XY-Chains are a sophisticated Sudoku solving technique that links together a series of bivalue cells (cells with exactly two candidates) to form a logical chain whose endpoints force an elimination. Each link in the chain connects two bivalue cells that share a candidate and can see each other. The chain starts at one cell and hops through a sequence of bivalue cells, with each successive cell sharing one candidate with the previous cell and introducing a new candidate for the next link.
The power of an XY-Chain lies in its endpoints. If the first cell in the chain has candidates {A, B} and the last cell has candidates {Y, A}, then the chain creates a logical either-or situation: the starting cell is either A or B, and the ending cell is either Y or A. If the start is A, then A is placed at the start. If the start is B, the chain propagates through all intermediate cells and forces A at the end. Either way, A must appear at one end or the other. Therefore, any cell that can see BOTH endpoints and contains candidate A can have A safely eliminated.
XY-Chains generalize the Y-Wing (XY-Wing) technique, which is simply an XY-Chain of length 3. Longer chains can produce eliminations that no simpler technique can find, making XY-Chains one of the most versatile tools in an expert solver's arsenal. Finding XY-Chains requires you to maintain an accurate candidate grid and systematically explore paths through bivalue cells. The technique rewards patience and careful tracking, and it often unlocks cascading eliminations that crack open even the most stubborn puzzles.
Try It Yourself
Walk through each step of the xy-chains technique on a real puzzle. Follow the instructions and try entering the correct value when prompted.
Identify bivalue cells to build an XY-Chain. Cell (1,1) has candidates {2,5}, cell (1,4) has candidates {5,6}, and cell (3,4) has candidates {1,6}. These three cells form a potential chain path.
Step-by-Step Guide
Update your pencilmark grid so all candidates are current and accurate.
Identify all bivalue cells — cells with exactly two remaining candidates.
Pick a starting bivalue cell and note its two candidates, say {A, B}.
From that cell, find a connected bivalue cell (same row, column, or box) that shares candidate B and has a second candidate C — forming the link {B, C}.
Continue extending the chain through additional bivalue cells, each sharing one candidate with the previous cell.
When the chain's final cell shares a candidate A with the starting cell, the chain is complete.
Identify any cell that can see BOTH the start and end of the chain and contains candidate A — eliminate A from those cells.
Check if the elimination creates naked singles or triggers further solving progress.
Think of a row of people passing a message along a line -- each person whispers one of two options to their neighbor. No matter what the first person says, the last person always ends up saying the same thing, so anyone who can hear both ends of the line knows that option is taken.
XY-Chains are valid because each bivalue cell acts as a binary switch: if one candidate is eliminated, the other is forced. Chaining these switches together means that assuming one value at the start cascades deterministically through every link to force a specific value at the end. Since the start cell must hold one of its two candidates, both branches of the dichotomy are explored, and both branches guarantee that candidate A appears at one endpoint or the other. Any cell seeing both endpoints is therefore excluded from holding A, because it would conflict with whichever endpoint contains it.
When to use: Use XY-Chains when the grid has many bivalue cells and simpler techniques like Y-Wing have been exhausted. Look for a path of bivalue cells where the first and last cells share a common candidate.
Common Mistakes to Avoid
Including non-bivalue cells (cells with three or more candidates) in the chain.
Every cell in an XY-Chain must have exactly two candidates. If a cell has three or more, it cannot participate as a link in the chain.
Building a chain where consecutive cells do not share exactly one candidate.
Each pair of adjacent cells in the chain must share exactly one candidate, with the other candidate being different. Verify each link carefully.
Eliminating a candidate from a cell that does not see both endpoints of the chain.
The elimination only applies to cells that can see BOTH the first and last cell of the chain. A cell seeing only one endpoint is not affected.
More Examples
See xy-chains applied in different puzzle configurations to strengthen your pattern recognition.
XY-Chain linking four bivalue cells
Highlighted cells show the xy-chains pattern
Practice Puzzles
Apply the xy-chains technique on these mini challenges. Tap a highlighted cell and enter the correct digit.
Quick Reference
- Pattern:
- A chain of bivalue cells where each cell shares a candidate with the next
- Action:
- Eliminate the candidate shared by the first and last cells from cells seeing both endpoints
- Look for:
- A sequence of bivalue cells connected by shared candidates