The XY-Chains Elimination sudoku solving technique extends the Rectangle Elimination technique by combining it with the XY-Chains technique. The Rectangle Elimination technique involves the following steps:

Identify a 3x3 square where all cells with a certain candidate are located in one row and one column within the square.
Find two cells outside of this square with the same candidate, one in the same row and one in the same column.
If there is a scenario where both of these outside candidates can be ON, all candidates inside the 3x3 square will be eliminated. This scenario is invalid, and we should remove the outside candidates that caused this scenario.

According to the XY-Chains Elimination technique, the two outside candidates should belong to an XY-Chain and have the same color. To avoid the elimination of all inside candidates, we need to remove one outside candidate that belongs to the XY-Chain, is located in the same row or column as the inside candidates, and has a weak link to the next element of the chain. This approach is similar to the Rectangle Elimination technique, which involves one strong and one weak link. The following image shows an example of the XY-Chain Elimination technique.

The 3x3 square in the center of the Sudoku puzzle has two candidates '4' marked in blue. They are located in row D and column 5. There are no more candidates '4' in this 3x3 square. The XY-Chain includes seven candidates marked in red, orange, and yellow. The candidate '4' in column 5, marked in red in cell H5, belongs to the chain. There is a weak link between the red '4' and the candidate '9' marked in orange in the same cell H5. If the '4' in H5 is ON, all orange candidates of the XY-Chain are OFF, and all yellow candidates of the XY-Chain are ON. This scenario would eliminate all candidates '4' in the central 3x3 square and make the Sudoku invalid. To avoid invalidating the Sudoku, we need to remove the red candidate '4' in cell E5.
The X-Chain Details:

H5(4) => H5(9) - Weak Link: If 4 in cell H5 is ON, then 9 in cell H5 is OFF.
H5(9) -> I5(9) - Strong Link: If 9 in cell H5 is OFF, then 9 in cell I5 is ON.
I5(9) => I8(9) - Weak Link: If 9 in cell I5 is ON, then 9 in cell I8 is OFF.
I8(9) -> D8(9) - Strong Link: If 9 in cell I8 is OFF, then 9 in cell D8 is ON.
D8(9) => D7(9) - Weak Link: If 9 in cell D8 is ON, then 9 in cell D7 is OFF.
D7(9) -> D7(4) - Strong Link: If 9 in cell D7 is OFF, then 4 in cell D7 is ON.

The image on the left presents another example of the XY-Chain Elimination technique. The top-left 3x3 square has three '3' candidates (marked in blue) located in row C and column 1. The candidate '3' marked in red in cell D3 belongs to an XY-Chain that has 13 candidates marked in red, orange, and yellow. If the candidate '3' in cell D1 is ON, all orange candidates are OFF, and all yellow candidates are ON. This scenario would eliminate all '3' candidates in the top-left 3x3 square. To avoid this scenario, we should remove the candidate '3' in cell D1.
The X-Chain Details:

D1(3) => D1(1) - Weak Link: If 3 in cell D1 is ON, then 1 in cell D1 is OFF.
D1(1) -> E3(1) - Strong Link: If 1 in cell D1 is OFF, then 1 in cell E3 is ON.
E3(1) => E6(1) - Weak Link: If 1 in cell E3 is ON, then 1 in cell E6 is OFF.
E6(1) -> B6(1) - Strong Link: If 1 in cell E6 is OFF, then 1 in cell B6 is ON.
B6(1) => B5(1) - Weak Link: If 1 in cell B6 is ON, then 1 in cell B5 is OFF.
B5(1) -> B5(9) - Strong Link: If 1 in cell B5 is OFF, then 9 in cell B5 is ON.
B5(9) => H5(9) - Weak Link: If 9 in cell B5 is ON, then 9 in cell H5 is OFF.
H5(9) -> H6(9) - Strong Link: If 9 in cell H5 is OFF, then 9 in cell H6 is ON.
H6(9) => H6(4) - Weak Link: If 9 in cell H6 is ON, then 4 in cell H6 is OFF.
H6(4) -> A6(4) - Strong Link: If 4 in cell H6 is OFF, then 4 in cell A6 is ON.
A6(4) => C4(4) - Weak Link: If 4 in cell A6 is ON, then 4 in cell C4 is OFF.
C4(4) -> C4(3) - Strong Link: If 4 in cell C4 is OFF, then 3 in cell C4 is ON.

by www.PuzzleMystery.com

XY-Chains Elimination