The New Sudoku Players' Forum

[StrmCkr]

(In fact fish are in solving power equivalent to T&E(rank0 fish) restricted to one digit.) ? this comment makes little to no sense to me.. can you shed some light on it as presently i'm muddled with:

i read that as a mathematical construct of nxn+k fish is trial and error? where rank0 is the nxn basic formation and is also trial and error.

how would it be trial and error if i have a data base of all fish constructs with 0 : 1 active if it has elms or doesn't. then i have a yes no answer if there is a fish or isn't and can look up all the triggers for said elim.

the rest i can follow.

[marek stefanik]

What I meant was that any elimination achievable by a fish is achievable by T&E(rank0 fish) restricted to the relevant digit and vice versa (same with direct rankX logic and T&E(rank0 logic)).
Consider the following state. 1r9c1 can be eliminated.

Code: Select all

,-------,-------,-------,
| . . . | 1 . 1 | . . 1 |
| . . . | 1 1 . | 1 . . |
| . . 1 | . . . | . . . |
:-------+-------+-------:
| 1 . . | . . . | . 1 1 |
| . 1 . | . 1 1 | . . . |
| 1 . . | . 1 . | . . 1 |
:-------+-------+-------:
| . 1 . | 1 . . | . 1 . |
| . 1 . | . 1 . | 1 . . |
| 1 . . | 1 . 1 | . . . |
'-------'-------'-------'

By T&E you would start with 1r9c1 and then try to reach a contradiction (here singles are enough).
Let's say you get 1b4, 1c6, 1b3, 1r8 and then 1b5 breaks. You can retrospectively build a fish:
1b5 \
1r8b5 \ c5
1r8b35 \ c57
1r8c6b35 \ r1c57
1r8c6b3445 \ r155c57
1r8c6b3445 \ r1559c1157b7
The first 5 are broken fish in each step of the T&E procedure, the last one links the uncovered candidates to the elimination.
Notice that the single b4\r5 is added twice to sufficiently cover r5c6.
(I could formulate it more generally as a proof by induction, if you need me to.)

Here you can also see the inverse – by assuming 1r9c1 you get the broken fish 1r8c6b3445 \ r155c57, then for example deleting 1b5 results in a rank0 fish eliminating all 1 candidates from b5.

Marek

[ghfick]

A POM finds the three exclusions. Andrew Stuart's solver and Philip Beeby's solver both find this POM. Given the state of this puzzle, the POM technique seems to be the easiest [ to me anyhow! ]. The POM is so closely related to many techniques, I guess.

[StrmCkr]

Yes fish was delevloped to explain templating omissions (pom) not all of them have fish, the ones that don't are known as no-fish.

Some of the no fish are known to have nxn+ fish solutions compared to nxn

Thanks, I get the comment trial and error tying to figure out which is the broken fish patteren

So far my initial qureey is correct and confirmed. No nxn fish.
my elims are sound mutiple ways to prove that. (3 way kraken chain off each cell in box 5, or the non colourable subgraph)

What I'm lacking is a true nxn+k solver, anyone have a fish for this in that scope.

[P.O.]

the resolution state:

Code: Select all

3     5     9     6     28    7     1     248   248           
128   128   6     1248  5     148   7     3     9             
4     7     128   9     128   3     6     28    5             
5     1234  123   124   7     9     8     124   6             
128   6     7     3     128   148   9     5     124           
9     1248  128   1248  6     5     3     7     124           
7     189   5     18    4     6     2     189   3             
6     1289  4     7     3     128   5     189   18             
128   1238  1238  5     9     128   4     6     7         


here is a map of possible templates for value 8 in the context of all other values
the numbers indicate how many templates have this cell as an element, for example cell 58 is an element of 11 possible templates for 8
it can be seen that none of these cells (56 65 78) are elements of any possible template for 8

Code: Select all

  0    0    0    0    3    0    0    4    6
  6    4    0    1    0    2    0    0    0
  0    0    3    0    7    0    0    3    0
  0    0    0    0    0    0   13    0    0
  1    0    0    0    3    9    0    0    0
  0    6    6    1    0    0    0    0    0
  0    0    0   11    0    0    0    2    0
  0    0    0    0    0    2    0    4    7
  6    3    4    0    0    0    0    0    0


[ghfick]

Hi P.O.,

It is interesting to see your software displaying 13 templates / patterns. For this puzzle, you can 'easily' show that 8 in r9c6 is not possible as no patterns with 8r9c6 are possible. That means that 8 must be in r9c123.

I notice that after a UR, one can use a POM again and even a third time. This is a fascinating puzzle.

Gordon

[P.O.]

hi Gordon,
this is what the map indicates, there are 13 possible templates for 8 and none of the cells (r7c2 r8c2 r9c6) are in them.

Code: Select all

Value:           8
Value Cell:      (34)
Candidate Cells: (5 8 9 10 11 13 15 21 23 26 37 41 42 47 48 49 56 58 62 65 69 71 72 73 74 75 78)
Union csets:     (5 8 9 10 11 13 15 21 23 26 37 41 42 47 48 49  . 58 62  . 69 71 72 73 74 75  .)
Complementary Sets: 13
(5 10 26 42 47 58 72 75)
(5 10 26 42 48 58 72 74)
(8 10 23 42 47 58 72 75)
(8 10 23 42 48 58 72 74)
(9 10 23 42 47 58 71 75)
(9 10 23 42 48 58 71 74)
(5 11 26 42 48 58 72 73)
(8 11 23 42 48 58 72 73)
(9 11 23 37 49 62 69 75)
(9 11 23 42 48 58 71 73)
(9 13 21 41 47 62 69 73)
(8 15 21 41 47 58 72 73)
(9 15 21 41 47 58 71 73)
Candidate 8 to be eliminated in cells: (56 65 78)


checking any resolution state with a template procedure without doing any template combination you find for each value all candidates that are in all possible templates for that value, candidates that can be set, and all candidates that are not in any of the possible templates for that value, candidates that can be eliminated.

[sultan vinegar 2]

Code: Select all

+------------------------+-----------------------+-------------+
| 3      5        9      | 6       2(8)   7      | 1  248  248 |
| 12(8)  12(8)    6      | 124(8)  5      14(8)  | 7  3    9   |
| 4      7        12(8)  | 9       12(8)  3      | 6  28   5   |
+------------------------+-----------------------+-------------+
| 5      1234     123    | 124     7      9      | 8  124  6   |
| 12(8)  6        7      | 3       12(8)  14(8)  | 9  5    124 |
| 9      124(8)   12(8)  | 124(8)  6      5      | 3  7    124 |
+------------------------+-----------------------+-------------+
| 7      19(-8)   5      | 1(8)    4      6      | 2  189  3   |
| 6      129(-8)  4      | 7       3      12(8)  | 5  189  18  |
| 12(8)  123(8)   123(8) | 5       9      12(-8) | 4  6    7   |
+------------------------+-----------------------+-------------+

anyone have a proper fish for this construct: i built it with 6* ERI and row 2,9

I've put it threw 3 different solvers and haven't found one for it yet. {might be part of the no fish or nxn+k family }

size 8: fish??
Fish: R29B124578 / r56c123456
doesn't make sense but the eliminations are solid confirmed via xsudo
the elms are solid and easy to check by iterating the 3cells of the ERI in box 5 { all 3 cells lead to the elms}

Hi, StrmCkr.

I visited here for the first time in years, and your post piqued my interest.

I do have what I call a fish for this construct, but the caveat is that I use a virtual truth derived from negative rank logic.

One reason that a single fish might be hard to find for this example could be that we need a Siamese fish. Note that for the three eliminations, once r9c6 <> 8, then by locked candidates one gets r78c2 <> 8 for free, and vice versa, once r78c2 <> 8, one gets r9c6 <> 8 for free. If the fish be two Siamese twins, then I've never worked out how to make the set, link, and rank count work — just try making one single fish to get all four eliminations for a (minimal) skyscraper. Does that mean they really are two separate fish not one?

One reason why virtual truths derived from negative rank logic might not be transformable to native inference sets (and hence why fish might represent a proper subset of single digit pattern overlay method, i.e. no-fish exist) could be that fish logic uses the principles of strong (at least one true) and weak (at most one true) sets, whereas the negative rank logic in my virtual truth uses the principle of conjugate (exactly one true) sets, thereby seemingly using the strong and weak properties at the same time which confuses the set, link and rank count.

Anyway, let me know whether you want to see my 'fish'. I'll need to remind myself how to post it!

[StrmCkr]

Yes a skyscraper is 2 Sashimi x wings under nxn logic. For 2 elims per.
The à.i.c chain "skyscraper" is based on nets 4 elims.

Siamese allows adding addition base sectors that can be exchanged 1:1 at no cost for extra elims, however to do it technically would be nxn+k reusing the base sectors for the base count.

Sure post the fish once u Remeber how to write out the chain or use the code tags.

[sultan vinegar 2]

Code: Select all

+------------------------+-----------------------+-------------+
| 3      5        9      | 6       2(8)   7      | 1  248  248 |
| 12(8)  12(8)    6      | 124(8)  5      14(8)  | 7  3    9   |
| 4      7        12(8)  | 9       12(8)  3      | 6  28   5   |
+------------------------+-----------------------+-------------+
| 5      1234     123    | 124     7      9      | 8  124  6   |
| 12(8)  6        7      | 3       12(8)  14(8)  | 9  5    124 |
| 9      124(8)   12(8)  | 124(8)  6      5      | 3  7    124 |
+------------------------+-----------------------+-------------+
| 7      19(-8)   5      | 1(8)    4      6      | 2  189  3   |
| 6      129(-8)  4      | 7       3      12(8)  | 5  189  18  |
| 12(8)  123(8)   123(8) | 5       9      12(-8) | 4  6    7   |
+------------------------+-----------------------+-------------+

Right, let's start with defining the virtual truth set V:
To prevent the negative-rank dark (odd number of conjugate links) loop {r2c1=r2c4=r6c4=r6c23=r5c1=r2c1},
we have V = {r2c26, r7c4, r9c1}, a.k.a. Guardians.

Now, our fish (a virtual finned mutant X-Wing) is:
VB7 / r9c26b8

I'll spare the full xsudo splitting the fish into regions of different rank because the set triplet at r9c1 is nothing more complicated as an endofin N in this case.

N false => fish true, two truths, four links => r9c6 <> 8 by three cover links r9c26b8.
N true => r9c6 <> 8 by line of sight cover link r9.

Therefore, r9c6 <> 8, and then by locked candidates, r78c2 <> 8.

[sultan vinegar 2]

And re: the skyscraper, one can write that as a single fish for all four eliminations if one uses two virtual cover sets.

[yzfwsf]

Right, let's start with defining the virtual truth set V:
To prevent the negative-rank dark (odd number of conjugate links) loop {r2c1=r2c4=r6c4=r6c23=r5c1=r2c1},
we have V = {r2c26, r7c4, r9c1}, a.k.a. Guardians.

Now, our fish (a virtual finned mutant X-Wing) is:
VB7 / r9c26b8

If you identify this as a fish, then my solver can find it, but it is called the Oddagon Forcing Chain.

[StrmCkr]

we have V = {r2c26, r7c4, r9c1}, a.k.a. Guardians.

yup, that would be the oddagon logic or broken-wing citation for no-fish patterns

Code: Select all

and then by locked candidates, r78c2 <> 8.

that's pretty much how the forcing chain breaks the fish down when iterating the ERi cells in box 5, the fish tags the r9c6 elimination leaving blr to finish the rest of the elims.

And re: the skyscraper, one can write that as a single fish for all four eliminations if one uses two virtual cover sets.

yes, that what we did in hodoku for Siamese fish ~ simply added on the 2nd cover sectors/ Finns for the eliminations. As a purest nxn that this thread was intended as the nxn formation for a skyscraper is still 2 Sashimi x-wings

another view is the almost fish + Almost fish

b12457 / c12345 => r2589c6 not covered
+
r9 / c9 => r9c123 not covered

combined :
the cells cover's of (r9- fish) not in the base are excluded => r78c2 {which is using r123 as cover's : from the link fish}
the cells of the covers of b12457 not in the base are excluded => r9c6 { which is using c6 from the linked fish}

more specifically if r9c6 is true for both 1 fish and the large fish, as its base for both fish then they are both reduced to n-1 cells for n vertexes.

which is a rule* i was trying to devise for my nxn+k fish after i noticed that
r1 covered by C123 {K+2} mimicked R1/b1 for a 1 fish { where r1/c123 =>> peers of the base * {union of the covers} = eliminations }

a system for converting sectors that could swap for alternative covers. but in the case of this almost fish the union in theory functions for r78c2 elim.

other then that's as far as i got the idea off the ground or out of excel, when i first was testing it like 6+ years ago.

[sultan vinegar 2]

Oh, by getting all four eliminations for the minimal skyscraper with two virtual cover links, I meant this:

Code: Select all

 .  X  .  | .  .  . | *  /  *       
 .  /  .  | .  .  . | .  /  .       
 *  /  *  | .  .  . | .  X  .     
----------+----------+---------     
 .  /  .  | .  .  . | .  /  .       
 .  /  .  | .  .  . | .  /  .       
 .  /  .  | .  .  . | .  /  .       
----------+----------+---------     
 .  /  .  | .  .  . | .  /  .       
 .  X  .  | .  .  . | .  X  .       
 .  /  .  | .  .  . | .  /  .


The strong sets of the base may be the standard native C28.
One weak set of the cover may be the standard native r8.
And we may define two virtual weak sets to complete the cover:
At most one of {r1c2, [r1c79, r3c13]} is true by line of sight (be it a row or a box), where the second element is a grouped node of four candidates.
Likewise, at most one of {r3c8, [r1c79, r3c13]} is true.

Virtual (un-finned?) X-Wing, two base sets, three cover links => r1c79, r3c13 <> X because these candidates are members of zero base sets and two cover links.

Let me have some time to think about your combining fish. My initial thought was how does one address e.g. two independent disjoint finned X-Wings? The rank of each individually is one, but when you mix all the bases and covers into the one soup, the rank becomes two so no eliminations ensue. However, regions of reduced rank exist for sub-fish, i.e. the original finned X-Wings à la Xsudo.

[sultan vinegar 2]

Right, let's start with defining the virtual truth set V:
To prevent the negative-rank dark (odd number of conjugate links) loop {r2c1=r2c4=r6c4=r6c23=r5c1=r2c1},
we have V = {r2c26, r7c4, r9c1}, a.k.a. Guardians.

Now, our fish (a virtual finned mutant X-Wing) is:
VB7 / r9c26b8

If you identify this as a fish, then my solver can find it, but it is called the Oddagon Forcing Chain.

Yes, I would call this a fish just as I would call an AIC that used a strong link (e.g. from an AUR) a chain.

If you stay on the one floor (Champagne's term IIRC) and produce a counting argument, then it's a fish for me. If you combine fish by moving between floors (in such a way that their respective remote fins are not independent), and produce a counting argument, then it's a flying fish a.k.a. Exocet.

But at the end of the day, it's just a name, isn't it.