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A "Franken" Swordfish

 
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keith



Joined: 19 Sep 2005
Posts: 3355
Location: near Detroit, Michigan, USA
PostPosted: Sat Jul 21, 2007 11:24 am    Post subject: A "Franken" Swordfish Reply with quote
Fish (X-wings, swordfish, etc) are composed of strong links in the rows (columns) that line up (as weak links) in the columns (rows). However, there is no reason one of the strong links should not "line up" in a box.

Here is an example:

DB012007
Code:

+-------------+-------------+--------------+
| 478 2   5   | 89  1   78  | 3    67  469 |
| 3   47  1   | 6   479 5   | 8    2   79  |
| 78  6   9   | 38  478 2   | 47   5   1   |
+-------------+-------------+--------------+
| 29c 8   6   | 4   25  3   | 579a 1   579#|
| 5   49b 3   | 7   6   1   | 49A  8   2   |
| 24  1   7   | 258 258 9   | 6    3   45  |
+-------------+-------------+--------------+
| 6   5   4   | 1   78  78  | 2    9   3   |
| 79C 3   8   | 259 259 4   | 1    67  56  |
| 1   79B 2   | 59  3   6   | 57   4   8   |
+-------------+-------------+--------------+

The strong links (on <9>) are a-A, b-B, and c-C.

They line up as c-a in R4, b-A in R5, and C-B in B7. (Some of these are strong links. They do not need to be.) The elimination is <9> in the cell marked #.

I am not at all an authority on this subject. I posted this example some time ago and was told it is a frankenfish. For more on fish, look at:

http://www.sudoku.com/boards/viewtopic.php?p=23576#23576

Keith

Last edited by keith on Mon Aug 13, 2007 8:36 pm; edited 1 time in total
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cgordon



Joined: 04 May 2007
Posts: 769
Location: ontario, canada
PostPosted: Mon Aug 13, 2007 6:38 pm    Post subject: Reply with quote
I don't get that. Why is the link c-a and not say c-# (eliminating a) ??
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keith



Joined: 19 Sep 2005
Posts: 3355
Location: near Detroit, Michigan, USA
PostPosted: Mon Aug 13, 2007 8:04 pm    Post subject: Reply with quote
cgordon wrote:
I don't get that. Why is the link c-a and not say c-# (eliminating a) ??

Well, I agree this is not the greatest example, because there are so many other things going on. But, the answer to your question is that # is not strongly linked to another cell in the pattern.

Here is the swordfish logic:

1. If c is <9>, # is not <9> (weak link in R4).

2. If c is not <9>, C is <9> (strong link in C1), B is not <9> (weak link in B7*), b is <9> (strong link in C2), A is not <9> (weak link in R5*), a is <9> (strong link in C7), # is not <9> (weak link in R4).

* these are actually strong links, but the logic requires only a weak link.

You can also see that the strong links c-b in B4 and a-A in C7 form a Franken X-wing, which also eliminates <9> in #.

You can also begin to see why these things are not very useful. a-A are the only two places for <9> in C7, and they are both in B6. So, the <9> in # is eliminated by the Column - Box interaction.

Keith
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Myth Jellies



Joined: 27 Jun 2006
Posts: 64
PostPosted: Tue Aug 28, 2007 6:31 pm    Post subject: Reply with quote
The example given in posts above was trivialized by the existance of locked candidates. I'll try to illustrate a nice theoretical frankenfish so you can really see what is going on.

Consider two sets containing three houses each. To keep things simple we are going to add the stipulation that the houses in each set do not intersect with each other (no cell in common). Lets consider...

Set A: r3, r5, b7
Set B: c1, c2, c7

Now let the constraint group, (9)r35b7/c127 equal the intersection of the two sets for a given digit (9).
Code:

 B   B   . | .   .   . | B   .   .
 B   B   . | .   .   . | B   .   .
 X   X   A | A   A   A | X   A   A
-----------+-----------+-----------
 B   B   . | .   .   . | B   .   .
 X   X   A | A   A   A | X   A   A
 B   B   . | .   .   . | B   .   .
-----------+-----------+-----------
 X   X   A | .   .   . | B   .   .
 X   X   A | .   .   . | B   .   .
 X   X   A | .   .   . | B   .   .

The grid above shows how the constraint set intersection works. The cells marked with an X represent the constraint group (or franken-swordfish). Cells marked with A or B belong solely to that labeled set. Cells marked with at period may or may not contain a nine--they do not matter. Since each set was made up of three houses, the constraint group is considered TRUE if exactly three of the X cells are actually a 9. Note that if the constraint group is true, then none of the cells marked with either A or B can be true. Conversely, if the constraint group is false then that means that at least one of the cells marked with an A, AND at least one of the cells marked with a B must be true.

Now if you knew that none of the cells marked with a B could be a 9, then you know the franken-swordfish must be true and that none of the cells marked with an A can be 9 either.

Frankenfish, or constraint groups, can be finned as well. Consider the following where / means not a 9 and # represents a fin which could contain a 9.

Code:

 /   /   . | .   .   . | /   .   .
 /   /   . | .   .   . | /   .   .
 X   X   A | A   A   A | X   A   A
-----------+-----------+-----------
 /   /   . | .   .   . | /   .   .
 X   X   A | A   A   A | X   A   A
 /   /   . | .   .   . | /   .   .
-----------+-----------+-----------
 X   X   A | .   .   . | /   .   .
 X   X  -A-| .   .   . | #   .   .
 X   X   A | .   .   . | /   .   .

In the above grid, either the frankenfish is true, or the fin in r8c7 is true. Either way r8c3 <> 9

Code:

 B   B   . | .   .   . | B   .   .
 B   B   . | .   .   . | B   .   .
 X   X   / | /   /   / | X   /   /
-----------+-----------+-----------
 B   B   . | .   .   . |-B-  .   .
 X   X   / | /   /   / | X   #   #
 B   B   . | .   .   . |-B-  .   .
-----------+-----------+-----------
 X   X   / | .   .   . | B   .   .
 X   X   / | .   .   . | B   .   .
 X   X   / | .   .   . | B   .   .

In this grid, either the frankenfish is true, or the fin in r5c89 is true. Either way, r46c7 <> 9

Knowing about these things is one thing, and actually seeing them is quite another. However the intersection idea works well for regular fish and simpler constraint groups (like block/block locked candidates). This is a nice background to have when you are ready to start including simple fish groups and constraint groups in chains/AICs. Note that many of the more complex fish and frankenfish turn out to also be chains involving x-wing groups and/or boxbox/lineline constraint groups.
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