
- How to alternate strong and weak links to build chains
- Rules for propagating true/false states in chains
- The "coloring" approach in chain reasoning
- Methods for drawing conclusions from chain endpoints
- Sudoku Glossary - Quick reference for terms used in this article
- XY-Wing Technique - Practical application of chain reasoning
- XY-Chain Technique - Extended application of bi-value cell chains