## A Short Fish Story (amended)

Advanced methods and approaches for solving Sudoku puzzles

### A Short Fish Story (amended)

In the beginning, Allan Barker wrote (recap here):
Code: Select all
`.  .  . | .  .  . | .  .  .8  8  . | .  .  . | .  8  . .  .  8 | .  .  . | .  .  8--------+---------+--------- .  .  8 | .  8  . | .  8  . 8  .  . | 8  .  . | .  8  . .  8  8 | .  8  . | 8  .  . --------+---------+--------- 8  .  8 | .  8  . | 8  8  8 .  8  . | .  8  . | .  8 -8 8  8  . | 8  .  . | 8  .  .`
5x8 Xsudo-Fish c124b36\r25689c89b7
(Edit: Fix typo)

Then hobiwan wrote(PM):
5x5 Fish r3478b4\c1359b9 + fr4c8 fr6c2 fr8c2 efr4c3
5x5 Fish r2347b4\c135b39 + fr2c2 fr4c8 fr6c2 efr4c3
6x6 Fish r23478b4\c13589b9 + fr2c2 fr6c2 fr8c2 efr4c3
6x6 Fish r23467c2\c35789b7 + fr2c1 efr2c2 efr6c2

Then Sudtyro2 wrote:
2x2 Fish r37\c39 + fr7c78 rfr7c15

Then Pat wrote:
1x1 Fish r7\c9 + fr7c78 rfr7c135

Then DAJ wrote:
Sashimi 1x1 Fish c7\r8 + fr79c7 rfr6c7

Then blue wrote:
7x11 Obi-Fish r33c2477b4\r56689c399b199
7x11 Obi-Fish r34c247b33\r2689c38999b59

Then Pat wrote:
7x7 Blue-Fish r334c2477\r689c3b156 + fr3c9 fr79c7

The End

How many ways can you scale a Fish in 2014?
Happy New Year!
Last edited by Sudtyro2 on Fri Jan 10, 2014 12:47 am, edited 1 time in total.
Sudtyro2

Posts: 604
Joined: 15 April 2013

Using base and cover sectors from Alan Barker's SLG ...

10x14 Obi-Fish c1224b333366\r2256689c889999b7
10x10 Blue-Fish c1224b333366\r2256689c88b7 + (4x)fr3c9
(Belated) Happy New Year (to All) !

P.S. Thanks Pat, for the honorable mention.
(Hoping it was more than just a tease !)
blue

Posts: 714
Joined: 11 March 2013

### Re: A Short Fish Story (amended)

You forgot that SV wrote

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`.  .  . | .  .  . | .  .  .8  8  . | .  .  . | .  8  . .  .  8 | .  .  . | .  .  8--------+---------+--------- .  .  8 | .  8  . | .  8  . 8  .  . | 8  .  . | .  8  . .  8  8 | .  8  . | 8  .  . --------+---------+--------- F  .  8 | .  8  . | 8  8  8 .  G  . | .  8  . | .  8  T 8  G  . | 8  .  . | 8  .  .`

{Rank 2 Mutant Squirmbag: c1247b3\r25689c9b9} == F -- G == {Rank 1 Mutant Swordfish: c27b3\r26c9b9} -- T -- loop.

When you throw all the above native truth/link sets into a soup, you get a 7 x 11 Obi-Fish:

c12247b33/r2256689c99b799, where the target is a member of 5 link-sets (r8c99b99).

I think that explains where the doubling/tripling up of certain native sets comes from; they are used multiple times in the chain, so appear multiple times!
sultan vinegar

Posts: 81
Joined: 27 August 2013

### Re: A Short Fish Story (amended)

A further example to the above:

The doubling up of certain sectors in Blue's 7 x 11 Obi-Fish r33c2477b4\r56689c399b199 can be understood when you break it up into a single digit kraken blossom type (with an extra bit of branching) elimination, based on the truth in column 2 (left column below):

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`r2c2 - r3c3 = r3c9 ----------------------------\ ||                                            |r6c2 - r6c7 = r79c7 -------------------------\ | ||                                           \|r8c2 --------------------------------------- r8c8 ||                                           /| ||                     // r4c3 - r3c3 = r3c9  |           r9c2 - r9c4 = r5c4 - r5c1                      /                                                          \\ r6c23 - r6c7 = r89c7 `

Add up every truth and link set used and you get Blue's 7 x 11 Obi-Fish: r33c2477b4\r56689c399b199. For example, r3 appears twice as a truth set; once in the r2c2 chain (r3c3 = r3c9), and once in the top branch of the r9c2 chain (r3c3 = r3c9).
sultan vinegar

Posts: 81
Joined: 27 August 2013

### Re: A Short Fish Story (amended)

Yes, these interpretations are interesting and are also quickly closing the “equivalence gap” between the various Fish types and solution methodologies. JC Van Hay, in particular, actually did multiple interpretations of this grid on the arcilla thread. Contributors in that thread also offered up numerous additional fish. The head post here was only meant to be a sampling.

BTW, blue had previously interpreted his two Obi-Fish examples here, both of them as Kraken 3x3 Fish.

OK, now you guys can help me interpret one of my own...
From the arcilla thread, I had also worked up another 1-Fish having a single remote fin:
1x1 Fish r3\c9 + rfr3c3

The fin has the following network diagram:
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`      --------------------r46c3    /                      ||r3c3-r3c9=r2c8-r45c8=r6c7-r6c2   |                       ||   |                      r5c1-r5c4=r9c4-r9c2   |                          \           ||        |                           ----------r79c1      \                                     ||      -----------------------------------r7c3                                          ||                                         r8c2-r8c9`

I can easily see the Obi-conversions for straight chains and even for SV's Kraken blossom, but how does one handle the interactions above? I only see two excess covers on the target (r8c9), so it would seem that we need k=1 logic. I keep getting too many cover sectors from the weak links and not enough base sectors from the strongs. Note that r3 appears as both a base and a cover and therefore seems to cancel itself out.
Sudtyro2

Posts: 604
Joined: 15 April 2013

Sudtyro2 wrote:OK, now you guys can help me interpret one of my own...

Try this.

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`(**)                                           b7 ||r7c3-r3c3=r3c9---------------------------r8c9  r3\c39 ||r79c1-r5c1*                                    b4\c1 ||    || ||   r46c3-r3c3=r3c9--------------------r8c9  r3\c39 ||    ||                           ||   r6c2-r6c7=r45c8-r2c8=r3c9----------r8c9  b36\r6c89 ||r9c2-r9c4=r5c4-r5c1*                           c4b4\r59 ||             || ||            r46c3-r3c3=r3c9-----------r8c9  r3\c39 ||             || ||            r6c2-r6c7=r45c8-r2c8=r3c9-r8c9  b36\r6c89 ||r8c2-------------------------------------r8c9  \r8`

--> 11x16 Obi-fish: r333c4b3344667\r56689c13338899999

For a stricter interpretation of your "single fin", two of the rows above, would have a "r3c9-r3c3=r3c9" a section in them,
leading to an redundant r33\r33 in the Obi-fish. [ The rows that start with "r6c2-" ]

The stuff above, is from this network diagram:

Code: Select all
`r7c3------------------ ||                    \ ||              r46c3 - r3c3=r3c9-------- ||               ||                       \r79c1---------   r6c2-r6c7=r45c8-r2c8=r3c9 - r8c9 ||            \  ||                       /r9c2-r9c4=r5c4 - r5c1                     / ||                                      /r8c2------------------------------------`

Parts of that were duplicated, as sultan vinegar was suggesting up above.
blue

Posts: 714
Joined: 11 March 2013

From above,

blue wrote:
Code: Select all
`r7c3------------------ ||                    \ ||              r46c3 - r3c3=r3c9-------- ||               ||                       \r79c1---------   r6c2-r6c7=r45c8-r2c8=r3c9 - r8c9 ||            \  ||                       /r9c2-r9c4=r5c4 - r5c1                     / ||                                      /r8c2------------------------------------`

--> 11x16 Obi-fish: r333c4b3344667\r56689c13338899999

There is also a 10x14 Obi-fish, using the same base and cover sectors.
There's nothing smaller, using those base & cover sectors -- no smaller Obi-Wahn fish, that is.

Obi-fish: r33c4b3344667\r56689c133889999 ... from this diagram, for example:

Code: Select all
`r9c12-r9c4 ||    || ||   r5c4-r5c7-r5c1 ||         ||   || ||        r6c7-r6c23 ||         ||   || ||         ||  r4c3-r3c3=r3c9 ||         ||                 \ ||        r4c8-----            \ ||                  \           \ ||  r6c2-r6c7=r45c8 - r2c8=r3c9 - r8c9  ||   ||                         / ||   ||   r3c9-----------------/ ||   ||    ||                 /r7c1-r5c1   ||                / ||   ||    ||               /r7c3-r46c3-r3c3             / ||                        /r8c2----------------------`
Last edited by blue on Mon Jan 13, 2014 8:46 am, edited 1 time in total.
blue

Posts: 714
Joined: 11 March 2013

### Re: A Short Fish Story (amended)

Nice work, blue, on the alternate network diagrams!

The trick appears to be to account for every independent path to the target cell, as you have shown from your Kraken columns.
But, what can you do starting with my 1x1 Fish and its single fin? I'm currently looking at all possible paths, but don't know exactly how to handle the "downstream" fork at r5c1 in the [b4] SIS. Do the "upstream" links count twice or just once? I can't quite nail a valid Obi-Fish as yet, but am seeing close alternates in the size range of 11x17 and 10x15.

Sudtyro2

Posts: 604
Joined: 15 April 2013

### Re: A Short Fish Story (amended)

Hi Sudtyro2,

Sudtyro2 wrote:But, what can you do starting with my 1x1 Fish and its single fin?

Actually my first network diagram was produced from yours, and the 1x1-fish links.

I started by adding the 1x1 part, like this:
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`      --------------------r46c3    /                      ||r3c3-r3c9=r2c8-r45c8=r6c7-r6c2 || \                      || ||  \                    r5c1-r5c4=r9c4-r9c2 ||   \                       \           || ||    \                       ----------r79c1 ||     \                                 || ||       -------------------------------r7c3 ||                                       || ||                                      r8c2-r8c9 ||                                          /r3c9----------------------------------------`

I fooled around with that for a while, before realizing I would't get anywhere with both forking and converging weak links.
So, I took the mirror image of your diagram and added the 1x1 fish again, like this:
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`               r46c3-------------------                ||                      \               r6c2-r6c7=r45c8-r2c8=r3c9-r3c3                ||                      / ||r9c2-r9c4=r5c4-r5c1                    /  || ||           /                       /   ||r79c1--------                        /    || ||                                 /     ||r7c3-------------------------------      r3c9-r8c9 ||                                          /r8c2----------------------------------------`

I flattened out the last part, to give:
Code: Select all
`               r46c3--------------------                ||                       \               r6c2-r6c7=r45c8-r2c8=r3c9 - r3c3=r3c9 - r8c9                ||                       /           /r9c2-r9c4=r5c4-r5c1                     /           / ||           /                        /           /     r79c1--------                         /           / ||                                  /           /r7c3--------------------------------            / ||                                            /r8c2------------------------------------------`

Next, I started separating the individual paths, like in my earlier post, and saw that the "r3c9 - r3c3=r3c9" section was leading to r3 appearing in both sector lists (base and cover). Then I made a small modification to get the first r3c9 in that section, to link directly to the elimination target:
Code: Select all
`               r6c2-r6c7=r45c8-r2c8=r3c9---------                ||                                \               r46c3--------------------           \                ||                       \          \                ||                        r3c3=r3c9 - r8c9                ||                       /          /r9c2-r9c4=r5c4-r5c1                     /          / ||           /                        /          /     r79c1--------                         /          / ||                                  /          /r7c3--------------------------------           / ||                                           /r8c2-----------------------------------------`

A little rearrangement, gave the final diagram:
Code: Select all
`r7c3------------------ ||                    \ ||              r46c3 - r3c3=r3c9-------- ||               ||                       \r79c1---------   r6c2-r6c7=r45c8-r2c8=r3c9 - r8c9 ||            \  ||                       /r9c2-r9c4=r5c4 - r5c1                     / ||                                      /r8c2------------------------------------`

I looks like you understood the rest.
blue

Posts: 714
Joined: 11 March 2013

### Re: A Short Fish Story (amended)

Thanks, blue, for the amazing morph!
And you only needed five “minor” tweeks to get from my finned 1-Fish network to your full [b7]SIS solution.

Well, just for the record, I counted five separate paths in the original fin network:
Code: Select all
`r3c3-r46c3...r5c1-r5c4...----r8c9r3c3-r46c3...r5c1-r79c1...---r8c9r3c3-r3c9...r5c1-r5c4...-----r8c9r3c3-r3c9...r5c1-r79c1...----r8c9r3c3-r7c3...-----------------r8c9 `

The strong\weak links total up to:
15x20 c44b3344446677777\r3355668888899c1133388

Then add in the unfinned 1x1 Fish r3\c9, which removes r3\r3, to get:
15x20 c44b3344446677777\r355668888899c11333889

The target cell has six excess covers to effect the elimination, but r8c8 has seven, so it would go, too. Plus the logic has “base triplets” (and worse) in b4 and b7, which would have to be covered and thus increase the overall k-rank. IOW, this logic is a hopeless mess. I did try some of Obi's conversion arithmetic to reduce the size, but doing that only made my head hurt more.

So, it would appear that the 1x1 Fish plus fin network is certainly valid, but then why doesn't it convert directly to an Obi-Fish?

OK, I wasn't getting anywhere either with that original r3c3 fin network, but it sure took me a lot longer to realize why. Those diverging weak links were the problem, where I kept adding in an extra b7 or b4 base sector for every new weak leak entering a SIS. The mirror-image network, as blue noted early on with its converging weak links, makes it much easier to identify the proper pathways and to correctly count the SIS-related base sectors.

I would add only that whenever redundant base\cover sectors appear (r3\r3, e.g.) Obi's arithmetic does allow one to simply subtract that sector from both sides of the Fish. So, one can actually finish this analysis early with blue's network below:
Code: Select all
`               r46c3--------------------                ||                       \               r6c2-r6c7=r45c8-r2c8=r3c9 - r3c3=r3c9 - r8c9                ||                       /           /r9c2-r9c4=r5c4-r5c1                     /           / ||           /                        /           /     r79c1--------                         /           / ||                                  /           /r7c3--------------------------------            / ||                                            /r8c2------------------------------------------`

Paths containing the “r3c9-r3c3=r3c9” section have the redundant r3\r3 contributions, which are then subtracted out in the final Obi-Fish. However, it does remain rather impressive that blue can further manipulate these network diagrams to avoid that final subtraction.
Sudtyro2

Posts: 604
Joined: 15 April 2013