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Hidden XY Chains |
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A Hidden XY-Chain is a series of cells that contain at least two candidates each, one being strongly connected to a candidate in the cell directly ahead and behind in the chain. If the linked candidates in the first and last cells are the same number, then any candidate in the puzzle that shares a unit with both of those can be eliminated.
Consider the example on this page. We'll consider the first cell in the chain to be the one that contains an 8 and a 9. The 8 is strongly connected to the 8 in the other 8,9 cell, then the 9 of that cell is strongly connected to the 9 in the 7,9 cell, then the 7 of that cell is strongly connected to the 7 in the 5,7 cell, then the 5 of that cell is strongly connected to the 5 in the cell directly to the left, and the 8 of that cell is strongly connected to the 8 in the final remaining cell. The connecting candidates in the first and last cells are both 8's. Since the 8 highlighted in green shares a unit with both these 8's, it can be eliminated.
Consider the following logic. If the first cell is an 8, then the 8 in green can be eliminated. However, if it is not an 8, then the second cell must be because it's the last remaining 8 in that box. Then, by the same logic, the next cell must be a 9, then the next a 7, and so on until the final cell is an 8, which also eliminates the 8 in green. Whether the first cell is or is not an 8, the 8 in green gets eliminated.
Hidden XY Chains can be very long in puzzles that have a lot of cells with only two candidates. To augment the program's execution speed, and since hidden XY-Chains can be engulfed in higher solving methods, Sudoku Snake stops searching once a Hidden XY-Chain is more than 7 cells long.
Note that hidden XY-Chains differ from naked XY-Chains in that the connecting candidates in the first and last cell cause the elimination, while in naked XY-Chains it's the left-over candidates that do.
Sudoku Snake gives Hidden XY-Chains a skill value of 1800.
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