| Lummox JR
| | Joined: 07 Sep 2005 | | Posts: 202 | | : | |
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 Posted: Sun Oct 16, 2005 1:17 am Post subject: |
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Just to clarify a little further, this is the crucial truth of all the uniqueness tests: The following is not a valid arrangement in a solution board, unless at least one of these cells is one of the original clues.
| Code: | . . 1|. 2 .
. . .|. . .
. . 2|. 1 . |
If this arrangement exists, only those cells can influence each other to be one value or another. If their values are all unknown at the beginning of the puzzle, nothing can ever constrain them to be either the above form or this one:
| Code: | . . 2|. 1 .
. . .|. . .
. . 1|. 2 . |
In order to pin down one of the values, there has to be another cell in its box, column, or row which will force it to be filled in. The only other cells which can do that are part of the pattern. If they too are unknowns, then you have a set which no amount of logic can ever pin down.
In theory there's also a 3x3 form of uniqueness test, where you have this sort of layout:
| Code: | . . 1|. 2 .|3 . .
. . 2|. 3 .|1 . .
. . 3|. 1 .|2 . . |
That should result in 12 different permutations. As you can tell, that has to be so rare as to border on the impossible, and it's not worth the effort to search for it. |
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