X-Cycle (Continuous) is a strategy dealing with loops of a single digit. It is also known as a "Continuous Nice Loop."
Unlike its discontinuous cousin which finds a contradiction, a continuous loop is a perfectly balanced structure where every link works in both directions. This stability turns Weak Links into Strong Links, allowing for eliminations.
[!NOTE] Real Example Pending: This strategy is an advanced theoretical concept of loop analysis. We are currently waiting for a pure example in our database. The following is a theoretical explanation.
Interactive Example
Click "Apply Logic" to see the strategy in action.
The Logic: The "Perfect Loop"
Imagine a chain of cells that only care about the number 7:
- Start: Cell A.
- Strong Link: If A is 7, Cell B is NOT 7. (And if B is 7, A is NOT 7).
- Weak Link: If B is NOT 7, Cell C is 7.
- ...Chain continues...
- Return: The chain loops back to A perfectly.
Because the loop has an even number of links and alternates perfectly between Strong (S) and Weak (W), it acts like a closed circuit. - If A is True, then B is False, C is True... - If A is False, then B is True, C is False... - Both states are valid!
The Power of Stability
Because both states are valid, every Weak Link in the loop effectively becomes a Strong Link. - Normally, "If B is 7, C is not 7" is weak logic (both could be false). - In a continuous loop, one of them MUST be 7. It's impossible for both to be false.
The Elimination: Since every pair of connected cells in the loop now has a strong relationship (one of them must be true), any cell that sees both of them cannot contain the candidate.
Visual Guide
(Strong) (Weak becomes Strong)
[A] ───────── [B] ══════════════ [C]
| |
| | (Strong)
└─────────────────────────────── [D]
(Weak becomes Strong)
In this loop of 7s: - One of B or C must be 7. - Therefore, any cell Z that sees both B and C cannot be a 7. - Eliminate 7 from Z.
Continuous vs. Discontinuous
| Feature | Discontinuous X-Cycle | Continuous X-Cycle |
|---|---|---|
| Result | Logic crashes (Contradiction) | Logic flows forever (Stable Loop) |
| Logic | "If Start is True, Start is False" | "Start is either True or False, both work" |
| Elimination | The Candidate at the Start/End | Candidates outside the loop (Peers) |
How to Spot It
- Single Digit Filter: Focus on one number (e.g., 7).
- Find Strong Links: Draw lines between cells that are strongly linked (only 2 candidates in a row/col/box).
- Connect with Weak Links: Use weak links (3+ candidates) to bridge gaps.
- Find a Loop: Try to form a closed ring.
- Check Even/Odd: Count the nodes. A continuous nice loop must have an even number of cells.
Common Mistakes
- Odd Number of Nodes: A loop with an odd number of cells is never continuous. It will always be discontinuous (a contradiction).
- Wrong Digit: This strategy works on ONE digit at a time. If you switch digits, it becomes an XY-Chain.
Comparison Table
| Strategy | Cycle Type | Link Type | Application |
|---|---|---|---|
| X-Wing | Continuous Loop (4 nodes) | Geometric | Basic Fish |
| Simple Coloring | Continuous/Discontinuous | Color-based | Visual Chaining |
| X-Cycle (Cont.) | Continuous Loop (Any length) | Logical Links | Flexible Loop |
Note: Determining that a Fish pattern (like X-Wing or Swordfish) is essentially a small, continuous X-Cycle is a great way to understand the underlying logic of Sudoku!
Related Strategies
- X-Cycle (Discontinuous): The contradictory version.
- Simple Coloring: A visual way to find these chains.