The New Sudoku Players' Forum

[StrmCkr]

Code: Select all

.----------------.-------------------.---------------------.
| 6   78   2     | 3     478    9    | 1     5       478   |
| 17  3    178   | 2     4578   578  | 789   4789    6     |
| 9   5    4     | 78    1      6    | 278   3       278   |
:----------------+-------------------+---------------------:
| 2   6    789   | 5     789    4    | 3     789     1     |
| 4   1    789   | 6789  2      3    | 5     6789    789   |
| 5   789  3     | 1     6789   78   | 2789  246789  24789 |
:----------------+-------------------+---------------------:
| 3   79   679   | 4     6789   2    | 789   1       5     |
| 8   4    15    | 79    3      15   | 6     279     279   |
| 17  2    15679 | 6789  56789  1578 | 4     789     3     |
'----------------'-------------------'---------------------'

regarding this discussion on fins and fin sectors...

I in full agree that the term "Fin" should be removed and left to nxn fish exclusively.

nxn+k fish use cover-sectors & additional cover-sectors exclusively keep the two types of fish separated as much as possible to prevent confusion.

load the above puzzle into hodoku: now search for 3x3 fish

Finned Franken Swordfish: 7 c16b4 r269 fr4c3 fr5c3 => r29c3<>7
Finned Franken Swordfish: 7 c16b4 r26b7 fr4c3 fr5c3 fr9c6 => r9c3<>7
Finned Franken Swordfish: 7 c16b4 r69b1 fr2c6 fr4c3 fr5c3 => r2c3<>7
Finned Mutant Swordfish: 7 c16b4 r26c3 fr9c1 fr9c6 => r9c3<>7
Finned Mutant Swordfish: 7 c16b4 r69c3 fr2c1 fr2c6 => r2c3<>7

this singular 3x3+1 fish represents all of these patterns at once: the direct summation of multiple nxn fish at once!!!
(7) 3x3+1 Base: C16B4, Cover: R269C3, Exclusions: R2R9C3

notice, that in nxn fish the fins cells are some times in the base and some versions they are in the cover.

can you tell me which + cover sector is specifically is the "fin sector"?
how about me which ones are not the + cover sectors?

all cover sectors can be treated as + sectors in a nxn+k fish

nxn+k fish has no PE as well, it lists only EE locations.

[Pat]

your puzzle should only need 7\8

r1458c1b27\r29c234589
to exclude r9c5

oops

r1c5 twice in base,
only once in cover
— not valid

can be fixed by adding \r9c5

( yes, that bizzare beast with
same house twice in the cover )

making 7\10

r1458c1b27\r299c2345589

to exclude r9c5

4 - 0 > 10 - 7

[Pat]

you remember the NoFish list!!!

perhaps we should re-open the NoFish discussion---

[ronk]

Reply to StrmCkr placed here on the Obi-Wahn thread titled About the arithmetic of Ultimate Fish.

[tarek]

I'm pleased to know that Forum members are passionate about this topic. I've read all the latest exchanges.

Obi-Wahn must hate Finned fish (He comes from a galaxy far far away after all!!!). So his attempt was to include all fish under a finless banner.

But his description of a "Fin sector" is that of a number ... & didn't specify which of the N+k sectors are the actual "Fin sectors". This As I mentioned makes it easier -At least from a programming point of view- to code the Nx(N+k) fish catcher algorithm.

Essentially his fish are Finless. His description of a Fin sector is possibly an attempt to bring closer his ideas to our Finned NxN oriented minds.

Finding the original NxN fish within that Nx(N+k) creature is going to take more work and might not be easy to spot at first glance.

I am with describing an Nx(N+k) arithmetic that would include all of the NxN finless fish and NxN finned fish reaching to Obi-Wahn's Nx(N+k) Finless fish.

Regarding Fins: What I'm trying to say is that I'm hoping that when we mention Fin(s), only Fin cells would come to our minds IOW Fin <---> Fin cell.

[tarek]

you remember the NoFish list!!!

perhaps we should re-open the NoFish discussion---

Mike Barker's discussion about the first list of NoFish http://forum.enjoysudoku.com/a-revival-of-broken-wings-t5225-46.html
I posted several other NoFish puzzles later that were lost with the old forum but reposted here http://forum.enjoysudoku.com/post201361.html#p201361
These are interesting NxN fish that were extracted using Hoduko http://forum.enjoysudoku.com/post210194.html#p210194

Lost with the old forum are the analysis of the NoFish when many complex NxN fish and some wonderful examples with remote fins were fished out.

[David P Bird]

Let me clear up one specific point here.

.. so all the eliminations are in PE cells

nxn+k fish has no PE as well, it lists only EE locations.

That’s a misconception:
By the definitions in the opening post Fins are cells or digits that belong to more base sectors than cover sectors, and PEs are those that belong to more cover sectors than base sectors

While a Nx(N+1) fish is being constructed both Fins and PEs will exist.
It's only when all Fins have been eliminated does the pattern become a valid Nx(N+k) fish
If extra cover sectors have been added to achieve this, then k will increase.
Those PEs that belong to > k more cover sectors than base sectors then become EEs.
However EEs are still PEs according to the definition of the term, and there will generally be other non-EE PEs in the pattern as well.

David

[blue]

This post is about the connection between NxN, finned or unfinned "UFG" fish, and Nx(N+k) "Obi-Wahn" fish.

UFG NxN fish:

N base sectors
N cover sectors
Fin candidates are (base sector) candidates that are in more base than cover sectors
PE candidates are (cover sector) candidates that are in more cover than base sectors
A PE candidate that can see all fin candidates, can be eliminated

Obi-Wahn Nx(N+k) fish:

N base sectors
N+k cover sectors
All candidates are in at least as many cover as base sectors.
A candidate that is in 'b' base and (b+m+k) cover sectors, for some 'b' and some m > 0, can be eliminated

Suppose that 'x' is a candidate that can be eliminated via an Nx(N+k) (Obi-Wahn) fish.

Since 'x' satisfies Obi-Wahn's exclusion rule, it's in 'b' base, and (b+m+k) cover sectors, for some 'b' and some m > 0.

If we strip out any choice of 'k' cover sectors containing 'x', these things happen:


1. We are left with N base and N cover sectors -- the foundation for an NxN "UFG" fish (finned or unfinned).
   
2. Base sector candidates that see 'x' through one of the 'k' cover sectors, can lose the property that they are in at least as many cover as base sectors. In UFG terminlogy, such candidates would be classified as "fin" candidates.
   Note: the fin candidates (if any), would all "see" candidate 'x'.
3. Since candidate 'x', at that point, would be in 'b' base and (b+m) cover sectors (with m > 0), it would be classified as a "PE" candidate.
With that, for the NxN (UFG) fish that would remain, candidate 'x' would be a PE candidate that can see all of the fin candidates.

Suppose that 'x' is a candidate that can be eliminated via a (finned or unfinned) NxN (UFG) fish.

Candidate 'x' is a PE candidate for the fish, so it's in 'b' base and (b+m) cover sectors, for some 'b' and some m > 0.

Candidates in the base sectors, are of two types:

• "fin" candidates -- in fewer cover than base sectors.
  
• "non-fin" candidates -- in at least as many cover as base sectors.
If we add cover sectors as required, allowing duplicates if necessary, linking 'x' to fin candidates as many times as necessary to ensure that each fin candidate is in at least as many cover as base sectors, then supposing that 'k' cover sectors were required, these things happen:


1. We get an Nx(N+k) "Obi-Wahn" fish -- i.e. one where every candidate satisfies the property that it is in at least as many cover as base sectors.
2. Candidate 'x' ends up in 'b' base, and (b+m+k) cover sectors, and since (b+m+k)-b = (m+k) > k, it satisfies the requirements of Obi-Wahn's exclusion rule.

There is something in common with the two views (of the same elimination).

It's the (N-b)x(N-b-m) base\cover problem that would remain if:


1. candidate 'x' was asserted
   
2. the 'b' base sectors containing 'x' were removed from the base set
3. the cover sectors containing 'x' were removed from the cover set
   
   1. (b+m) of them for the NxN case
   2. (b+m+k) of them for the Nx(N+k) case
4. (some of) the base sector candidates that can see 'x' were elimminated
   1. fin candidates, and candidates that see 'x' through one of the (b+m) cover sectors containing 'x', for the NxN (UFG) case
   2. candidates that can see 'x' through one of the (b+m+k) cover sectors containing 'x', for the Nx(N+k) (Obi-Wahn) case
For that problem, the cover sector count is less less than the base sector count, and each base sector candidate is in at least as many cover as base sectors. With that, we can claim that the (reduced) base\cover problem has no solutions ... "proof required" ... and with that it would follow that candidate 'x' can't be true in any solution to the puzzle.

Note: on re-reading the bit above, it seems that there's no guarantee that with step (4) above, you would get exactly the same reduced base\cover problem, if you started with a UFG and Obi-Wahn fish that are related by one of the conversions above. That's due to the fact that for the Obi-Wahn fish, the (b+m+k) cover sectors that contain 'x', might also contain base sector candidates that don't have the "fin candiate" designation for the UFG fish. It isn't important, really ... that the two reduced problems might be different. Neither one would have a solution, and for the same reasons.

[daj95376]

Returning to differences in complexities: Between the NxN fish & Obi-Wahn's Nx(N+k) fish there will exist some fish logic where you have fin cells within fin sectors & some fin cells that are left exposed without fin sectors. In that situation: candidate cells where ( cover_sector_count - base_sector_count ) > k need to be peers of all the fin cells that have been left exposed without fin sectors. The choice is then left to the solver: a 7x7 Mutant finned fish with endo-fins ... To a 7x9 finned fish to a 7x11 fish with no exposed fin cells.

To remove some of the confusion with naming some elements related to Fin(s) between NxN fish & Nx(N+k) fish; I'm hoping that we remove the use of the term "fin sectors". In a situation where k-covers (new name) & fin cells not within k-covers exist in the same Nx(N+k) that does not fulfil Obi-Wahn's strict rules; the use of the word "fin" can be confusing. Dropping the the term "fin sector" in favour of "k-sector or k-cover" will restore the "Fin" term to a non confusing state which can be used in any fish related conversation without abiguity.

Whew! Travel for a few days and I return to discover that this thread has grown like a weed.

I did some research and it may help on the issue of "k" and fin sectors. Obi-Wahn never used "k" in his post. I know that I used N\(N+2) later in Obi-Wahn's thread, but I'm not sure why. Maybe it was from something that I read in another thread. In any event, I'm guessing that Obi-Wahn's use of fin sectors was simply his way of (indirectly) acknowledging that any additional cover sectors must pass through fin cells present in an (underlying) finned NxN fish. Given this possibility, I'd prefer to continue using fin sectors and not change the name. For sure, in order to create an Nx(N+k) fish, I always start with my solver's finned NxN fish and manually add additional cover sectors that pass through the elimination cell(s) and the fin cells. I consider these additional cover sectors to represent Obi-Wahn's fin sectors.

I believe that I went too far in saying finned Nx(N+k) fish. I now see three categories of fish:

Code: Select all

*) unfinned NxN     fish
*)   finned NxN     fish
*)          Nx(N+k) fish   (that meet Obi-Wahn's arithmetic)


I'm not convinced that a hybrid fish -- 7x9 for my example -- is acceptable. Others seem to think it's acceptable. I'll try not to argue.

[daj95376]

Finned Franken Swordfish: 7 c16b4 r269 fr4c3 fr5c3 => r29c3<>7
Finned Franken Swordfish: 7 c16b4 r26b7 fr4c3 fr5c3 fr9c6 => r9c3<>7
Finned Franken Swordfish: 7 c16b4 r69b1 fr2c6 fr4c3 fr5c3 => r2c3<>7
Finned Mutant Swordfish: 7 c16b4 r26c3 fr9c1 fr9c6 => r9c3<>7
Finned Mutant Swordfish: 7 c16b4 r69c3 fr2c1 fr2c6 => r2c3<>7

this singular 3x3+1 fish represents all of these patterns at once: the direct summation of multiple nxn fish at once!!!
(7) 3x3+1 Base: C16B4, Cover: R269C3, Exclusions: R2R9C3

notice, that in nxn fish the fins cells are some times in the base and some versions they are in the cover.

can you tell me which + cover sector is specifically is the "fin sector"?

Here are the, respective, fin sectors for your fish.

Code: Select all

c16b4/r269  +   c3
c16b4/r26b7 + r9c3
c16b4/r69b1 + r2c3
c16b4/r26c3 + r9     (note: Obi-Wahn also performs r2c3<>7)
c16b4/r69c3 + r2     (note: Obi-Wahn also performs r9c3<>7)


The 1st, 4th, and 5th Nx(N+1) fish are identical ... and perform eliminations r29c3<>7.

It's also apparent that the 2nd and 3rd NxN fish are less than desirable starting places because they create Nx(N+2) fish.

This takes us back to my suggestion that you find Nx(N+1) fish before Nx(N+2) fish.

[daj95376]

can be fixed by adding \r9c5

( yes, that bizzare beast with
same house twice in the cover )

making 7\10

r1458c1b27\r299c2345589

to exclude r9c5

4 - 0 > 10 - 7

Congratulations! You've demonstrated that my original 7x7 fish isn't the best choice as a foundation for creating an Nx(N+k) fish for this elimination. In my solver listing (hidden in a post above), I have two other 7x7 fish listed with this base set. Both of them can be extended to your 7/10 fish.

[David P Bird]

Working from a NxN fish that has a PE cell seeing all the fin cells, the rule to transform it into a Nx(N+k) fish is:
Add d cover sets to the houses common to each of the fins and the excludable PE cells, where d = base sectors – cover sectors containing the fin cell.

Using Obi-Wahn's rules it's then possible to construct the hidden pattern for the resultant fish. As the elimination is already known, this is the only reason I can think of for going through this process.

[Edit] Corrected the reversed PE and fin terms throughout this post - as picked up by ronk

[ronk]

can be fixed by adding \r9c5

( yes, that bizzare beast with
same house twice in the cover )

making 7\10

r1458c1b27\r299c2345589

to exclude r9c5

4 - 0 > 10 - 7

Congratulations! You've demonstrated that my original 7x7 fish isn't the best choice as a foundation for creating an Nx(N+k) fish for this elimination. In my solver listing (hidden in a post above), I have two other 7x7 fish listed with this base set. Both of them can be extended to your 7/10 fish.

That's the same fish that I posted here sans hiccups where ...

Code: Select all

+-------------------+-----------------------+---------------------+
| 6     8(7)  2     | 3       48(7)   9     | 1     5       48(7) |
| 1(7)  3     18    | 2       458(7)  58(7) | 789   489     6     |
| 9     5     4     | 8(7)    1       6     | 278   3       278   |
+-------------------+-----------------------+---------------------+
| 2     6     89(7) | 5       89(7)   4     | 3     89(7)   1     |
| 4     1     89(7) | 689(7)  2       3     | 5     689(7)  89(7) |
| 5     789   3     | 1       6789    78    | 2789  246789  24789 |
+-------------------+-----------------------+---------------------+
| 3     9(7)  69(7) | 4       6789    2     | 789   1       5     |
| 8     4     15    | 9(7)    3       15    | 6     29(7)   29(7) |
| 1(7)  2     1569  | 6789    5689-7  1578  | 4     789     3     |
+-------------------+-----------------------+---------------------+

     7 Truths = {7R1458 7C1 7B27}
     8 Links = {7r29 7c234589}
     1 Elimination --> r9c5<>7

[ronk]

... a fin cell seeing all the PE cells ...
... excludable fin cell ...

Looks like you have your fin cells and PE cells reversed ... again.

[daj95376]

making 7\10

r1458c1b27\r299c2345589

to exclude r9c5

4 - 0 > 10 - 7

That's the same fish that I posted sans hiccups where ...

Code: Select all

+-------------------+-----------------------+---------------------+
| 6     8(7)  2     | 3       48(7)   9     | 1     5       48(7) |
| 1(7)  3     18    | 2       458(7)  58(7) | 789   489     6     |
| 9     5     4     | 8(7)    1       6     | 278   3       278   |
+-------------------+-----------------------+---------------------+
| 2     6     89(7) | 5       89(7)   4     | 3     89(7)   1     |
| 4     1     89(7) | 689(7)  2       3     | 5     689(7)  89(7) |
| 5     789   3     | 1       6789    78    | 2789  246789  24789 |
+-------------------+-----------------------+---------------------+
| 3     9(7)  69(7) | 4       6789    2     | 789   1       5     |
| 8     4     15    | 9(7)    3       15    | 6     29(7)   29(7) |
| 1(7)  2     1569  | 6789    5689-7  1578  | 4     789     3     |
+-------------------+-----------------------+---------------------+

     7 Truths = {7R1458 7C1 7B27}
     8 Links = {7r29 7c234589}
     1 Elimination --> r9c5<>7

They are not the same fish. Pat covered cells r1c5 and r9c1 twice in order to neutralize their being in two base sectors. You covered them once (each) and still need them as fin cells to prevent eliminations in cells other than r9c5. After tarek's post, I nicknamed this a hybrid fish.

Addendum:

Here is the Obi-Wahn base/cover cell counts for your 7x(7+1) fish.

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 7x8-Fish r1458c1b27\r29c234589        <> 7    r9c5
 +-----------------------------------------------+
 |  2/0 1/1 1/1  |  2/1 2/1 2/0  |  1/0 1/1 1/1  |
 |  1/1 0/2 0/2  |  1/2 1/2 1/1  |  0/1 0/2 0/2  |
 |  1/0 0/1 0/1  |  1/1 1/1 1/0  |  0/0 0/1 0/1  |
 |---------------+---------------+---------------|
 |  2/0 1/1 1/1  |  1/1 1/1 1/0  |  1/0 1/1 1/1  |
 |  2/0 1/1 1/1  |  1/1 1/1 1/0  |  1/0 1/1 1/1  |
 |  1/0 0/1 0/1  |  0/1 0/1 0/0  |  0/0 0/1 0/1  |
 |---------------+---------------+---------------|
 |  2/0 1/1 1/1  |  0/1 0/1 0/0  |  0/0 0/1 0/1  |
 |  3/0 2/1 2/1  |  1/1 1/1 1/0  |  1/0 1/1 1/1  |
 |  2/1 1/2 1/2  |  0/2 0/2 0/1  |  0/1 0/2 0/2  |
 +-----------------------------------------------+


When this is combined with the candidate grid, you perform the following eliminations if you do not treat r1c5 and r9c1 (because of its counts) as fin cells.

Code: Select all

 +------------------------------------+
 |  .  7  x  |  .   7  .  |  .  x  7  |
 |  7  .  .  |  x   7  7  |  7  .  .  |
 |  .  .  .  |  7   .  .  |  7  .  7  |
 |-----------+------------+-----------|
 |  .  x  7  |  x   7  .  |  .  7  x  |
 |  .  x  7  |  7   .  .  |  .  7  7  |
 |  .  7  .  |  .   7  7  |  7  7  7  |
 |-----------+------------+-----------|
 |  .  7  7  |  .   7  .  |  7  .  .  |
 |  .  .  .  |  7   x  x  |  .  7  7  |
 |  7  x  x  | *7  *7  7  |  . *7  .  |
 +------------------------------------+


But Templates says that the only (single-digit) elimination is r9c5<>7.

This matches the general response that I gave Pat here when she initially presented the same 7x8-fish.