A chain of bivalue cells where consecutive pairs share a candidate and a unit. The endpoints drive eliminations.
Chain bivalue cells end to end, each consecutive pair sharing one candidate and one unit, so a value at one end forces values all the way along. Compare the outside candidate of the first cell with that of the last: if they match, eliminate that digit from any cell seeing both endpoints.
An XY-Chain generalizes the XY-Wing from three cells to any number of bivalue cells linked together. Each cell in the chain has exactly two candidates, and consecutive cells share one candidate and one unit. The chain creates a logical path: if the first cell is digit X, the next must be Y, the next must be Z, and so on.
The elimination comes from the chain's two endpoints. Look at the "outside" candidate of each endpoint, the candidate that is not shared with its neighbor in the chain. If both endpoints' outside candidates are the same digit, that digit can be eliminated from any cell that sees both endpoints.
The logic mirrors XY-Wing: the first cell is either its outside candidate or its shared candidate. Following the chain of forced values, one of the two endpoints must contain the elimination digit. Any cell seeing both endpoints is therefore blocked.
Find bivalue cells and try to link them. Each consecutive pair must share one candidate and one unit (row, column, or box). The shared candidate is the "link" digit, and it alternates being the forced value down the chain.
At the first cell, identify the "outside" candidate (not shared with cell 2). At the last cell, identify the "outside" candidate (not shared with the second-to-last cell). If these outside candidates are the same digit, you can eliminate that digit.
Eliminate the common outside digit from any cell that shares a unit with both the first and last cells in the chain.
Example 1: Short XY-Chain (4 cells)
Step 1: Build the chain: R1C2{2,4} connects to R1C3{4,6} (share digit 4, same row). R1C3{4,6} connects to R5C3{3,6} (share digit 6, same column). R5C3{3,6} connects to R5C4{2,3} (share digit 3, same row).
Step 2: The outside candidate of R1C2 is 2 (not shared with R1C3). The outside candidate of R5C4 is 2 (not shared with R5C3). Both outside candidates are digit 2.
Step 3: Eliminate 2 from any cell seeing both R1C2 and R5C4. R5C2 sees R1C2 (column 2) and R5C4 (row 5). Remove 2 from R5C2.
Example 2: Long XY-Chain (8 cells)
Step 1: Build the chain: R2C5{5,7} → R2C6{2,7} → R5C6{2,7} → R5C9{6,7} → R4C9{6,7} → R4C1{5,7} → R4C7{4,5} → R5C7{4,5}.
Step 2: The outside candidate of R2C5 is 5. The outside candidate of R5C7 is 5 (not shared with R4C7, where the link is on 4). Both outside candidates are digit 5.
Step 3: Eliminate 5 from any cell seeing both R2C5 and R5C7. R5C5 sees both. Remove 5 from R5C5.
The outside candidate is the one candidate of an end cell that is not shared with its neighbour in the chain. Each end cell has two candidates: one is the link into the chain, the other points outwards. Only when both ends’ outward candidates are the same digit does the chain eliminate anything.
Yes — an XY-Wing is an XY-Chain of length three. The pivot is the middle cell and the two wings are the endpoints, whose outside candidates are the shared wing digit. Once you can build chains of four, five or more bivalue cells, the XY-Wing becomes the smallest case of a general pattern.
Every cell in an XY-Chain must be bivalue — exactly two candidates, no more. A cell with three candidates breaks the forcing logic, because knowing it is not one digit no longer tells you which digit it is. The two-candidate restriction is what makes each link pass a definite value along.
XY-Chain eliminations happen outside the chain, in cells that share a unit with both the first and the last cell. The cells in the middle of the chain are untouched; they are the machinery, not the target. A cell that sees only one endpoint is unaffected, since it must see both to be blocked.
There is no fixed limit — a chain can be any length, provided every cell is bivalue and each consecutive pair shares a candidate and a unit. Chains of three or four cells are the common ones; longer chains turn up rarely, but they can break positions that nothing shorter touches.
An XY-Chain links bivalue cells and moves between different digits as it goes, whereas Simple Coloring follows one digit only, chaining the pairs of cells that are that digit’s only options in a unit. XY-Chain eliminations fall on cells seeing both endpoints; Simple Coloring’s fall on cells seeing both colours.
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