Solving Tips
X-Cycle Technique: Single-Digit Closed Loop Chain Elimination
X-Cycle is a powerful single-digit chain technique. It tracks a candidate across multiple cells through alternating strong links and weak links, ultimately forming a closed loop, which allows for candidate elimination.
Core Principle:
X-Cycle analyzes a single candidate. When a digit forms a closed loop through alternating strong and weak links across multiple cells, at least one endpoint of any weak link must be true. Therefore, any cell that can see both endpoints of a weak link can have that candidate eliminated.
X-Cycle analyzes a single candidate. When a digit forms a closed loop through alternating strong and weak links across multiple cells, at least one endpoint of any weak link must be true. Therefore, any cell that can see both endpoints of a weak link can have that candidate eliminated.
X-Cycle Diagram: Strong links (solid) and weak links (dashed) alternating to form a closed loop
Before reading this article, it's recommended to understand the basic concepts of strong and weak links, which form the foundation for understanding X-Cycle.
Strong and Weak Links Review
Before learning X-Cycle, let's review the definitions of strong and weak links:
S
Strong Link: When a candidate appears in exactly two cells within a unit (row/column/box), these two cells form a strong link. The key property is: if one is false, the other must be true.
W
Weak Link: When two cells can see each other (in the same row/column/box) and both contain a candidate, they have a weak link. The key property is: if one is true, the other must be false (but not vice versa).
Key Understanding: Strong links can act as weak links!
Strong link logic is "if one is false, the other is true," but it also satisfies "if one is true, the other is false." Therefore, strong links inherently have weak link properties. In X-Cycles, when a weak link connection is needed, a strong link can fulfill that role. This means in actual X-Cycles, you may see multiple consecutive strong links (like 3 strong links in a row), because some of them are actually acting as weak links.
Strong link logic is "if one is false, the other is true," but it also satisfies "if one is true, the other is false." Therefore, strong links inherently have weak link properties. In X-Cycles, when a weak link connection is needed, a strong link can fulfill that role. This means in actual X-Cycles, you may see multiple consecutive strong links (like 3 strong links in a row), because some of them are actually acting as weak links.
X-Cycle Closed Loop Structure
The essence of X-Cycle is: multiple cells connected through alternating strong and weak links forming a closed loop.
1
Loop Formation: Select a candidate, find multiple cells containing that candidate, and connect them end-to-end through strong and weak links to form a loop.
2
Alternating Connection: Ideally, strong and weak links should alternate. But since strong links can act as weak links, you may see multiple consecutive strong links in practice (where some strong links serve as weak links).
3
Key Reasoning: Track state changes around the loop:
- Assume one endpoint A of a weak link is false (not that digit)
- Through strong link deduction, the next cell must be true
- Through weak link deduction, the cell after that must be false
- Continue alternating until reaching the other endpoint B of the weak link
- If A is false, following the chain means B must be true
4
Elimination Conclusion: The two endpoints A and B of a weak link cannot both be false (at least one must be true). Therefore, any other cell that can see both A and B cannot be that digit and can have that candidate eliminated.
Example 1: X-Cycle for Digit 8
Let's look at the first example showing an X-Cycle formed by digit 8.
Figure 1: X-Cycle - Digit 8 forms a closed loop at R3C6, R3C9, R6C9, R6C3, R5C2, R5C6, eliminating candidate 8 from R7C9
Analysis Process
1
Identify candidate 8 distribution: Find where digit 8 appears and analyze the link relationships between cells.
2
Trace X-Cycle path (alternating strong/weak):
- R3C6 ═══ R3C9 (Row 3 strong link: 8 appears only in these two cells)
- R3C9 ─── R6C9 (Column 9 weak link: cells see each other)
- R6C9 ═══ R6C3 (Row 6 strong link: 8 appears only in these two cells)
- R6C3 ─── R5C2 (Box 4 weak link: actually a strong link acting as weak)
- R5C2 ═══ R5C6 (Row 5 strong link: 8 appears only in these two cells)
- R5C6 ─── R3C6 (Column 6 weak link: actually a strong link acting as weak)
3
Apply elimination rule:
- Weak link endpoints: R3C9 and R6C9
- At least one contains digit 8
- R7C9 can see both cells (same column)
- Therefore R7C9 cannot be 8
Conclusion:
X-Cycle: Digit 8 forms a closed loop at R3C6, R3C9, R6C9, R6C3, R5C2, R5C6.
Action: Eliminate candidate 8 from R7C9.
X-Cycle: Digit 8 forms a closed loop at R3C6, R3C9, R6C9, R6C3, R5C2, R5C6.
Action: Eliminate candidate 8 from R7C9.
Example 2: X-Cycle for Digit 4
Let's look at another example showing an X-Cycle formed by digit 4.
Figure 2: X-Cycle - Digit 4 forms a closed loop at R2C3, R2C4, R9C4, R9C1, R6C1, R4C3, eliminating candidate 4 from R3C4
Analysis Process
1
Identify candidate 4 distribution: Find where digit 4 appears and analyze the link relationships between cells.
2
Trace X-Cycle path (alternating strong/weak):
- R2C3 ─── R2C4 (Row 2 weak link: cells see each other)
- R2C4 ═══ R9C4 (Column 4 strong link: 4 appears only in these two cells)
- R9C4 ─── R9C1 (Row 9 weak link: actually a strong link acting as weak)
- R9C1 ═══ R6C1 (Column 1 strong link: 4 appears only in these two cells)
- R6C1 ─── R4C3 (Box 4 weak link: actually a strong link acting as weak)
- R4C3 ═══ R2C3 (Column 3 strong link: 4 appears only in these two cells)
3
Apply elimination rule:
- Weak link endpoints: R2C3 and R2C4
- At least one contains digit 4
- R3C4 can see both cells (same column as R2C4, same box as R2C3)
- Therefore R3C4 cannot be 4
Conclusion:
X-Cycle: Digit 4 forms a closed loop at R2C3, R2C4, R9C4, R9C1, R6C1, R4C3.
Action: Eliminate candidate 4 from R3C4.
X-Cycle: Digit 4 forms a closed loop at R2C3, R2C4, R9C4, R9C1, R6C1, R4C3.
Action: Eliminate candidate 4 from R3C4.
How to Find X-Cycles?
Finding X-Cycles requires a systematic approach:
1
Choose a target digit: Select one candidate to analyze (usually choose digits with moderate occurrence count).
2
Find strong links: In each unit (row/column/box), if the target digit appears in exactly two cells, mark it as a strong link.
3
Try to build a closed loop: Starting from any strong link endpoint, alternate between strong and weak links (or use strong links as weak links) to try forming a closed loop.
4
Look for elimination targets: Find weak links in the loop and check if any other cells can see both endpoints of the weak link.
Important Notes:
- X-Cycle only analyzes a single candidate
- Strong links require the target digit to appear exactly twice in the unit
- The loop must be closed, eventually returning to the starting point
- Strong links can act as weak links, so multiple consecutive strong links may appear
- X-Cycle is an advanced technique; master simpler chain techniques like Skyscraper first
Technique Summary
Key points for applying X-Cycle:
- Analysis target: Single candidate
- Core structure: Alternating strong and weak links forming a closed loop
- Key understanding: Strong links can act as weak links, so you may see multiple consecutive strong links
- Elimination rule: At least one endpoint of a weak link must be true; cells that can see both endpoints can eliminate that digit
Related Techniques:
X-Cycle is an advanced single-digit chain technique. Recommended learning order:
Skyscraper → Multi-Strong Links → X-Cycle
After mastering these techniques, you'll be able to handle most chain reasoning scenarios.
X-Cycle is an advanced single-digit chain technique. Recommended learning order:
Skyscraper → Multi-Strong Links → X-Cycle
After mastering these techniques, you'll be able to handle most chain reasoning scenarios.
Practice Now:
Start a Sudoku game and try to discover X-Cycle patterns! Since finding them manually is complex, it's recommended to first use the calculator's hint feature to familiarize yourself with this pattern.
Start a Sudoku game and try to discover X-Cycle patterns! Since finding them manually is complex, it's recommended to first use the calculator's hint feature to familiarize yourself with this pattern.