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XY-Chains Parallel Elimination
The XY-Chains Parallel Elimination technique is similar to the XY-Chains Elimination sudoku solving technique. The XY-Chains Parallel Elimination technique involves the following steps:

  • Identify a 3x3 square where all cells with a certain candidate are located in two rows or two columns within the square.
  • Find two cells outside of this square with the same candidate, one in the same rows or the same columns.
  • 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 Parallel Elimination technique, the two outside candidates should belong to an XY-Chain and have the same color. To avoid the elimination of all inside 3x3 square 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 XY-Chains Parallel Elimination technique.

The following image shows an example of the XY-Chain Elimination technique.
XY-Chains Parallel Elimination Example The top right 3x3 square has four candidates '3' (marked in blue) in columns 7 and 8. The XY-Chain has three candidates ‘3’ in cells E7 and I8, and three candidates ‘5’ in cells E7, I7, and I8. Both chain candidates ‘3’ are in columns 7 and 8. Candidate '3' marked in red in cell I8 has a weak link to the next chain candidate '5' marked in orange in cell I8. If the candidate ‘3’ in I8 is ON, all orange candidates are OFF, and all yellow candidates are ON. In this case, the candidates ‘3’ in columns 7 and 8 are ON. This scenario would eliminate all candidates ‘3’ in the top right 3x3 square and make the sudoku puzzle invalid. Therefore, we should remove the candidate ‘3’ in cell I8.
The X-Chain Details:
  • I8(3) => I8(5) - Weak Link: If 3 in cell I8 is ON, then 5 in cell I8 is OFF.
  • I8(5) -> I7(5) - Strong Link: If 5 in cell I8 is OFF, then 5 in cell I7 is ON.
  • I7(5) => E7(5) - Weak Link: If 5 in cell I7 is ON, then 5 in cell E7 is OFF.
  • E7(5) -> E7(3) - Strong Link: If 5 in cell E7 is OFF, then 3 in cell E7 is ON.
XY-Chains Parallel Elimination Example An image on the left shows another example of the XY-Chains Parallel Elimination technique. The central 3x3 square has four candidates '5' (marked in blue) located in columns 5 and 6. The XY-Chain has thirteen elements. Candidate '5' (marked in red) in cell H6 has a weak link with candidate '5' (marked in orange) in cell H8. If candidate ‘5’ in H6 is ON, all orange elements of the XY-Chain are OFF, and all yellow candidates are ON. In this case, candidate '5' marked in yellow in cell A5 is ON as well, and all candidates ‘5’ in the central 3x3 square are eliminated. This scenario is invalid, and candidate ‘5’ in H6 should be removed.
The X-Chain Details:
  • H6(5) => H8(5) - Weak Link: If 5 in cell H6 is ON, then 5 in cell H8 is OFF..
  • H8(5) -> G8(5) - Strong Link: If 5 in cell H8 is OFF, then 5 in cell G8 is ON.
  • G8(5) => G8(8) - Weak Link: If 5 in cell G8 is ON, then 8 in cell G8 is OFF.
  • G8(8) -> A8(8) - Strong Link: If 8 in cell G8 is OFF, then 8 in cell A8 is ON.
  • A8(8) => A8(7) - Weak Link: If 8 in cell A8 is ON, then 7 in cell A8 is OFF.
  • A8(7) -> A2(7) - Strong Link: If 7 in cell A8 is OFF, then 7 in cell A2 is ON.
  • A2(7) => A2(5) - Weak Link: If 7 in cell A2 is ON, then 5 in cell A2 is OFF.
  • A2(5) -> A5(5) - Strong Link: If 5 in cell A2 is OFF, then 5 in cell A5 is ON.