XY-Cycle (Continuous) is a rare and beautiful version of the XY-Chain. While a Discontinuous cycle creates a contradiction (proving a number is impossible), a Continuous cycle creates a stable, self-sustaining loop.
In a Continuous loop, every cell affects its neighbors perfectly. This "validates" the loop and turns every logical connection within it into a Strong Link.
[!NOTE] Real Example Pending: This strategy is so rare that we are currently waiting for a pure example to appear in our database. The following is a theoretical explanation of the logic.
Interactive Example
Click "Apply Logic" to see the strategy in action.
The Logic: The "Nice Loop"
Imagine a chain of bivalue cells (cells with 2 candidates) that loops back on itself perfectly:
- Start: Cell A {1, 2}
- Link: Cell B {2, 3}
- Link: Cell C {3, 4}
- Link: Cell D {4, 1}
- Return: Back to Cell A {1, 2}
If you trace the logic: - If A = 1 → D = 4 → C = 3 → B = 2. - If A = 2 → B = 3 → C = 4 → D = 1.
In both possible realities, the relationship holds. The loop is stable.
The Eliminations
Because the loop is stable, two powerful things happen:
-
Weak Links become Strong: Any two cells in the loop connected by a digit (e.g., A and B connected by '2') are now "locked". One of them must be 2.
- Elimination: Any cell outside the loop that sees both A and B cannot be a 2.
-
Locked Candidates: If a cell in the loop technically had a 3rd candidate (making it not a true bivalue cell), but that candidate would break the loop logic, it can be eliminated. (This turns a non-bivalue loop into a bivalue one).
Visual Guide
Row 1 Row 2
[A] {1,2} ─── [B] {2,3}
| |
| |
[D] {4,1} ─── [C] {3,4}
The Loop: A(1) → D(4) → C(3) → B(2) → A(1)...
Elimination Example: - Look at the link between A and B (the digit 2). - One of them is 2. - Any cell Z that "sees" both A and B cannot contain 2.
Continuous vs. Discontinuous
| Feature | Discontinuous | Continuous |
|---|---|---|
| Result | Logic crashes (Contradiction) | Logic flows forever (Stable Loop) |
| Outcome | Eliminates candidate at the start/end of chain | Eliminates candidates outside the chain |
| Rarity | Very Rare | Extremely Rare |
How to Spot It
- Find XY-Chains: Start building chains of bivalue cells.
- Look for Loops: Try to connect the end of your chain back to the start.
- Check Parity: Does the loop match up? (e.g., ending with '1' connects to a start of '1').
- If it matches → Continuous (Stable).
- If it mismatches (e.g., ending with '1' connects to a start of '2') → Discontinuous (Contradiction).
Comparison Table
| Strategy | Cycle Type | Link Result | Elimination |
|---|---|---|---|
| XY-Chain | Open (No loop) | Endpoint Inference | At start/end |
| Discontinuous XY-Cycle | Closed (Broken) | Contradiction | At the break point |
| Continuous XY-Cycle | Closed (Perfect) | All Links Strong | Outside the loop |
Tips for Beginners
- Coloring Helps: Use two colors (e.g., Green/Blue) to trace the chain. If Green connects to Blue perfectly at the start, it's continuous.
- Look for Clusters: These loops often form in dense clusters of bivalue cells.
- Don't Force It: If a chain gets complicated, move on. The most useful chains are often short (4-6 links).
Common Mistakes
- Misinterpreting the Loop: Thinking a Discontinuous loop is Continuous. Double-check the parity (odd vs even links)!
- Wrong Elimination: Eliminating inside the loop. In a continuous loop, the cells inside are safe (they become strongly linked). You eliminate candidates from outside that see the strong links.
Related Strategies
- XY-Chain: The open-ended version of this loop.
- XY-Cycle (Discontinuous): The contradictory version.