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

Advanced methods and approaches for solving Sudoku puzzles

### Re: Re:

Sudtyro2 wrote:
daj95376 wrote:
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` Sashimi 1-Fish c7\r8 with fin cells r79c7 and single remote fin cell r6c7`

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

It's a network that's best described as sequences of assignments:

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` r6c7=8, r78c8=8                                 =>  r8c9<>8           ||         r2c8=8, r3c3=8, r5c1=8, r9c4=8, r8c2=8  =>  r8c9<>8`

[Edit: removed an assignment that should have been an elimination. Thanks Pat !]
Last edited by daj95376 on Fri Jul 05, 2013 10:19 am, edited 1 time in total.
daj95376
2014 Supporter

Posts: 2624
Joined: 15 May 2006

### RE: Kraken

Sudtyro2 wrote:
Pat wrote:
1-fish

r7\c9——

Awesome catch, Pat!

thanks
but you did all the work
all i did was reduce the size of your fish
— down to the absolute minimum

incidentally i m very ignorant on Kraken,
expecting the experts to tell me i'm all wet,
show i'm swimming up the wrong tree

Pat

Posts: 3713
Joined: 18 July 2005

### Re: RE: Kraken

Pat wrote:
all i did was reduce the size of your fish
— down to the absolute minimum

OK, Pat and daj...your 1-Fish are amazing, so let me try a cast in this pond...

Pat...following up on your example, I now take arcilla's CPR list, look at the row-3 (instead of row-7) column entries (39), keep only the 9, and treat the (3) as a potential fin. This gives:
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`1-Fish r3\c9 + fr3c3`

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` .  .  . |  .  .  8 |  .  .  . 8  8  . |  .  .  . |  .  8  .  .  . #8 |  .  .  . |  .  . *8 r3 base---------+----------+---------  .  .  8 |  .  8  . |  .  8  .  8  .  . |  8  .  . |  .  8  .  .  8  8 |  .  8  . |  8  .  . ---------+----------+---------  8  .  8 |  .  8  . |  8  8  8  .  8  . |  .  8  . |  .  8 -8  8  8  . |  8  .  . |  8  .  .`

Daj...following your example, I try an assignment-sequence network on the fin (if I understand how it works, and since I can't do the chains yet). This seems to give:
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`r3c3=8, r2c8=8, r6c7=8, r5c1=8, r4c5=8, r9c4=8, r8c2=8 => r8c9<>8`

Can this 1-finned 1-Fish possibly be right?
Sudtyro2

Posts: 652
Joined: 15 April 2013

### Re: RE: Kraken

Sudtyro2 wrote:_

Pat...following up on your example, I now take arcilla's CPR list, look at the row-3 (instead of row-7) column entries (39), keep only the 9, and treat the (3) as a potential fin. This gives:
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`1-Fish r3\c9 + fr3c3`

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` .  .  . |  .  .  8 |  .  .  . 8  8  . |  .  .  . |  .  8  .  .  . #8 |  .  .  . |  .  . *8 r3 base---------+----------+---------  .  .  8 |  .  8  . |  .  8  .  8  .  . |  8  .  . |  .  8  .  .  8  8 |  .  8  . |  8  .  . ---------+----------+---------  8  .  8 |  .  8  . |  8  8  8  .  8  . |  .  8  . |  .  8 -8  8  8  . |  8  .  . |  8  .  .`

Daj...following your example, I try an assignment-sequence network on the fin (if I understand how it works, and since I can't do the chains yet). This seems to give:
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`r3c3=8, r2c8=8, r6c7=8, r5c1=8, r4c5=8, r9c4=8, r8c2=8 => r8c9<>8`

Can this 1-finned 1-Fish possibly be right?

Your Kraken fish appears correct to me. FWIW, you can remove one of your assignments.

r3c3=8, r2c8=8, r6c7=8, r5c1=8, r4c5=8, r9c4=8, r8c2=8 => r8c9<>8

I should mention that there are few who would use this approach ... unless the elimination leads to a significant reduction in the complexity of the grid for the puzzle.

FWIW, I would opt for something like:

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` finned Kraken Starfish r3c248b4\r2589c3 w/fin cells r3c9,r7c8  =>  r8c9<>8 where the Kraken cells r4c8,r6c2 trivially generate the desired elimination (8)r4c8 \          - (8)r6c7 = (8)r79c7 - (8)r8c9 (8)r6c2 /`

Since I'm not an expert on Kraken Fish, I suspect there are numerous smaller fish whose Kraken cells trivially generate the desired elimination.

Aside Note on the exemplar:

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`  finned Kraken Starfish r3c248b4\r2589c3 w/fin cells (#) and Kraken cells (@)  +-----------------------------------+  |  .  /  *  |  /  .  .  |  .  /  .  |  |  *  X  *  |  X  *  *  |  *  X  *  |  |  /  /  X  |  /  /  /  |  /  /  #  |  |-----------+-----------+-----------|  |  /  /  X  |  /  .  .  |  .  @  .  |  |  X  /  *  |  X  *  *  |  *  X  *  |  |  /  @  X  |  /  .  .  |  .  /  .  |  |-----------+-----------+-----------|  |  .  /  *  |  /  .  .  |  .  #  .  |  |  *  X  *  |  X  *  *  |  *  X **  |  |  *  X  *  |  X  *  *  |  *  X  *  |  +-----------------------------------+`

It's not readily apparent from the above that the Kraken cells might trivially lead to an elimination in r8c9.
daj95376
2014 Supporter

Posts: 2624
Joined: 15 May 2006

### Re: RE: Kraken

daj95376 wrote:
Your Kraken fish appears correct to me. FWIW, you can remove one of your assignments.
r3c3=8, r2c8=8, r6c7=8, r5c1=8, r4c5=8, r9c4=8, r8c2=8 => r8c9<>8

Thanks daj...sometimes I can't even see an obvious conjugate pair (r59c4).

daj95376 wrote:
I should mention that there are few who would use this approach ... unless the elimination leads to a significant reduction in the complexity of the grid for the puzzle.
FWIW, I would opt for something like:
Code: Select all
` finned Kraken Starfish r3c248b4\r2589c3 w/fin cells r3c9,r7c8  =>  r8c9<>8 where the Kraken cells r4c8,r6c2 trivially generate the desired elimination(8)r4c8 \          - (8)r6c7 = (8)r79c7 - (8)r8c9 (8)r6c2 /`

Since I'm not an expert on Kraken Fish, I suspect there are numerous smaller fish whose Kraken cells trivially generate the desired elimination.

What I'm seeing so far is a simple trade-off: The smaller the fish, the more numerous and/or complex the Kraken fins. From a manual-solver standpoint, however, I like the smaller fish (via arcilla's lists) and feel more comfortable working the AIC nets and your assignment sequences. I might want to start a new topic somewhere else about those AIC nets. I can't find suitable chains for the last two 1-Fish (yours and mine).

BTW, per the intent of the lead post, we've all managed to go from a 5-Fish to a 1-Fish size reduction (or 6-to-1 ratio, counting hobiwan's two Whales). That 6-to-1 reduction is about the same as the reduction seen in my (limited) stock-share values during the 2008 market crash.
Sudtyro2

Posts: 652
Joined: 15 April 2013

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

Hi Sudtyro2 and All,

I hesitated a long time to post the following as it may be considered as OT.
However...
Sudtyro2 wrote:Well, I can bait a hook, but I rarely catch a fish.
I don't look for fish anymore as fish theory, however highly valuable, became very intricated, complex, hard to apply and last but not least, only 99,9999... % rigorous.
But ...
For each digit, I always look for all the exclusions based only on the set of candidates for the digit in question.
The method I am using is very simple, easy and fast (~1 min in the worst case) and it certainly has been described elsewhere (but I don't know where).
When I am posting, I nevertheless try to translate as simply as possible all these exclusions in "Fish" language, but I generally only state the dimensions .
And finally, "Fish" theory will never gives all the eliminations in all the cases, except maybe by a stupid computer to which an undefined number of "rules" has to be given !

So be patient, if you want to read further, as I will run into some details, but not all.

The problem : to find all the exclusions in the following set of candidates for the digit 8.
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`.  .  . |  .  .  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  .  .`
Some preliminary observations :
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`In the B/B-Plot, the digit 8 is present due to 2 clusters :                                     8r3c3      8r2c8                                         \\    //the bilocal 8r5c4=8r9c4 and the V-cluster 8r3c9Furthermore, there are a lot of ALC : 8B1,5,6,8 and 8C7,9!`

Hidden Text: Show
In this particular case, the best starting point is 8C4.

If r9c4=8, then the set of candidates reduces to (E=exclusion)
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` .  .  . |  .  .  8 |  .  .  . 8  8  . |  .  .  . |  .  8  . .  .  8 |  .  .  . |  .  .  8---------+----------+--------- .  .  8 |  .  8  . |  .  8  . 8  .  . | E8  .  . |  .  8  . .  8  8 |  .  8  . |  8  .  .---------+----------+--------- 8  .  8 |  . E8  . |  8  8  8 .  8  . |  . E8  . |  .  8  8E8 E8  . |  8  .  . | E8  .  .   Now, either r6c7=8, giving rise to    .  .  . |  .  .  8 |  .  .  .   E8  8  . |  .  .  . |  .  8  .    .  .  8 |  .  .  . |  .  .  8   ---------+----------+---------    .  . E8 |  .  8  . |  . E8  .    8  .  . | E8  .  . |  . E8  .    . E8 E8 |  . E8  . |  8  .  .   ---------+----------+---------   E8  .  8 |  . E8  . | E8  8  8    .  8  . |  . E8  . |  .  8 e8   E8 E8  . |  8  .  . | E8  .  . ---> CASE I   where (e)8r8c9 is the last exclusion by a 2-Fish(FXW/K/ER) in the unsolved boxes B1397.   Or r7c7=8, giving rise to    .  .  . |  .  .  8 |  .  .  .    8 E8  . |  .  .  . |  . E8  .    .  . E8 |  .  .  . |  .  .  8   ---------+----------+---------    .  .  8 |  .  8  . |  . E8  .   E8  .  . | E8  .  . |  .  8  .    . E8  8 |  .  8  . | E8  .  .   ---------+----------+---------   E8  . E8 |  . E8  . |  8 E8 E8    .  8  . |  . E8  . |  . E8 E8   E8 E8  . |  8  .  . | E8  .  . ---> CASE II   with no further exclusion (unsolved boxes : B45; 2 solutions).`
If r5c4=8, then the set of candidates reduces to
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` .  .  . |  .  .  8 |  .  .  . 8  8  . |  .  .  . |  .  8  . .  .  8 |  .  .  . |  .  .  8---------+----------+--------- .  .  8 |  . E8  . |  .  8  .E8  .  . |  8  .  . |  . E8  . .  8  8 |  . E8  . |  8  .  .---------+----------+--------- 8  .  8 |  .  8  . |  8  8  8 .  8  . |  .  8  . |  .  8  8 8  8  . | E8  .  . |  8  .  .   The Skyscraper(R34) leads to further exclusions :    .  .  . |  .  .  8 |  .  .  .    8  8  . |  .  .  . |  .  E8  .    .  . E8 |  .  .  . |  .  .  8   ---------+----------+---------    .  .  8 |  . E8  . |  .  8  .   E8  .  . |  8  .  . |  . E8  .    .  8  8 |  . E8  . |  8  .  .   ---------+----------+---------    8  .  8 |  .  8  . |  8  8 E8    .  8  . |  .  8  . |  .  8 E8    8  8  . | E8  .  . |  8  .  . ---> CASE III   Here, there are no more exclusions.   Proof : If r4c8=8, then there are no other exclusions than 8r78c8 in the remaining unsolved boxes    (unfavorable pattern of unsolved boxes and no Locked Candidates). And 8r78c8 are used with 8r6c7.`
Now the key point : 8r9c9 is the only common exclusion in the three cases. It is the only candidate that doesn't belong to at least one solution for the digit 8.

The simplest interpretation (not always as easy to find; it generally takes me more than 1 min in the worst case as I am a slow solver) is obtained via a forcing chain as in Allan Barker's Starfish :
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`+-------------+-----------+---------------+| .    .    . | .    .  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    .    .   |+-------------+-----------+---------------+`
In English : r8c9=8->r2c8=8;FXW(8C14) and r6c7=8; C2 is devoid of 8 :=> -8r8c9
or
by using a Forbidding Matrix (giving all the details of the previous) :
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`r8c9r3c9=r2c8     r2c1=r5c1=r79c1          r5c4=r9c4  -------> FXWing(C14)     r45c8==========r6c7r8c2=r2c2======r9c2=r6c2`
or
on a single line :
r3c9=r2c8-[r2c2 and r2c1=FXW(C14)-r9c2 and r45c8=r6c7-r6c2]=r8c2

Another interpretation by reading the forbidding matrix from the last row :
Kraken Column 8C2 --> r8c2=r3c9 :=> -8r8c9
r2c2-r2c8=r3c9
||
r6c2-r6c7=r45c8-r2c8=r3c9
||
r8c2
||
r9c2-FXWing(r59c4.r579c1)=r2c1-r2c8=r3c9

Notes :
From the rows of the Forbidding Matrix, one reads the base sets of the "Fish" : B3,C1,C4,B6,C2 respectively and
from the columns of the Forbidding Matrix, one reads the cover sets : C9R8(in the first column giving the exclusion) and R2C8,R5,B7R9,R6 respectively.
I let you find the fins as well as the Path of Overlapping Linksets and the subsequent calculations of all the "local ranks" as I don't use them!

Finally, if I had to name this method, I would be very cautious as any name I used in the past led to confusion.

Best Regards,

JC.
JC Van Hay

Posts: 719
Joined: 22 May 2010

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

JC Van Hay wrote:Hi Sudtyro2 and All,
I hesitated a long time to post the following as it may be considered as OT.

Hi JC...thanks for your post, although it's probably a bit OT. The key point of my lead post was to look at the possibility of big reductions in fish size based on examination of arcilla's lists. But, in my case I already had advance knowledge of the CEC, which allowed me to choose which cover-sector(s) I needed from arcilla's lists.

Your approach seems to go directly for the single-digit grid exclusions right off, although your example grids do indicate some multiple initial choices for assignment (placement), as well as intermediate use of some short AICs and even a 2-Fish (if I understand your notation), in the overall exclusion process, leading to r8c9<>8.

Perhaps we could pursue these issues in a separate topic?
Sudtyro2

Posts: 652
Joined: 15 April 2013

JC Van Hay wrote:
For each digit, I always look for all the exclusions based only on the set of candidates for the digit in question.

The method I am using is very simple, easy and fast (~1 min in the worst case)
and it certainly has been described elsewhere (but I don't know where).

Pat

Posts: 3713
Joined: 18 July 2005

### POM and "templates"

in the templates discussion,
Myth Jellies (2005.Sep.20) wrote:

This is similar to the Pattern Overlay Method that I use to solve SudoKu by hand.

I describe the method in
www.sudoku.org.uk/discus/messages/2/231.html?1127148223

also in Mike Barker's list,
Pattern Overlay Method (POM) is listed with a broken link

? any suggestions re the broken link ?

Pat

Posts: 3713
Joined: 18 July 2005

### Re: POM and "templates"

Pat wrote:? any suggestions re the broken link ?

Give up on the broken link and use the Sudopedia Mirror description.
daj95376
2014 Supporter

Posts: 2624
Joined: 15 May 2006

### Re: A few loose ends

It was noted in a post on this thread on Jul 6, 2013 that I couldn't find suitable AICs for the remote fins on two of the sample 1-Fish (my own and DAJ's). Well, AIC is apparently not the correct term...more like “assignment-network diagram” similar to those to be discussed below. But, in the process of later working up those diagrams, it became clear that many other finned 1-Fish and N-Fish solutions were also available, and that network diagrams could be easily found for each remote fin of all the Fish configurations tried. In fact, this led to a rather startling observation regarding the grid and its candidate-elimination cell (CEC):

EVERY candidate in the ENTIRE grid can see (weakly link to) the CEC, either directly or remotely!

The direct links are simply those involving peers of the CEC. For all other candidates the remote links have network diagrams comprising implication streams with branching strong-inference sets (SIS) where needed.

The grid in question is repeated here. The CEC is indicated with a (-).
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` .  .  . |  .  .  . |  .  .  . 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  .  .`

Listed below, as hidden text, are the network diagrams for every candidate in the grid that is not a peer to the CEC. The special symbol (||) links elements in a SIS, which itself can contain only one truth. Each weak-inference link emanates unidirectionally from an assignment, which simply forces any downstream peer to be false. Most of the implication streams are relatively short and easy to trace mentally. Only three from the hidden list are somewhat complex and were derived by first sketching the assignment/elimination sequence on a copy of the grid:

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`Candidate r2c8 is one remote fin for 2-Fish c58\r48 + fr7c8 rfr25c8 rfr67c5.Assignment network:r2c8=8, r3c3,r6c7=8, r5c1=8, r9c4=8, r8c2=8 => r8c9<>8  Candidate r3c3 is the remote fin for 1-Fish r3\c9 + rfr3c3.Assignment network:r3c3=8, r2c8=8, r6c7=8, r5c1=8, r9c4=8, r8c2=8 => r8c9<>8Candidate r6c7 is the remote fin for DAJ's Sashimi 1-Fish c7\r8 + fr79c7 rfr6c7.Assignment network:r6c7=8, r78c8=8                                => r8c9<>8          ||        r2c8=8, r3c3=8, r5c1=8, r9c4=8, r8c2=8 => r8c9<>8`

Hidden Text: Show
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`r2c1-r3c3=r3c9r2c2-r3c3=r3c9                ---------------r7c3               /                ||    -r2c12=r3c3-r4c3            ||   /             ||             ||r2c8-r45c8=r6c7-r6c23           ||                 ||             ||                r5c1-r5c4=r9c4-r9c2                    \           ||                     ----------r79c1                                                   ||                               r8c2      --------------------r46c3    /                      ||r3c3-r3c9=r2c8-r45c8=r6c7-r6c2   |                       ||   |                      r5c1-r5c4=r9c4-r9c2   |                          \           ||        |                           ----------r79c1      \                                     ||      -----------------------------------r7c3                                          ||                                         r8c2r4c3-r3c3=r3c9r4c5-r5c4=r9c4-r9c2                ||               r8c2                ||               r6c2-r6c7=r79c7                ||               r2c2-r3c3=r3c9r4c8-r2c8=r3c9r5c1-r5c4=r9c4-r9c2                ||               r8c2                ||               r6c2-r6c7=r79c7                ||               r2c2-r3c3=r3c9r5c4-r4c5      ||     r4c8-r2c8=r3c9      ||     r4c3-r3c3=r3c9r5c8-r2c8=r3c9r6c2-r6c7=r79c7r6c3-r6c7=r79c7r6c5-r6c7=r79c7     r78c8      ||r6c7-r45c8            --------------r7c3   |  ||            /                ||   | r2c8-r2c12=r3c3-r4c3            ||    \                 ||             ||      ---------------r6c23           ||                      ||             ||                     r5c1-r5c4=r9c4-r9c2                         \           ||                          ----------r79c1                                                        ||                                    r8c2r7c1-r89c2      ||      r6c2-r6c7=r79c7      ||     r2c2-r2c8=r3c9r7c3-r89c2                or        r7c3-r3c3=r3c9         ||      r6c2-r6c7=r79c7      ||     r2c2-r2c8=r3c9r7c5-r4c5      ||     r4c8-r2c8=r3c9      ||     r4c3-r3c3=r3c9r9c1-r9c2      ||     r8c2      ||     r6c2-r6c7=r79c7      ||     r2c2-r3c3=r3c9r9c2-r9c4=r78c5-r4c5                 ||                r4c8-r2c8=r3c9                 ||                r4c3-r3c3=r3c9r9c4-r9c2      ||     r8c2      ||     r6c2-r6c7=r79c7      ||     r2c2-r3c3=r3c9`

So, a question for the experts:
Is this just one big whale of a fluke (or vv, maybe )?

Either way, it seems there is an enormous pool of NxN Fish that will produce the desired elimination. The main consideration for an N-Fish is that one of its cover sectors must include the CEC. All potential fins from the N base sectors can already see the CEC, so the elimination from the basic Fish pattern is guaranteed. For row/col 1-Fish alone, I count 14 available for this CEC.

Edit to add r7c3 alternate chain in the hidden-text list (courtesy daj95376, 01/15/2014).
Last edited by Sudtyro2 on Thu Jan 16, 2014 7:40 pm, edited 1 time in total.
Sudtyro2

Posts: 652
Joined: 15 April 2013

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

You left out r8c9 (It can't see itself directly, but ...)

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`r8c9-r3c9=r3c3-r2c2                ||               r9c2-r9c4=r5c4-r5c1                ||             ||               r8c2           r4c3-r3c3=r3c9                ||             ||               r6c2           r6c23-r6c7=r79c7                   \               /                     -------------`
blue

Posts: 784
Joined: 11 March 2013

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

To turn a net into a NxM fish the base set can be composed of the houses where strong links are used and the cover set of those where weak ones are. Trying that on this puzzle I found I needed 10 cover sectors, so I could be generous with the base sectors, which I got up to 7. As one sector was common to both sets, this reduced to 9 cover sectors and 6 base sectors, ie difference or k-rank of 3. But this was too great for any cell to be shown as an elimination.

My original sets were r34b34678\r4689c135899 where the common sector is r4. Column 9 must be duplicated in the cover set as r3c9 is contained by two base sectors, r3 and b3.

Using Obi-Wahn's transformations, this then simplifies to r3457b3\c135899b49 (5 base and 8 cover sectors) but the k-rank stays the same, and still there are no eliminations.

I'm therefore coming to believe that, unless anyone can do any better, this puzzle may serve as a counter-example to show that NxM fish can't replicate all net eliminations.
David P Bird
2010 Supporter

Posts: 1040
Joined: 16 September 2008
Location: Middle England

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

blue wrote:You left out r8c9 (It can't see itself directly, but ...)

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`r8c9-r3c9=r3c3-r2c2                ||               r9c2-r9c4=r5c4-r5c1                ||             ||               r8c2           r4c3-r3c3=r3c9                ||             ||               r6c2           r6c23-r6c7=r79c7                   \               /                     -------------`

R8c9 is the CEC and was omitted on purpose. In the standard Rx for a basic row/column NxN Fish, the CEC, by definition, is not a member of any base sector, so it will never be a potential fin and need to "see" itself.
BTW, your network diagram seems to have a conflict: r3c3 appears twice, once as true and once as false.
Sudtyro2

Posts: 652
Joined: 15 April 2013

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

Sudtyro2 wrote:R8c9 is the CEC and was omitted on purpose. In the standard Rx for a basic row/column NxN Fish, the CEC, by definition, is not a member of any base sector, so it will never be a potential fin and need to "see" itself.

Right. It was meant to be "food for thought", mainly -- and an interesting twist on your observations.

Sudtyro2 wrote:BTW, your network diagram seems to have a conflict: r3c3 appears twice, once as true and once as false.

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`                r8c2                 ||                r2c2----------                 ||            \r8c9-r79c7=r6c7-r6c2       r4c8-r2c8=r9c3                 ||         ||                r9c2-r9c4  r4c3-r3c3=r9c3                      ||    ||                     r5c4--r4c5`

On a more serious note, and getting back to the question that you posed: if the original puzzle had an X-Wing, for example, where all of the eliminations had been done ... then the X-Wing candidates couldn't have remote weak links to the target.
Edit: the statement above has been retracted.

Not knowing the "rules" for the net-based (remote) weak links, I can't say much more than that.
I'm hoping to see some comments from the "experts", myself.
Last edited by blue on Fri Aug 23, 2013 5:55 am, edited 1 time in total.
blue

Posts: 784
Joined: 11 March 2013

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