XY-Chains

An XY-Chain is a chain of bi-value cells where consecutive cells share a candidate. The chain alternates between digits, creating a logical path that forces one of the endpoints to be true.

Note: XY-Chains are a subset of Alternating Inference Chains — they use only bi-value cells for linking.

How It Works

The Building Blocks

XY-Chains use only bi-value cells — cells with exactly two candidates. Each cell acts as a logical switch: if one candidate is false, the other must be true.

Chain Structure

The chain connects cells by their shared candidates:

Cell 1 {A, B} → Cell 2 {B, C} → Cell 3 {C, D} → Cell 4 {D, E}

Cell 1: Start with A, exit on B
Cell 2: Enter on B (shared), exit on C
Cell 3: Enter on C (shared), exit on D
Cell 4: Enter on D (shared), end with E

Each step:

Enter a cell on one digit
Exit on the other digit (the cell's "switch" property)
Connect to next cell via shared candidate in same unit

The Logic

If the chain starts with digit X and ends with digit X:

Either the start cell contains X, OR
The chain propagates through, forcing the end cell to contain X

Result: At least one endpoint must be X. Any cell seeing both endpoints cannot contain X.

Chain Notation (Eureka)

XY-Chains use Eureka notation — the standard format for writing chains:

Symbol	Meaning
`=`	Strong link (within bi-value cell)
`-`	Weak link (between cells in same unit)
`(digit)cell`	Candidate and position

Example: (7=4)r2c3-(4=1)r7c3-(1=2)r8c3-(2=4)r8c2-(4=3)r8c7-(3=7)r2c7

Reading: "If R2C3 is not 7, it's 4. That 4 sees R7C3, so R7C3 isn't 4, making it 1. That 1 sees R8C3..."

In XY-Chains:

Strong links (=) always occur within bi-value cells (one candidate being false forces the other true)
Weak links (-) connect cells that share a unit (same row, column, or box)

Example 1

Look at this 6-cell XY-Chain:

The Chain:

R2C3: {4, 7} — start with 7
R7C3: {1, 4} — linked by 4, continue with 1
R8C3: {1, 2} — linked by 1, continue with 2
R8C2: {2, 4} — linked by 2, continue with 4
R8C7: {3, 4} — linked by 4, continue with 3
R2C7: {3, 7} — linked by 3, end with 7

Both endpoints have 7. R2C5 sees both R2C3 and R2C7 (same row), so 7 is eliminated from R2C5.

puzzle: S9B022y0i1a3q3q08011i0f012i4e66092e050b082q030z3n02360d090z095w4o6g041j0o1i0d2g060p9l2f057s080z0o4406bs4m7n070d030f0r09170z020h070i0s0l074f470u060e0g08050w1s1i7y7q01
mode: guided
technique: XY-Chains
initial:
  layers:
    hints: true
    focus: true
  focus:
    enabled: true
    biValue: true
steps:
  - text: >
      Look for bi-value cells (cells with exactly 2 candidates). These form the links of an XY-Chain.
    state:
      focus:
        enabled: true
        biValue: true

  - text: >
      R2C3 has {4, 7}. If it's not 7, it must be 4. This 4 connects to R7C3.
    hint: subtle
    technique: XYC
    state:
      selection:
        cell: R2C3
      focus:
        enabled: true
        digits: [4, 7]

  - text: >
      R7C3 has {1, 4}. If R2C3 is 4, then R7C3 can't be 4, so R7C3 must be 1.
    hint: subtle
    technique: XYC
    state:
      selection:
        cells: [R2C3, R7C3]
      focus:
        enabled: true
        digits: [1, 4]

  - text: >
      Follow the chain: R8C3 {1,2} → R8C2 {2,4} → R8C7 {3,4} → R2C7 {3,7}.
    hint: obvious
    technique: XYC
    state:
      selection:
        cells: [R2C3, R7C3, R8C3, R8C2, R8C7, R2C7]

  - text: >
      Both endpoints (R2C3 and R2C7) contain 7. One of them MUST be 7.
    hint: obvious
    technique: XYC

  - text: >
      R2C5 sees both endpoints (same row). Since one endpoint must be 7, R2C5 cannot be 7.
    hint: detailed
    technique: XYC
    state:
      selection:
        cells: [R2C3, R2C5, R2C7]
      focus:
        enabled: true
        digits: [7]
settings:
  showCandidates: true
  showControls: true
  showDescription: true
  navigation: numbered

Example 2

A 5-cell XY-Chain with a different structure:

Eureka notation: (3=7)r2c3-(7=1)r5c3-(1=9)r6c2-(9=8)r6c8-(8=3)r6c1

The Chain:

R2C3: {3, 7} — start with 3
R5C3: {1, 7} — linked by 7, continue with 1
R6C2: {1, 9} — linked by 1, continue with 9
R6C8: {8, 9} — linked by 9, continue with 8
R6C1: {3, 8} — linked by 8, end with 3

Both endpoints have 3. R2C1 sees both R2C3 (row) and R6C1 (column), and R4C3 sees both endpoints (column/box). Eliminate 3 from both.

puzzle: S9B0F080E040I020G0A0C2A0B2E0E0H0A090406090R0R07030F0H0B0E4U8A7U1I010G1UB6022Q062B0H0B090Z030D467N021I0D051FB60G020G0H0A0F0C040509168A7U0B0G080C0F010A03060I0E0D020G08
mode: guided
technique: XY-Chains
initial:
  layers:
    hints: true
    focus: true
  focus:
    enabled: true
    biValue: true
steps:
  - text: >
      This puzzle has a 5-cell XY-Chain. Start by identifying bi-value cells.
    state:
      focus:
        enabled: true
        biValue: true

  - text: >
      R2C3 has {3, 7}. We'll use 3 as our starting digit.
      In notation: (3=7)r2c3
    hint: subtle
    technique: XYC
    state:
      selection:
        cell: R2C3
      focus:
        enabled: true
        digits: [3, 7]

  - text: >
      R5C3 has {1, 7} and shares Column 3 with R2C3. The 7 links them.
      Notation so far: (3=7)r2c3-(7=1)r5c3
    hint: subtle
    technique: XYC
    state:
      selection:
        cells: [R2C3, R5C3]
      focus:
        enabled: true
        digits: [1, 7]

  - text: >
      Continue through R6C2 {1,9} → R6C8 {8,9} → R6C1 {3,8}.
      Full chain: (3=7)r2c3-(7=1)r5c3-(1=9)r6c2-(9=8)r6c8-(8=3)r6c1
    hint: obvious
    technique: XYC
    state:
      selection:
        cells: [R2C3, R5C3, R6C2, R6C8, R6C1]

  - text: >
      Both endpoints (R2C3 and R6C1) contain 3. One of them MUST be 3.
    hint: obvious
    technique: XYC

  - text: >
      R2C1 sees both endpoints — same row as R2C3, same column as R6C1.
      R4C3 sees both via column and box. Eliminate 3 from both cells.
    hint: detailed
    technique: XYC
    state:
      selection:
        cells: [R2C1, R2C3, R4C3, R6C1]
      focus:
        enabled: true
        digits: [3]
settings:
  showCandidates: true
  showControls: true
  showDescription: true
  navigation: numbered

Chain Types

Open Chain (Most Common)

A chain with two distinct endpoints. If both endpoints have the same digit, eliminate that digit from cells seeing both.

Closed Chain (Loop)

When a chain returns to its starting cell, it forms a loop. Closed XY-Chains have additional elimination rules similar to X-Cycles.

Tips

Find bi-value cells first — These are your building blocks
Look for shared candidates — Cells must share a digit to connect
Track the alternating digits — Enter on one, exit on the other
Check both endpoints — Same digit at both ends enables elimination
Use Focus Mode — Highlight bi-value cells to spot potential chains

Relationship to Other Techniques

Technique	Link Type	Digits
Simple Colouring	Strong (conjugate pairs)	Single digit
X-Cycles	Strong + Weak	Single digit
XY-Chain	Bi-value cells	Multiple digits
3D Medusa	Strong + Bi-value	Multiple digits

XY-Chains are essentially X-Cycles extended to multiple digits, using bi-value cells as the linking mechanism.

More Puzzles

Related Techniques

X-Cycles — Single-digit chains with strong/weak links
Simple Colouring — Single-digit strong link chains
3D Medusa — Multi-digit colouring extension
Naked Pair — Bi-value cells in the same unit