An XY-Chain links together a series of bi-value cells, each sharing one candidate with the next. If the first and last cells share a common digit, any cell seeing both endpoints cannot hold that digit.
Understanding the Concept
Each cell in the chain has exactly two candidates. Adjacent cells share a house and one candidate (the "link").
Example 4-cell chain: {1,3} -> {3,7} -> {7,9} -> {9,1}. The chain starts with 1 and ends with 1.
Reasoning: if the first cell is NOT 1 (it must be 3), then cell 2 is NOT 3 (must be 7), then cell 3 is NOT 7 (must be 9), then cell 4 is NOT 9 (must be 1). So the last cell is 1.
Conversely, the first cell could be 1 directly.
Either way, at least one endpoint holds digit 1. So any cell seeing both endpoints cannot be 1.
Chains can be any length. Longer chains are harder to spot but follow the same alternating logic.
XY-Chains generalize XY-Wings, which are length-3 chains with specific structure.
Examples
4-Cell XY-Chain Eliminating Digit 4
Chain: R0C1{4,6} -> R0C7{6,2} -> R8C7{2,9} -> R8C1{9,4}. Links: 6 connects cells 1-2 (same row 0), 2 connects cells 2-3 (same column 7), 9 connects cells 3-4 (same row 8). Starts and ends with digit 4. Both endpoints are in column 1. Any other cell in column 1 with candidate 4 loses it.
Pro Tips
- •Focus on bi-value cells — they are the building blocks of every XY-Chain
- •Adjacent chain cells must share a house (row, column, or box) and one candidate
- •The key insight: the starting and ending digits must match for an elimination
- •Start with short chains (3-4 cells) and build up to longer ones
- •An XY-Wing is just a 3-cell XY-Chain with the pivot in the middle
