XY-Cycle (Discontinuous) is an extreme-level chain strategy that uses a powerful logical principle: if assuming something is TRUE leads back to proving it must be FALSE, then the assumption was wrong.

This strategy chains through bivalue cells (cells with exactly 2 candidates), creating a loop that contradicts itself.

Real Example Walkthrough

In the example puzzle above, the solver finds a 7-cell chain that creates a contradiction:

The Chain Path:

R1C1=7(ON) ~R1C1=6(OFF) -R1C2=6(ON) ~R1C2=9(OFF) -R9C2=9(ON) ~R9C2=1(OFF) -R7C3=1(ON) ~R7C3=4(OFF) -R6C3=4(ON) ~R6C3=9(OFF) -R6C4=9(ON) -R1C4=9(OFF) ~R1C4=7(ON) -R1C1=7(OFF) ← CONTRADICTION!

Step-by-Step Translation:

The Contradiction: We assumed R1C1=7, but the chain proves R1C1≠7. Impossible!

Result: Eliminate 7 from R1C1.

Understanding the Chain Notation

Weak Links (~)

Weak links happen within a single bivalue cell: - Cell has candidates {6, 7} - If cell = 7, then cell ≠ 6 - Notation: R1C1=7(ON) ~R1C1=6(OFF)

Strong Links (-)

Strong links happen between different cells: - R1C1 and R1C2 are the only cells with 6 in Row 1 - If R1C1 ≠ 6, then R1C2 must be 6 - Notation: R1C1=6(OFF) -R1C2=6(ON)

Why "Discontinuous"?

The "discontinuity" is the break in the pattern that creates the contradiction.

Visual Representation

``` Chain through bivalue cells:

[R1C1]─────weak─────[R1C1]
{6,7}               same cell
Start               (bivalue toggle)
  │                     │
  │ Assume 7           ↓
  ▼                   6 OFF
7 ON                    │
  │                     │
  └──────strong─────────┘
        (to R1C2)

...chain continues through 7 cells...

[R1C4]─────strong────[R1C1]
{7,9}                 {6,7}
  │                     │
7 ON ──────────────→ 7 OFF
                        ↑
                  CONTRADICTION!

```

Step-by-Step: How to Find XY-Cycles

1. Identify bivalue cells: Mark all cells with exactly 2 candidates
2. Start at any bivalue cell: Pick a candidate to assume TRUE
3. Follow the chain:
4. Weak link (~): Go to the other candidate in the same cell
5. Strong link (-): Jump to a connected cell in the same row/column/box
6. Look for a loop: Does the chain return to the starting cell?
7. Check for contradiction: Does it return with the opposite state (ON→OFF)?
8. Eliminate: If contradiction found, eliminate the starting candidate

The Logic Behind It

This is a form of proof by contradiction:

1. Hypothesis: Assume candidate X is in cell A
2. Deduction: Follow logical chain through the grid
3. Contradiction: Chain returns saying X is NOT in cell A
4. Conclusion: The hypothesis must be FALSE
5. Action: Eliminate X from cell A

It's logically airtight — if assuming TRUE leads to FALSE, the assumption was wrong.

Chain Building Tips

Finding Strong Links

Strong links exist when a candidate appears in only 2 cells within a region: - Only 2 cells have 6 in Row 1 → strong link between them - Only 2 cells have 9 in Column 2 → strong link between them - Only 2 cells have 4 in Box 7 → strong link between them

Following Weak Links

In a bivalue cell {A, B}: - If A is ON → B is OFF - If B is OFF → A is ON

This is automatic within any bivalue cell.

Building the Chain

1. Start with a candidate assumption
2. Toggle within the cell (weak link)
3. Jump to a connected cell (strong link)
4. Repeat until you return to start or get stuck

Comparison with Related Strategies

XY-Cycle vs XY-Chain

The XY-Chain finds eliminations between endpoints. The XY-Cycle finds an elimination at the starting cell.

Common Mistakes

"I can't find where to start"

Try any bivalue cell. If no loop forms, try another starting candidate or cell. Many starting points won't work — that's normal.

"My chain doesn't return to the start"

Not every chain forms a cycle. If you reach a dead end, try: - A different starting cell - A different starting candidate - A different direction through the grid

"I found a loop but no contradiction"

If the chain returns with the same state (ON→ON or OFF→OFF), that's a continuous cycle, not discontinuous. Continuous cycles have different elimination rules.

When XY-Cycles Apply

• ✅ Grid has many bivalue cells (2 candidates each)
• ✅ Strong links exist (candidates appearing in only 2 cells per region)
• ✅ A chain can be built that loops back to the start
• ✅ The loop returns with the opposite state (contradiction)

Tips for Beginners

1. Master XY-Chain first: It's simpler and teaches the same link concepts
2. Mark bivalue cells: Highlight all cells with exactly 2 candidates
3. Practice chain notation: Get comfortable with ON/OFF states
4. Use pencil marks: Write down the chain as you explore
5. Don't give up: XY-Cycles are rare but powerful when found

Why This Strategy Works

The strategy is based on propositional logic:

• If P → Q and Q → R and R → ¬P
• Then P → ¬P
• Which means P must be FALSE

In Sudoku terms: - If "R1C1=7" leads through a chain to "R1C1≠7" - Then "R1C1=7" is impossible - Eliminate 7 from R1C1

Related Strategies

Chain Strategies

• XY-Chain — Open chain between bivalue cells
• X-Cycle (Discontinuous) — Single-candidate cycle
• Remote Pair — Chain of cells with same two candidates

Coloring Strategies

• Simple Coloring — Two-color candidate tracking
• 3D Medusa — Multi-digit coloring

Wing Strategies

• Y-Wing — 3-cell chain pattern
• XYZ-Wing — Extended Y-Wing
• W-Wing — Paired bivalue cells with strong link