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Squirmbag + 1

 
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Marty R.



Joined: 12 Feb 2006
Posts: 5121
Location: Rochester, NY, USA

PostPosted: Fri Apr 14, 2006 3:47 pm    Post subject: Squirmbag + 1 Reply with quote

I had a 6-fish 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 6-fish be paired with a swordfish? I just don't get it.

"# 5-fish : squirmbag - For 99 Sudoku, there's no in point naming higher-order (>4) fish, since every N-fish comes paired with a 9-N fish whose effect is the same (thus any 5-fish is paired with a jellyfish; any 6-fish with a swordfish; any 7-fish with an x-wing; any 8-fish with a hidden or naked single). Nevertheless, a 5-fish is occasionally called a squirmbag.
# 6+ fish : 6-gronk, 7-gronk.. [23] - these patterns are only useful for Sudoku larger than 99."
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Steve R



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

PostPosted: Fri Apr 14, 2006 7:40 pm    Post subject: Squirmbag + 1 Reply with quote

Marty, if you still have your 6-fish, 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 6-fish was based on rows, there would be another fish based on the non-target 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 5-fish 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 6-fish could be a swordfish (no X yet inserted), an X-Wing (a single X previously entered) or even, in theory, a single square.

Steve
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Marty R.



Joined: 12 Feb 2006
Posts: 5121
Location: Rochester, NY, USA

PostPosted: Sat Apr 15, 2006 3:59 pm    Post subject: Reply with quote

Steve, thank you very much for taking the time and effort for that explanation.

I don't still have the 6-fish. All my puzzles are done with pencil and paper and there's no trail.
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