X-Cycles

X-Cycles are chains of alternating strong and weak links for a single digit. They extend Simple Colouring by incorporating weak links, enabling more powerful elimination patterns.

Note: This technique uses chain colouring to display candidates. Square (▢) and Diamond (◇) shapes represent the two colour partitions.

Link Types

Strong Link (Conjugate Pair)

Two cells in a unit where the digit can only appear in those two positions.

If one cell is FALSE, the other must be TRUE
Notation: A ═══ B

Weak Link

Two cells in the same unit with the digit as a candidate.

If one cell is TRUE, the other must be FALSE
Notation: A ─ ─ ─ B

Cycle Types

X-Cycles come in three forms, each with different elimination rules:

1. Continuous Loop (Nice Loop)

A closed chain where links strictly alternate between strong and weak throughout.

A ══ B ── C ══ D ── A (strong → weak → strong → weak)

Elimination Rule: Any candidate that "sees" (shares a unit with) two cells connected by a weak link can be eliminated.

2. Discontinuous Loop

A closed chain where one junction has two links of the same type (breaking the alternation).

Two Weak Links at Junction:

A ── B ══ C ── A (weak → strong → weak)

The junction (A) has two weak links. Rule: A MUST contain the digit.

Two Strong Links at Junction:

A ══ B ── C ══ A (strong → weak → strong)

The junction (A) has two strong links. Rule: A CANNOT contain the digit (eliminate it).

3. X-Chain (Open Chain)

A chain that doesn't form a loop — it has two endpoints.

A ══ B ── C ══ D (endpoints: A and D)

Rule: If both endpoints are on strong links, eliminate the digit from any cell that sees BOTH endpoints.

How to Read X-Cycles

Following the chain:

Start at any cell with the digit as candidate
Follow strong links (if one is OFF → other is ON)
Then weak links (if one is ON → other is OFF)
Alternate between strong and weak
Return to start (loop) or reach a dead end (chain)

The alternating pattern creates logical implications that lead to eliminations.

Example: Continuous Loop (Rule 1)

This example demonstrates a Continuous Loop on digit 8. The chain forms a perfect cycle with alternating strong and weak links, eliminating candidates from cells that see both sides of a weak link.

Pattern Analysis:

Digit: 8
Chain cells: R3C2, R3C7, R7C2, R8C3, R8C9, R9C7
Links: 6 alternating links forming a closed loop
Eliminations: R3C3~~8, R3C9~~8, R7C3~~8, R7C9~~8

puzzle: S9B82020401124I0607BA82064648074K0401BA0743470906044I024M02040605090103080701030504080702090608070906020301050404434548120907064K03054407010609044406090744044K4I0301
mode: guided
technique: X-Cycles
initial:
  layers:
    hints: true
steps:
  - text: >
      X-Cycles extend Simple Colouring by adding weak links between cells. Look for digit 8 — it has both strong links (conjugate pairs) and weak links across the grid.
    hint: subtle
    technique: XC

  - text: >
      Enable the chain display to see links for digit 8. Solid lines show strong links (conjugate pairs), dashed lines show weak links.
    hint: subtle
    technique: XC
    state:
      graph:
        type: singleDigitChain
        digits: [8]

  - text: >
      Start building the chain at R3C2 and R7C2. These form a conjugate pair in column 2 — a strong link.
    hint: subtle
    technique: XC
    state:
      selection:
        cells: [R3C2, R7C2]
      graph:
        type: singleDigitChain
        digits: [8]

  - text: >
      From R7C2, there's a weak link to R8C3 (same box 7). Then R8C3 and R8C9 form a conjugate pair in row 8 — another strong link.
    hint: subtle
    technique: XC
    state:
      selection:
        cells: [R3C2, R7C2, R8C3, R8C9]
      graph:
        type: singleDigitChain
        digits: [8]

  - text: >
      Continue: R8C9 has a weak link to R9C7 (box 9). Then R9C7 and R3C7 form a conjugate pair in column 7 — strong link.
    hint: obvious
    technique: XC
    state:
      selection:
        cells: [R3C2, R3C7, R7C2, R8C3, R8C9, R9C7]
      graph:
        type: singleDigitChain
        digits: [8]

  - text: >
      Finally, R3C7 connects back to R3C2 via a weak link (row 3). The loop is complete: strong → weak → strong → weak → strong → weak → back to start.
    hint: obvious
    technique: XC
    state:
      selection:
        cells: [R3C2, R3C7, R7C2, R8C3, R8C9, R9C7]

  - text: >
      Rule 1 applies: eliminate 8 from any cell that sees BOTH endpoints of a weak link. R3C3 sees R3C2 (row 3) and R8C3 (column 3). Eliminate R3C3~8.
    hint: detailed
    technique: XC
    state:
      selection:
        cells: [R3C3]

  - text: >
      Similarly, R3C9 sees R3C7 and R8C9. R7C3 sees R7C2 and R8C3. R7C9 sees R8C9 and R9C7 (though indirectly). All four cells lose candidate 8.
    hint: detailed
    technique: XC
    state:
      selection:
        cells: [R3C3, R3C9, R7C3, R7C9]
settings:
  showCandidates: true
  showControls: true
  showDescription: true
  coordinateFormat: rncn
  navigation: numbered

The continuous loop creates a "squeeze" effect: cells caught between adjacent weak links are eliminated because one end or the other must be true.

Example: Strong Junction (Rule 2)

A Discontinuous Loop with Strong Junction has two consecutive strong links meeting at one cell. This forces the junction cell to contain the digit.

Pattern Analysis:

Digit: 1
Chain cells: R9C1, R7C3, R5C3, R5C8, R9C8
Junction: R9C1 (two strong links meet here)
Result: R9C1 = 1 (place the digit)

puzzle: S9B0H7N04050C078J1H0L9E0B03060A0D9E080E062B0E0I080B2B0C047Y1Q8K018Q0E0H077O057U0L077U080C0L069F08AB0B8I030D0E7N023E370H050I1F1N037U058I0C0G01028Q0H7R7Q0H040B06057N0G
mode: guided
technique: X-Cycles
initial:
  layers:
    hints: true
steps:
  - text: >
      Rule 2 applies to discontinuous loops where two STRONG links meet at one cell. Look for digit 1 — it has conjugate pairs that form a cycle with a strong-strong junction.
    hint: subtle
    technique: XC

  - text: >
      Enable the chain display to see links for digit 1. Notice that R9C1 is connected to two different strong links.
    hint: subtle
    technique: XC
    state:
      graph:
        type: singleDigitChain
        digits: [1]

  - text: >
      Trace the loop: R9C1 ══ R7C3 (strong in box 7), then R7C3 ── R5C3 (weak in column 3).
    hint: subtle
    technique: XC
    state:
      selection:
        cells: [R9C1, R7C3, R5C3]
      graph:
        type: singleDigitChain
        digits: [1]

  - text: >
      Continue: R5C3 ══ R5C8 (strong in row 5), then R5C8 ── R9C8 (weak in column 8).
    hint: obvious
    technique: XC
    state:
      selection:
        cells: [R9C1, R7C3, R5C3, R5C8, R9C8]
      graph:
        type: singleDigitChain
        digits: [1]

  - text: >
      Finally, R9C8 ══ R9C1 (strong in row 9). But wait — R9C1 already has a strong link to R7C3! Two strong links meet at R9C1.
    hint: obvious
    technique: XC
    state:
      selection:
        cells: [R9C1, R7C3, R5C3, R5C8, R9C8]

  - text: >
      With a strong junction, the colouring creates a CONFLICT at R9C1 — it would need to be both colours. Rule 2 resolves this: R9C1 MUST be 1.
    hint: detailed
    technique: XC
    state:
      selection:
        cells: [R9C1]
settings:
  showCandidates: true
  showControls: true
  showDescription: true
  coordinateFormat: rncn
  navigation: numbered

A strong junction means two "if-false-then-true" implications converge on one cell — it must be true.

Example: Weak Junction (Rule 3)

A Discontinuous Loop with Weak Junction has two consecutive weak links meeting at one cell. This forces the junction cell to NOT contain the digit.

Pattern Analysis:

Digit: 1
Chain cells: R3C3, R3C7, R7C7, R7C2, R8C3
Junction: R3C3 (two weak links meet here)
Elimination: R3C3~1

puzzle: S9B4J0G060212B604BMBB4I09040A120G4O0648020N4N040FB6BNBM07BM0F4K03070ABOBWBG0704460E090B460106830P150F0804880G7S030L0I0G0T06454C050F080L0I0T0E07030T0D0E070H0L03067O7P
mode: guided
technique: X-Cycles
initial:
  layers:
    hints: true
steps:
  - text: >
      Rule 3 applies to discontinuous loops where two WEAK links meet at one cell. Look for digit 1 — it forms a cycle with a weak-weak junction.
    hint: subtle
    technique: XC

  - text: >
      Enable the chain display to see links for digit 1. Notice that R3C3 is connected to weak links on two sides.
    hint: subtle
    technique: XC
    state:
      graph:
        type: singleDigitChain
        digits: [1]

  - text: >
      Trace the loop: R3C3 ── R3C7 (weak in row 3), then R3C7 ══ R7C7 (strong in column 7).
    hint: subtle
    technique: XC
    state:
      selection:
        cells: [R3C3, R3C7, R7C7]
      graph:
        type: singleDigitChain
        digits: [1]

  - text: >
      Continue: R7C7 ── R7C2 (weak in row 7), then R7C2 ══ R8C3 (strong in box 7).
    hint: obvious
    technique: XC
    state:
      selection:
        cells: [R3C3, R3C7, R7C7, R7C2, R8C3]
      graph:
        type: singleDigitChain
        digits: [1]

  - text: >
      Finally, R8C3 ── R3C3 (weak in column 3). But R3C3 already has a weak link to R3C7! Two weak links meet at R3C3.
    hint: obvious
    technique: XC
    state:
      selection:
        cells: [R3C3, R3C7, R7C7, R7C2, R8C3]

  - text: >
      With a weak junction, the colouring creates a CONFLICT at R3C3 — it would need to be both colours. Rule 3 resolves this: R3C3 CANNOT be 1.
    hint: detailed
    technique: XC
    state:
      selection:
        cells: [R3C3]
settings:
  showCandidates: true
  showControls: true
  showDescription: true
  coordinateFormat: rncn
  navigation: numbered

A weak junction means two "if-true-then-false" implications converge on one cell — it must be false.

Complexity

X-Cycles are expert-level because:

Requires understanding both strong and weak links
Chains can be long and span the entire grid
Multiple cycle types with different rules
Must correctly identify the cycle type before eliminating

Relationship to Other Techniques

Technique	Links Used	Pattern
Simple Colouring	Strong only	Two-colour chains
X-Cycles	Strong + Weak	Alternating chains/loops
3D Medusa	Strong + Bi-value	Multi-digit chains

X-Cycles sits between Simple Colouring (strong links only) and 3D Medusa (multiple digits).

Tips

Start with Simple Colouring — If that doesn't work, try X-Cycles
Look for conjugate pairs — These are your strong links
Add weak links carefully — They must alternate with strong links
Identify the cycle type — Different rules apply to each
Use the chain visualiser — The app can show strong and weak links

Visual Indicators

In Lazy Sudoku's graph display:

Strong chains show only strong links (Simple Colouring)
Weak chains show both strong and weak links (X-Cycles)

Toggle between modes to see different chain structures.

More Puzzles

Related Techniques

Simple Colouring — Strong links only
X-Wing — Simpler pattern using conjugate pairs
3D Medusa — Multi-digit extension