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Marty R.
Joined: 12 Feb 2006 Posts: 5475 Location: Rochester, NY, USA

Posted: Fri Apr 14, 2006 3:47 pm Post subject: Squirmbag + 1 


I had a 6fish yesterday and was researching to see if it had a name. The Wikipedia entry indicated there's no point in giving it a name in a 9x9 puzzle. Could someone explain that to me? Why would my 6fish be paired with a swordfish? I just don't get it.
"# 5fish : squirmbag  For 9×9 Sudoku, there's no in point naming higherorder (>4) fish, since every Nfish comes paired with a 9N fish whose effect is the same (thus any 5fish is paired with a jellyfish; any 6fish with a swordfish; any 7fish with an xwing; any 8fish with a hidden or naked single). Nevertheless, a 5fish is occasionally called a squirmbag.
# 6+ fish : 6gronk, 7gronk.. [23]  these patterns are only useful for Sudoku larger than 9×9." 

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Steve R
Joined: 24 Oct 2005 Posts: 289 Location: Birmingham, England

Posted: Fri Apr 14, 2006 7:40 pm Post subject: Squirmbag + 1 


Marty, if you still have your 6fish, it should be easy enough to find its dual.
The general scheme is this. Suppose, in an n x n puzzle, you have an m x m fish for X based on rows. The nature of the fish is that it also identifies m columns. These columns are the target columns in the sense that X can be eliminated from the target columns except where they intersect the base rows.
Now think about the other n – m columns, those which are not targets. Where can X be placed? Only outside the base rows. So now we have n – m columns in which X can be placed only in the n – m rows which are not base rows. That is, we have an (n – m) x (n – m) fish based on columns. This is the dual fish. If your 6fish was based on rows, there would be another fish based on the nontarget columns.
This may be a little abstract. Bear in mind that fish are nothing to do with boxes (unless thy have fins); they otherwise relate entirely to columns and rows. So you can permute the columns and rows until a 5fish based on rows looks like this:
Code: 

 X X X X X . . . . 
 X X X X X . . . . 
 X X X X X . . . . 
 X X X X X . . . . 
 X X X X X . . . . 
 . . . . . . . . . 
 . . . . . . . . . 
 . . . . . . . . . 
 . . . . . . . . . 


The fish excludes X from the unmarked cells in columns 1 to 5 so all the Xs in rows 6 to 9 must fall in columns 6 to 9. Marking the possible cells with X gives a 4 fish based on columns 6 to 9:
Code: 

 . . . . . . . . . 
 . . . . . . . . . 
 . . . . . . . . . 
 . . . . . . . . . 
 . . . . . . . . . 
 . . . . . X X X X 
 . . . . . X X X X 
 . . . . . X X X X 
 . . . . . X X X X 


Once the eliminations are made, reverse the permutations and fill in the framework of the boxes to retrieve the puzzle proper.
I have simplified the position a little in an attempt at clarity. Once the principle is grasped, the next stage is to take account of any Xs which have already been inserted in the puzzle. If r Xs have been inserted, the puzzle becomes (n – r) x (n – r) for fishy purposes so the dual of an m x m fish is of dimension (n – r – m) x (n – r – m).
The dual of your 6fish could be a swordfish (no X yet inserted), an XWing (a single X previously entered) or even, in theory, a single square.
Steve 

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Marty R.
Joined: 12 Feb 2006 Posts: 5475 Location: Rochester, NY, USA

Posted: Sat Apr 15, 2006 3:59 pm Post subject: 


Steve, thank you very much for taking the time and effort for that explanation.
I don't still have the 6fish. All my puzzles are done with pencil and paper and there's no trail. 

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