a new (?) view of fish (naked or hidden)

Advanced methods and approaches for solving Sudoku puzzles

Re: a new (?) view of fish (naked or hidden)

Postby re'born » Sat Jun 30, 2007 3:15 pm

denis_berthier wrote:
arcilla wrote:A while ago, when learning about FISHes, it occurred to me that one could look at them in a different way.
[…]
The idea is, that for most 'normal' fishes (for instance where boxes do not come into play, as in 'franken' types, if I understand them correctly), if you administer them in a certain way, they are equivalent to naked or hidden pairs, triples, quads, etc. The administration is simple. Make a list of nine cells, like a Sudoku row. In cell n (1..9) write the (row-)numbers (1..9) of the cells in column n that contain the candidate.
[…]
An X-wing shows up as a naked pair […] and a sword fish as a hidden triple.


Arcilla, I was not aware of your remark when I wrote my book, "The Hidden Logic of Sudoku". Otherwise, I would have cited you. You had a great idea. Unfortunately, as I can see from the posts in this thread, nobody seems to have really understood it or pushed it further.

Based on more general ideas of symmetry, I came upon quite the same idea, formalised it. More generally, I introduced rn- and cn- spaces and an associated extended sudoku board to deal with them. I also extended the idea to chains, where it allows introducing completely new types of chains (hidden chains).


Actually I understood quite well and even was writing down a version of your rn and cn spaces. Here are a couple of my missing posts from this thread:

rep'nA (now re'born) wrote:arcilla,
This is a very creative approach. Kudos! It gives me hope that I might find a jellyfish or finned swordfish on my own someday.
I have a rather remedial question for you (or anyone else kind enough to point out the obvious to me).
arcilla wrote:As far as I can see finned fish are equivalent to almost locked sets (and can therefore be attacked analogously).

Here is an example from the newly formed Ultimate Fish Guide:

Code: Select all
5.....1..2...16.8...35....7...4.8.6...6....9..85..32...5.64.9....91.....3......14
 *--------------------------------------------------------------------*
 | 5      479-6  478    | 23789  2789   2479   | 1      234    2369   |
 | 2      479    47     | 379    1      6      | 345    8      359    |
 |#14689 *1469   3      | 5      289    249    |*46     24     7      |
 |----------------------+----------------------+----------------------|
 | 179    12379  127    | 4      2579   8      | 357    6      35     |
 | 47     2347   6      | 27     257    1      | 34578  9      358    |
 | 479    8      5      | 79     6      3      | 2      47     1      |
 |----------------------+----------------------+----------------------|
 | 178    5      1278   | 6      4      27     | 9      237    238    |
 | 4678   2467   9      | 1      3      27     | 678    5      268    |
 | 3     *267    278    | 2789   2789   5      |*678    1      4      |
 *--------------------------------------------------------------------*

The row placements of 6 (which should be viewed as a column vector) are
R_6: [(29)(6)(127)], [(8)(3)(5)], [(4)(1279)(27)]

I added the []'s to help denote the blocks which seems important to me.

R_6[3,9] (the third and ninth entries) form an almost locked set, with the 1 in R_6[3] being the obstruction to a naked pair (and hence the corresponding x-wing). The conclusion, it seems to me, is that in general you could remove a 2 from any of R_6[1,2] since any elimination must occur in the same block as the obstruction and must also be in the same triple as the obstruction (the triples are {1,2,3}, {4,5,6}, {7,8,9}, though this is just a numerical consequence of needing to be in the same block as the obstruction).

First, is this the correct way to view the elimination of the 2 in R_6[1] (or equivalently of the 6 from r1c2)?

Second, if instead you take the column placements of 6:

C_6: [(38)(1389)(5)], [(7)(6)(2)], [(389)(4)(18)],

then we see an almost locked set in C_6[1,7] with 9 being the obstruction to the naked pair. Again, I would think that this implies that one could remove an 8 from any other entry in the third block of C_6 (since that is where the obstruction is), namely remove the 8 from C_6[9].
Code: Select all
 *--------------------------------------------------------------------*
 | 5      4679   478    | 23789  2789   2479   | 1      234    2369   |
 | 2      479    47     | 379    1      6      | 345    8      359    |
 |*14689  1469   3      | 5      289    249    |*46     24     7      |
 |----------------------+----------------------+----------------------|
 | 179    12379  127    | 4      2579   8      | 357    6      35     |
 | 47     2347   6      | 27     257    1      | 34578  9      358    |
 | 479    8      5      | 79     6      3      | 2      47     1      |
 |----------------------+----------------------+----------------------|
 | 178    5      1278   | 6      4      27     | 9      237    238    |
 |*4678   2467   9      | 1      3      27     | *678   5     -268    |
 | 3      267    278    | 2789   2789   5      | #678   1      4      |
 *--------------------------------------------------------------------*

Of course, that is exactly what the theory of finned x-wings tells you can be done and while it is not mentioned in that thread, it very well could have been (of course, it isn't needed after the first elimination). So this is very nice and was terribly easy for me to spot using your numerology. My question is how should I see the first elimination using C_6 (or the second elimination using R_6)? It seems I could use almost hidden sets and get the deduction, but is there a faster way?

To be more precise about how I would see it, consider again

R_6: [(29)(6)(127)], [(8)(3)(5)], [(4)(1279)(27)].

R_6[3,8,9] is an almost hidden set (with hidden candidates 1 and 7) with the obstruction to a hidden pair being the 7 in R_6[9]. Then you can remove any 9 from an entry in the third block, i.e., a 9 from R_6[8]. Is that how you would see it?

and
rep'nA (now re'born) wrote:Here is another example taken from the Ultimate Fish Guide.
# Sashimi Swordfish digit 3
Code: Select all
003400000000025009040700060801000090070050010060000703080006020600170000000003500
.------------------.------------------.------------------.
| 259   259   3    | 4     6     189  | 128   578   12578|
| 7     1     6    | 38    2     5    | 348  #348   9    |
| 259   4     8    | 7    *139   19   |-123   6     125  |
:------------------+------------------+------------------:
| 8    *235   1    | 236  *34    7    | 246   9     245  |
| 2349  7     249  | 23689 5     2489 | 2468  1     248  |
| 2459  6     2459 | 289   1489  12489| 7     458   3    |
:------------------+------------------+------------------:
| 349   8     479  | 5     49    6    | 1349  2     147  |
| 6    *2359  2459 | 1     7     2489 | 3489 *348   48   |
| 1     29    2479 | 289   489   3    | 5     478   6    |
'------------------'------------------'------------------'


It took me quite a while to make this deduction initially, not because of the logic, that is pretty straight forward, but because even if somebody tells you that the puzzle solves with a Sashimi Swordfish and even if somebody tells you "hint, hint, look at 3", it may still be difficult to reel in the fish. On the other hand, converting the '3' placements into the Arcilla matrix, we get:

C_3: [(57),(48),(1)], [(245),(34),(9)], [(2378),(28),(6)]

It is reasonably easy to spot the ALS in C_3[2,5,8] where the 2 in C_3[8] is an obstruction to the ALS being a (degenerate) naked triple. Therefore, we can remove any 1 or 3 from different entries in the same block, i.e., we can remove the 3 from C_3[7]. This translates to deducing r3c7 <> 3. On the other hand, the not-so-bright examiner (e.g., me) will ask why we didn't take the 3 in C_3[5] to be the obstruction. This is perfectly legitimate and implies that we can remove any 1 or 2 from different entries in the same block, i.e., we can remove the 2 from C_3[4]. This translates to deducing r2c4 <> 3 and corresponds to the sashimi swordfish:

# Sashimi Swordfish digit 3
Code: Select all
.------------------.------------------.------------------.
| 259   259   3    | 4     6     189  | 128   578   12578|
| 7     1     6    |-38    2     5    | 348  *348   9    |
| 259   4     8    | 7    #139   19   | 123   6     125  |
:------------------+------------------+------------------:
| 8    *235   1    | 236  *34    7    | 246   9     245  |
| 2349  7     249  | 23689 5     2489 | 2468  1     248  |
| 2459  6     2459 | 289   1489  12489| 7     458   3    |
:------------------+------------------+------------------:
| 349   8     479  | 5     49    6    | 1349  2     147  |
| 6    *2359  2459 | 1     7     2489 | 3489 *348   48   |
| 1     29    2479 | 289   489   3    | 5     478   6    |
'------------------'------------------'------------------'

Naturally, either deduction implies the other so you don't need both, but it is very pleasing to see both fish constructed so easily, especially for someone like me who historically bought his fish at the fish market, pre-gutted, sliced and diced.
re'born
 
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Postby udosuk » Fri Jul 20, 2007 2:04 am

I've recently applied this approach on a puzzle posted here.

I don't know if it's acceptable as non-T&E logic because the move is over 2 different digits (all examples above are over a single digit only). But if we're willing to extend these analysis to 3 or more digits perhaps we can solve a lot more puzzles "logically", even including the SE 10+ ones.

Just a random thought.:idea:
udosuk
 
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Postby Mike Barker » Sat Jul 21, 2007 8:24 pm

Udosuk, that's a very interesting elimination. So much so, that I've modified by solver to convert puzzles to RN and CN spaces. Starting to evaluate what techniques are applicable in these spaces, I am finding a disappointing few so far. Hidden singles map to naked singles, but boxes are missed. Of course there's Arcilla's mapping between fish and locked sets. Denis covers more in his book. I've not looked at all possibilities, but was curious about your elimination in RC space. My best take on it is that it is a Kraken Column Swordfish (c357/r458, fins=r7c5|r9c7) on 1's. Three strong links on 3's (r5c3=3=r1c3-3-, r7c5=3=r3c5-3-, r9c7=3=r9c6-3-) connect the fins and a cover set, r5 which is really only r5c3, to the candidate elimination cell, r1c6.
Code: Select all
+---------------------+------------------------+---------------------+
|    1   2467   3467@ | 24678       5  2678-3  |   678   4678     9  |
|    5  24679   4679  |     1    2478   26789  |   678      3   467  |
|  369   4679      8  |  4679     347$    679  |     2      5     1  |
+---------------------+------------------------+---------------------+
| 2389      5    179* |  2789    1278*      4  |   679   2679   367  |
|    4    179  1379*@ |     5       6    1279  |   379    279     8  |
| 2689   6789    679  |     3     278    2789  |     4      1     5  |
+---------------------+------------------------+---------------------+
|  689  14689      2  |  4678  13478*$   1678  |     5  46789  3467  |
|  689      3   1469* | 24678   12478*      5  | 16789* 46789   467  |
|    7   1468      5  |   468       9    1368% | 1368*%   468     2  |
+---------------------+------------------------+---------------------+

I'm really looking forward to finding out what else can be done in RN and CN spaces.
Mike Barker
 
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Joined: 22 January 2006

Postby daj95376 » Sun Jul 22, 2007 3:01 am

Let's try Mike's explanation from a slightly different perspective. All of the 3s can be colored using simple coloring -- say Blue and Green. In particular, [r5c3], [r7c5], and [r9c7] will be the same color -- say Blue. Now, there's also a Sashimi Swordfish c357\r458 in <1> with fin cells [r7c5] and [r9c7]. Now, either one of the fin cells must be true or the Sashimi cell [r5c3] must be true. Thus, we can conclude that one of three Blue cells must be false for <3> -- forcing all of the Green cells true for <3>. Impressive!!!

An alternate approach.

Code: Select all
Blue cells for <3> => [r8c7]=1 => [r4c3]=1 => [c5] void of <1> => Green cells for <3>
*-----------------------------------------------------------------------------*
|  1       2467   G3467   |  24678   5      B23678  |  678     4678    9      |
|  5       24679   4679   |  1       2478    26789  |  678     3       467    |
| B369     4679    8      |  4679   G347     679    |  2       5       1      |
|-------------------------+-------------------------+-------------------------|
| G2389    5       179    |  2789    1278    4      |  679     2679   B367    |
|  4       179    B1379   |  5       6       1279   | G379     279     8      |
|  2689    6789    679    |  3       278     2789   |  4       1       5      |
|-------------------------+-------------------------+-------------------------|
|  689     14689   2      |  4678   B13478   1678   |  5       46789  G3467   |
|  689     3       1469   |  24678   12478   5      |  16789   46789   467    |
|  7       1468    5      |  468     9      G1368   | B1368    468     2      |
*-----------------------------------------------------------------------------*
daj95376
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Postby ravel » Sun Jul 22, 2007 6:06 pm

Since the cn- and rn-views are just other representations of the same information you also have in the normal sudoku grid, they cannot provide new techniques by themselves. Each technique you use there, has an equivalent in the normal grid.

But some things are easier to spot and describe.

One is, that x-wing, swordfish and jellyfish appear as pair, triple and quad.

The other nice thing i know of, are the "hidden xy-wings". Here is a sample:
Suppose you have a grid with
Code: Select all
 .  1A  2B |  .
 .  1C  .  |  1D
 .  .   .  |  12E
-------------------
 .  .   2F |  2G
(A-G being arbitrary numbers > 2) with strong links for 1 in column 2 and 4 and for 2 in columns 3 and 4.

Then there is a nice loop:
[r1c2]=1=[r2c2]-1-[r2c4]=1=[r3c4]=2=[r4c4]-2-[r4c3]=2=[r1c3]-2-[r1c2]
I.e. either r1c2=1 or r1c2<>2, in either case r1c2<>2.

In cn-view strong links in columns come as bivalue cells and you simply have:
Code: Select all
     c1  c2  c3 | c4
d1:   .  12  .  | 23
d2:   .  X   14 | 34

with the "xy-chain"
either d1c2=1
or d1c2=2 => d1c4=3 => d2c4=4 => d2c3=1
=> d2c2<>1 <=> r1c2<>2.

The drawback: strong links in boxes are hard to spot. If you have a strong link for 1 between r1c2 and r2c1 instead of r2c2, you can make the same deduction, but the cn-view might look like
Code: Select all
     c1    c2   c3  | c4
d1: 24589 14578 479 | 23
d2:   .     X   14  | 34


For me the views would become interesting, when there is a free program available, where you can (graphically) switch between the views to study a grid. But i would not do that manually on paper (in fact i dont solve much puzzles on paper, that need such advanced techniques).
ravel
 
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Postby Bob Hanson » Tue Dec 23, 2008 1:12 am

There's a discussion of this at the Sudoku Assistant site http://www.stolaf.edu/people/hansonr/sudoku

The idea also extends to almost-locked sets in a very interesting way.
Bob Hanson
 
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Postby PIsaacson » Tue Jan 20, 2009 6:19 pm

ravel wrote:For me the views would become interesting, when there is a free program available, where you can (graphically) switch between the views to study a grid.

Download Ruud's SudoCue: { broken links } www.sudokuvault.com/SudoCueV310.msi or www.sudokuvault.com/SudoCueV320.msi

I'm using the beta 3.20 version with nrczt chains - I think that's the main difference. Use the VIEW pulldown to switch between RC/RN/CN space views.
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Re: a new (?) view of fish (naked or hidden)

Postby Sudtyro2 » Thu Jun 27, 2013 10:02 pm

There's a country-music lyric that goes, “He can't even bait a hook.” Well, I can bait a hook, but I rarely catch a fish. The special lists discussed in the above posts by arcilla (et al) are certainly helping to change that situation for me, as a die-hard manual solver.

Aside from finding fish directly, an interesting by-product of the method is that one can sometimes replace a very large (and complex) existing fish with a much smaller basic one by judiciously applying the lists. A 5-Fish to 2-Fish example reduction follows.

The grid below was taken from Allan Barker's website in 2008 as an (advanced) sample problem for his Set/Linkset method. It was equivalent to a 5-Fish with three extra cover sectors, c124b36\r25689c89b7, and therefore of raw rank 3. Allan next used some complicated (for me!) linkset-triplet logic to reduce the rank from 3 to 1. The intersection of linksets (sectors) r8 and c9 at the CEC (candidate-elimination cell) then provided for the indicated elimination.
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  .  . |  8  .  .

Being unable to fully grasp that linkset-triplet logic, I then turned to
daj95376 and hobiwan for some fishing t(r)ips. Both provided help, and hobiwan also offered up the following Kraken fish having the same CEC.
Code: Select all
Finned Franken Squirmbag: r3478b4\c1359b9 + fr4c8 fr6c2 fr8c2 efr4c3
Finned Franken Squirmbag: r2347b4\c135b39 + fr2c2 fr4c8 fr6c2 efr4c3
Finned Franken Whale: r23478b4\c13589b9 + fr2c2 fr6c2 fr8c2 efr4c3
Finned Mutant  Whale: r23467c2\c35789b7 + fr2c1 efr2c2 efr6c2

These fish certainly do the job, but they are still large in size, involve endofins(ef), and two of the fish place the CEC in a base sector (but also at the intersection of two covers). Fairly complex fish for a manual solver.

Just recently, I decided to try out arcilla's lists on Allan's grid. List headings below are RPC (row numbers, per col) and CPR (col numbers, per row). The CEC position is bolded in both lists:

RPC: (2579)(2689)(3467)(59)(4678)( - )(679)(24578)(378)
CPR: ( - )(128)(39)(358)(148)(2357)(135789)(2589)(1247)

For a basic fish of rows and columns only, the CEC will have to be covered by either r8 or c9. Column 9 is a good first choice for the cover since it contains only two additional rows, r37, beyond the CEC's own r8. We now examine the CPR list for those two rows. CPR rows 3 and 7 both contain the required col-9 entry (underlined above), but notice that row 3 has only two column entries, (39), while row 7 has six, (135789), one of which is a 3. So, by simply treating the row-7 entries (1578) as spoilers (obstacles), one has then created the arcilla-equivalent of an AAAALS, where the LS is a (39) pair. This immediately translates to a basic 2-Fish (X-Wing) with four fins: r37\c39 + fr7c1578. Two fins (r7c78) are regular, and two (r7c15) are remote. The resulting grid is shown below. (*) marks a covered base cell, and (#) marks a fin.
Code: Select all
 .  .  . |  .  .  8 |  .  .  .
 8  8  . |  .  .  . |  .  8  .
 .  . *8 |  .  .  . |  .  . *8  r3 base
---------+----------+---------
 .  .  8 |  .  8  . |  .  8  .
 8  .  . |  8  .  . |  .  8  .
 .  8  8 |  .  8  . |  8  .  .
---------+----------+---------
#8  . *8 |  . #8  . | #8 #8 *8  r7 base
 .  8  . |  .  8  . |  .  8 -8
 8  8  . |  8  .  . |  8  .  .

The Fish pattern is strongly linked to its Fin (fin group) as [Fish]=[Fin]. So, if each (true) fin can see the CEC, then the Fish deduction is valid. The regular fins, r7c78, see the CEC directly. Suitable AICs (hopefully) for the two remote fins are shown below. Symbol (||) signifies a Strong-Inference Set (SIS) formed by either a row or a column. SIS are easy to use and appear to be well documented in forming AIC nets.
Code: Select all
r7c1-r89c2
      ||
      r6c2-r6c7=r79c7[or r45c8-r2c8=r3c9]
      ||
      r2c2-r3c3=r3c9
r7c5-r4c5
      ||   
     r4c8-r2c8=r3c9
      ||
     r4c3-r3c3=r3c9

And there you have it...a basic row-based 2-Fish with four fins, courtesy of arcilla's CPR list! OK readers...is this for real or have I somehow completely missed the boat? If not, then brace yourselves for Parts II and III of this fish story.
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Re: a new (?) view of fish (naked or hidden)

Postby daj95376 » Fri Jun 28, 2013 3:40 am

Sudtyro2 wrote:
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  .  . |  8  .  .

Code: Select all
 .  .  . |  .  .  8 |  .  .  .
 8  8  . |  .  .  . |  .  8  .
 .  . *8 |  .  .  . |  .  . *8  r3 base
---------+----------+---------
 .  .  8 |  .  8  . |  .  8  .
 8  .  . |  8  .  . |  .  8  .
 .  8  8 |  .  8  . |  8  .  .
---------+----------+---------
#8  . *8 |  . #8  . | #8 #8 *8  r7 base
 .  8  . |  .  8  . |  .  8 -8
 8  8  . |  8  .  . |  8  .  .


You appear to have derived:

Code: Select all
 Kraken X-Wing: r37\c39 w/fin cells r7c78 and Kraken cells r7c15  =>  r8c9<>8

An alternate with only one Kraken cell:

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  .  . | #8  .  .

 Kraken Franken Swordfish c27b3\r268 w/fin cells r3c9,r7c78 and Kraken cell r9c2  =>  r8c9<>8

The alternate can also be viewed as a Kraken Column [c2] with memory:

Code: Select all
 (8)r2c2 - r2c8 = r3c9                               - (8)r8c9
  |
 (8)r6c2 - r6c7               = r79c7                - (8)r8c9
  |
 (8)r8c2                                             - (8)r8c9
  |
 (8)r9c2 - r9c4 = r5c4 - r5c1 = r46c3 - r3c3 = r3c9  - (8)r8c9
         - r6c2 ............./
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Re: a new (?) view of fish (naked or hidden)

Postby Sudtyro2 » Fri Jun 28, 2013 6:53 pm

Thanks, daj, for the positive feedback and for the alternate 3-Fish example.

The main advantage of the arcilla-list approach appears to be that small, basic fish (rows and columns only) can be developed manually and very quickly, and that fins (both regular and remote) are simply the “almost” elements of the arcilla-equivalent locked subsets. The only drawback, so far, seems to be dealing with extra remote (Kraken) fins and their requisite AICs (which I actually enjoy doing). I can illustrate that issue by continuing with the rest of the fish story...

To review, we're working with a grid having one CEC:
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  .  . |  8  .  .

The arcilla CPR list has already given us a row-based 2-Fish (X-Wing) with four fins (two were remote): r37\c39 + fr7c1578.

Now, a 2-Fish with column bases may also be available by choosing r8 (instead of c9) to cover the CEC and then applying the RPC list.

RPC: (2579)(2689)(3467)(59)(4678)( - )(679)(24578)(378)

Two row-8 entries (out of the three underlined above) are needed for a 2-Fish, so one must choose from among RPC cols 2, 5 and 8. Four row-pair combinations are possible: (28),(48),(68),(78). Starting with the (28) combination, RPC row entries 69 from col 2 and row entries 457 from col 8 are all treated as spoilers, for the arcilla-equivalent of an AAAAALS, where the LS is the (28) pair. This is another basic 2-Fish (X-Wing) with five fin cells: c28\r28 + fr69c2 fr457c8. One fin (r7c8) is regular, and the remaining four are remote (Kraken). The resulting grid is shown below. (*) marks a covered base cell, and (#) marks a fin.
Code: Select all
    c2                    c8
 .  .  . |  .  .  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  .  . |  8  .  .

The one regular fin (r7c8) sees the CEC directly. AICs for the four remote fins are listed below. Note that the SIS shown here was also used in the previous row-based 2-Fish.
Code: Select all
r4c8-r2c8=r3c9
r5c8-r2c8=r3c9
r6c2-r6c7=r79c7
r9c2-r9c4=r78c5-r4c5
                 ||
                r4c8-r2c8=r3c9
                 ||   
                r4c3-r3c3=r3c9

OK, so that gives us a column-based 2-Fish, but what about the other possible row-pair combinations, (48),(68),(78)? I tried all three, and each one of these row-pairs does generate a potential multi-fin 2-Fish, but there is a single fin in each case that cannot see the CEC (or at least I couldn't find a suitable chain). The (48) combination is a good example. The 2-Fish with five fins would be: c58\r48 + fr257c8 fr67c5. However, fr2c8 cannot see the CEC, so the Fish deduction is invalid. [EDIT: Per PM with daj95376, remote fin r2c8 does, in fact, see the CEC via an assignment sequence. Suitable fin chains were also later found for the (68) and (78) row-pair combinations.]

One solution to the problem is simply to eliminate the offending fin by adding an additional cover, r2, and thus moving up to a larger 3-Fish. The RPC list can generate this new fish directly from the entries in cols 2,5,8 as an equivalent AAAALS, where the LS is the (248) triple, and the remaining entries, (69) in col 2, (67) in col 5, and (57) in col 8, are the spoilers. This translates to a 3-Fish with six fins: c258\r248 + fr69c2 fr67c5 fr57c8, as shown below.
Code: Select all
    c2         c5         c8
 .  .  . |  .  .  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  .  . |  8  .  .

One fin (r7c8) is regular, and the other five are remote (Kraken):
Code: Select all
r5c8-r2c8=r3c9
r6c2-r6c7=r79c7
r6c5-r6c7=r79c7
r7c5-r4c5
      ||   
     r4c8-r2c8=r3c9
      ||
     r4c3-r3c3=r3c9
r9c2-r9c4=r78c5-r4c5
                 ||
                r4c8-r2c8=r3c9
                 ||   
                r4c3-r3c3=r3c9

Note that four of the five remote fins have already appeared in the previous 2-Fish examples. Only remote fin (r6c5) is new.

The arcilla fishing hole is small, but very productive...

As an addendum, note that parts of the AICs developed for the fins in the 2-Fish examples can also be used to directly construct a single AIC net for the original grid that achieves the desired elimination, r8c9<>8: [EDIT: Corrected cell sequencing of the first line]
Code: Select all
           r6c2-r6c7=r79c7
            ||
           r2c2-r3c3=r3c9
            ||
           r8c2
            ||
           r9c2-r9c4=r78c5-r4c5
                            ||
                           r4c8-r2c8=r3c9
                            ||   
                           r4c3-r3c3=r3c9
Last edited by Sudtyro2 on Sat Jul 27, 2013 4:17 pm, edited 2 times in total.
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RE: Kraken

Postby Pat » Wed Jul 03, 2013 1:58 pm

Sudtyro2 wrote:——small, basic fish (rows and columns only)
can be developed manually and very quickly——

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  .  . |  8  .  .

——a row-based 2-Fish (X-Wing)
with four fins (two were remote):
    r37\c39
    with fins r7c1578

    2-Fish (X-Wing) is convenient, yes;
    but, go one further:

      1-Fish (hidden single in row)

      r7\c9
      with fins r7c13578
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Re: RE: Kraken

Postby Sudtyro2 » Wed Jul 03, 2013 6:04 pm

Pat wrote:
    2-Fish (X-Wing) is convenient, yes;
    but, go one further:

      1-Fish (hidden single in row)

      r7\c9
      with fins r7c13578


Awesome catch, Pat!
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Postby daj95376 » Wed Jul 03, 2013 6:24 pm

Pat wrote:r7\c9 __ with fins r7c13578

Hmmm! Who says that the 1-Fish has to actually have an intersection between the base set and the cover set in the candidate grid?

Code: Select all
 Sashimi 1-Fish c7\r8 with fin cells r79c7 and single remote fin cell r6c7

This is essentially equivalent to Kraken Column [c7] for <8>, because one of the fin cells must be true.
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Re: Re:

Postby Sudtyro2 » Thu Jul 04, 2013 11:18 am

daj95376 wrote:
Code: Select all
 Sashimi 1-Fish c7\r8 with fin cells r79c7 and single remote fin cell r6c7

This is essentially equivalent to Kraken Column [c7] for <8>, because one of the fin cells must be true.


I'm weak on Sashimi-anything, but how does the (true) remote fin (r6c7) see the CEC? I can't find a suitable chain.
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re: Kraken

Postby Pat » Thu Jul 04, 2013 11:22 am

Pat wrote:

    r7\c9
    with fin r7c78
    and Kraken r7c1,r7c3,r7c5

but my personal preference
would be:

    r7\b9
    with Kraken r7c1,r7c3,r7c5
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