XY-Chains

XY-Chains use bi-value cells (cells with exactly two candidates) to build chains across multiple digits. Unlike X-Chains that track one digit, XY-Chains hop between digits through shared candidates.

What is an XY-Chain?

An XY-Chain is a chain of bi-value cells where:

• Each cell has exactly two candidates
• Adjacent cells share exactly one candidate
• The shared candidate creates a weak link (if one cell has it, the neighbor doesn't)
• Each cell also has a strong link within itself (if not X, then Y)

The result: You can trace implications through the chain, leading to eliminations.

The Bi-Value Cell Advantage

A bi-value cell like [3,7] has a built-in strong link:

• If the cell is not 3, it MUST be 7
• If the cell is not 7, it MUST be 3

This internal strong link is what powers XY-Chains.

Chain Structure

Consider three bi-value cells:

Cell A: [3,7]     Cell B: [7,5]     Cell C: [5,9]

They chain because:

• A and B share 7
• B and C share 5

Chain notation:

A[3-7] ─── B[7-5] ─── C[5-9]

The dash inside brackets shows the cell's two candidates. The line shows connection via shared candidate.

The Logic

Trace through the chain:

If A = 3:

• A is not 7
• B sees A, and they share 7, but A isn't 7, so no constraint on B from this
• Actually, the weak link is: if A = 7, then B ≠ 7

Let me reframe:

Start: Assume A's "left" candidate (3) is false.

• A = 7 (internal strong link)
• B sees 7 through the chain, so B ≠ 7
• B = 5 (internal strong link in B)
• C sees 5 through the chain, so C ≠ 5
• C = 9 (internal strong link in C)

The endpoints:

• Chain starts with candidate 3 (A's first option)
• Chain ends with candidate 9 (C's forced value if 3 is false)

Either:

• A = 3, OR
• C = 9

At least one endpoint is true. Cells seeing both the start and end candidates can be eliminated.

Elimination Rule

If an XY-Chain goes from candidate X at the start to candidate Y at the end:

• Either the start cell has X
• Or the end cell has Y

Any cell that:

• Has both X and Y as candidates
• Sees the start cell AND sees the end cell

Can have the common candidate eliminated... wait, let me be precise.

Standard XY-Chain elimination:

If the chain is A[X-...] to Z[...-Y]:

• Either A = X, or Z = Y
• Cells that see both A and Z, and contain the "endpoint" digits, can be eliminated

Actually, the elimination depends on what X and Y are:

• If X ≠ Y: Cells seeing A (with candidate X) and seeing Z (with candidate Y) — not directly useful unless X = Y
• If X = Y (same digit at both ends): Any cell seeing both A and Z can have X eliminated

The power case: When the starting and ending candidates are the same digit, cells seeing both endpoints can have that digit eliminated.

Same-Digit Endpoints

The most useful XY-Chains have the same digit at both ends:

A[3-7] ─── B[7-5] ─── C[5-3]
     ↑                    ↑
   starts with 3       ends with 3

Logic:

• If A ≠ 3: A = 7 → B ≠ 7 → B = 5 → C ≠ 5 → C = 3
• If A = 3: chain doesn't propagate, but A = 3

Either A = 3 or C = 3. At least one endpoint has 3.

Elimination: Any cell that sees BOTH A and C, and has candidate 3, can have 3 eliminated.

Worked Example

Grid has these bi-value cells:

R1C4: [2,8]
R1C9: [8,5]
R4C9: [5,6]
R4C3: [6,2]

Build the chain:

R1C4[2-8] ─── R1C9[8-5] ─── R4C9[5-6] ─── R4C3[6-2]

Connections:

• R1C4 and R1C9: share row 1, connected by 8
• R1C9 and R4C9: share column 9, connected by 5
• R4C9 and R4C3: share row 4, connected by 6

Endpoints:

• Start: 2 (from R1C4)
• End: 2 (from R4C3)

Same digit! This is the power case.

Elimination:

• Either R1C4 = 2, or R4C3 = 2
• Cells seeing both R1C4 and R4C3, with candidate 2, can be eliminated

What sees both?

• R1C3: same row as R1C4, same column as R4C3 ✓
• R4C4: same row as R4C3, same column as R1C4 ✓
• Box 4 cells might see both if positioned right

If R1C3 or R4C4 have candidate 2, eliminate it!

Finding XY-Chains

Step 1: Identify Bi-Value Cells

Scan the grid for cells with exactly two candidates. List them.

Step 2: Build Connection Graph

For each pair of bi-value cells that see each other (same row, column, or box):

• Check if they share exactly one candidate
• If yes, they can connect in a chain

Step 3: Search for Same-Digit Endpoints

Build chains that start and end with the same digit. These produce eliminations.

Step 4: Find Elimination Targets

Cells seeing both endpoints, with the endpoint digit as a candidate.

Chain Length

XY-Chains can be any length:

3 cells (XY-Wing):

A[X-Y] ─── B[Y-Z] ─── C[Z-X]

This is exactly an XY-Wing! The XY-Wing is a minimal XY-Chain.

4 cells:

A[X-Y] ─── B[Y-Z] ─── C[Z-W] ─── D[W-X]

5+ cells: Keep extending as long as you find connecting bi-value cells.

Longer chains: More elimination potential, but harder to find.

XY-Chain vs. XY-Wing

XY-Wing is the smallest XY-Chain with same-digit endpoints.

Branching Chains

Chains don't have to be linear. A cell can branch:

        ┌─── B[5-9] ─── C[9-3]
A[3-5] ─┤
        └─── D[5-7] ─── E[7-3]

Both branches end in 3. Complex analysis can still produce eliminations.

Practice Exercise

Find the XY-Chain:

Bi-value cells:

• R2C1: [1,4]
• R2C6: [4,9]
• R5C6: [9,7]
• R5C1: [7,1]

R2C1[1-4] ─── R2C6[4-9] ─── R5C6[9-7] ─── R5C1[7-1]
         row 2        col 6         row 5

Common Mistakes

Mistake 1: Non-bi-value cells in chain

Every cell in an XY-Chain must have exactly 2 candidates. Cells with 3+ candidates break the internal strong link.

Mistake 2: Wrong shared candidate

Adjacent cells must share EXACTLY ONE candidate. If they share zero or two, they don't connect.

Mistake 3: Different endpoint digits

Chains with different digits at start and end (X...Y where X ≠ Y) don't produce simple eliminations.

Mistake 4: Missing visibility

Elimination targets must see BOTH endpoints. Check row, column, and box for each.

Quick Reference

XY-Chain definition:

• Chain of bi-value cells
• Adjacent cells share one candidate
• Endpoints ideally have the same digit

Building chains:

1. Find all bi-value cells
2. Connect cells that see each other and share one digit
3. Build paths that start and end with the same digit

Elimination rule:

• Same digit at both ends
• Eliminate that digit from cells seeing both endpoints

Special case:

• 3-cell XY-Chain with same endpoints = XY-Wing