Wednesday, February 07, 2007

Fishy-Squirm Bag

Squirm-Bag
Jelly-Fish can be extended to five rows and columns. It's been named Squrim-Bag. I know of no necessary example in a solving sequence and thanks to Florian Fischer for explaining why.
Let's take the Jelly-Fish example above. Columns 2,4,6,8 are such that the 4's are restrained to rows 1,5,7,9. The 4 in each column must be placed in one of these rows, and therefore they must occupy all of them and within the yellow cells. So the 4's in rows 1,5,7,9 cannot appear in columns 1,3,5,7,9.
What are we saying? 4's in columns 2,4,6,8 are restricted to rows 1,5,7,9. That is equivalent to saying that there is no 4 in the intersection of columns 2,4,6,8 and rows 2,3,4,6,8. And that is equivalent to say that 4's in the five rows 2,3,4,6,8 are restricted to the five columns 1,3,5,7,9. And that describes a Squirm-Bag. Indeed, you can conclude that the five 4's must occupy all columns 1,3,5,7,9 within rows 2,3,4,6,8 and therefore the 4's in columns 1,3,5,7,9 cannot appear outside of these rows, that is in rows 1,5,7,9.
Generally, a size-N fish has a size-(9-N) fish counterpart. You just need to complement the set of rows and the set of columns and reason the other way round (column-wise or row-wise). Both are equivalent. Therefore you don't need to go further than size 4 Jelly-Fish in a standard 9x9 Sudoku.

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