## The Ultimate FISH Guide

Advanced methods and approaches for solving Sudoku puzzles

### Re: The Ultimate FISH Guide

1. It would be helpful if everyone talked about the 7-fish in the actual puzzle instead of some talking about the proposed exemplar.

2. There is also continued confusion about the N\(N + k) of the actual finned fish versus that of the algorithm used to find it. For example, while the later might be 7\(7 + 4) for Obi-Wahn's algorithm, it is 7\(7 + 2) for the actual finned fish.
ronk
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### Re: The Ultimate FISH Guide

Pat, Thanks for following my points!

David P Bird wrote:
DAJ wrote:DPB, You found one of the 11 other 7x7 fish present in my grid.

Not at all. The grid you presented seemed invalid to me because the 6 occupied cells in r9 consisted of 1 PE cell and 5 fin cells all of which could be considered false. Removing the r9 cover sector to make it fit my 'vertex cell in every base and cover sector' rule then exposed one of your original fish.

I admit I haven't explored these lop-sided fish, but it would seem to me that either this rule should be applied or such situations be detected and provided for in the the elimination-finding formulae.

What was behind my two previous posts is the concept that a general fish finding algorithm could be based on lop-sided principles but should be able to detect when a hit is found for k=0. However, I was too brief and never specifically made the point.

I was originally fixated on the point that my unfinned 7x7 fish would eliminate all of the candidates in [b8]. When I marked the candidate grid, I missed the fact that base sector [r9] didn't contain a candidate in any of the vertex cells -- which I assume is your point. In fact there was a candidate in r9c3 when I eliminated smaller fish and left only the 7-fish to be discussed. However, my solver found the 7x7 fish even after the eliminations for the smaller fish.

Hidden Text: Show
Code: Select all
` +-----------------------+ | . . 2 | 3 . 9 | 1 5 . | | . . . | 2 . . | . . 6 | | 9 . 4 | . . 6 | . . . | |-------+-------+-------| | 2 6 . | 5 . 4 | 3 . . | | . . . | . 2 . | . . . | | 5 . 3 | 1 . . | . . . | |-------+-------+-------| | 3 . . | 4 . . | . 1 5 | | 8 . . | . . . | 6 . . | | . 2 . | . . . | 4 . 3 | +-----------------------+   c2b1  Hidden Pair                     =  35   r23c2 r3      Hidden Pair                     =  15   r3c25 r8      Hidden Pair                     =  15   r8c36 +--------------------------------------------------------------------------------+ |  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       | +--------------------------------------------------------------------------------+ # 89 eliminations remain3-Fish c16b4\r269               fF  021\300 <> 7    r29c3  w/fin cells = r45c33-Fish c16b4\r69b1              fF  021\201 <> 7    r2c3   w/fin cells = r45c3,r2c63-Fish c16b4\r26b7              fF  021\201 <> 7    r9c3   w/fin cells = r45c3,r9c65-Fish r38c16b6\r269c49         fm  221\320 <> 7    r2c8   w/fin cells = r3c7,r458c85-Fish r38c167\r269c4b9         Bm  230\311 <> 7    r2c8   w/fin cells = r3c795-Fish r38c16b6\r69c489         fm  221\230 <> 7    r2c8   w/fin cells = r2c16,r3c75-Fish r38c16b6\r69c49b3        fm  221\221 <> 7    r2c8   w/fin cells = r2c1,r2c6,r458c85-Fish r38c167\r69c4b39         Am  230\212 <> 7    r2c8   w/fin cells = r2c16,r3c77-Fish r18c16b467\r269c2389     Am  223\340 <> 7    r9c5   w/fin cells = r1c5,r8c4,r9c17-Fish r18c167b47\r269c23b39    Am  232\322 <> 7    r9c5   w/fin cells = r1c5,r8c4,r9c17-Fish r1458c1b27\r29c23489     Bm  412\250 <> 7    r9c5   w/fin cells = r9c1,r14c57-Fish r1458c16b7\r29c2389b5    Am  421\241 <> 7    r9c5   w/fin cells = r1c5,r8c4,r9c17-Fish r18c16b467\r26c23589     Bm  223\250 <> 7    r9c5   w/fin cells = r8c4,r9c167-Fish r18c167b47\r26c235b39    Bm  232\232 <> 7    r9c5   w/fin cells = r8c4,r9c167-Fish r1458c1b27\r2c234589     Bm  412\160 <> 7    r9c5   w/fin cells = r1c5,r9c17-Fish r1458c16b7\r2c23589b5    Bm  421\151 <> 7    r9c5   w/fin cells = r8c4,r9c167-Fish r18c16b467\r26c2389b8    Bm  223\241 <> 7    r9c5   w/fin cells = r1c5,r9c17-Fish r18c167b47\r26c23b389    Bm  232\223 <> 7    r9c5   w/fin cells = r1c5,r9c17-Fish r1458c1b27\r2c23489b8    Bm  412\151 <> 7    r9c5   w/fin cells = r9c1,r14c57-Fish r1458c16b7\r2c2389b58    Bm  421\142 <> 7    r9c5   w/fin cells = r1c5,r9c1`

[Edit: dropped final paragraph. The wording didn't correctly match my thoughts.]
Last edited by daj95376 on Sun Aug 04, 2013 6:13 pm, edited 1 time in total.
daj95376
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### Re: The Ultimate FISH Guide

ronk wrote:1. It would be helpful if everyone talked about the 7-fish in the actual puzzle instead of some talking about the proposed exemplar.

2. There is also continued confusion about the N\(N + k) of the actual finned fish versus that of the algorithm used to find it. For example, while the later might be 7\(7 + 4) for Obi-Wahn's algorithm, it is 7\(7 + 2) for the actual finned fish.

Your source of proof for this claim ???

As I told StrmCkr:

In an NxN fish, a cell is a fin cell when its base/cover sector counts are 2/0. Adding a single cover sector may change the counts to 2/1, but does not change the fact that the cell is still a fin cell. You simply added an extraneous cover sector and calling it a valid Nx(N+1) fish. This adds complexity without adding value because the fin cells must still be incorporated into determining the eliminations !

[Edit: corrected my sentence syntax.]
Last edited by daj95376 on Sun Aug 04, 2013 5:20 pm, edited 1 time in total.
daj95376
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### Re: The Ultimate FISH Guide

tarek wrote:I'm not a fan of Nx(N+k) fish either but it has been very useful in explaining some eliminations that eluded NxN fish. We have since recovered some NxN fish with sharper fishing tools and remote fins but the complexities of Mutant Finned Giant fish may prove that a simpler Nx(N+k) is a better option!!!

Assuming that you are talking about vanilla Sudoku, would you please provide an example for this claim.

I also wait for someone to provide an example for this hypothesis. I'm not going to hold my breath.
daj95376
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### Re: The Ultimate FISH Guide

daj95376 wrote:
ronk wrote:2. There is also continued confusion about the N\(N + k) of the actual finned fish versus that of the algorithm used to find it. For example, while the latter might be 7\(7 + 4) for Obi-Wahn's algorithm, it is 7\(7 + 2) for the actual finned fish.

Your source of proof for this claim ???

My grids have only one physical r9 on them. Same for c5. Are yours somehow different?
ronk
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### Re: The Ultimate FISH Guide

daj95376 wrote:
tarek wrote:I'm not a fan of Nx(N+k) fish either but it has been very useful in explaining some eliminations that eluded NxN fish. We have since recovered some NxN fish with sharper fishing tools and remote fins but the complexities of Mutant Finned Giant fish may prove that a simpler Nx(N+k) is a better option!!!

Assuming that you are talking about vanilla Sudoku, would you please provide an example for this claim.

I've used the variant excuse once, don't push it now
At the start it was useful to find eliminations that can't be explained by fish at that time. But as we we became more competent with our NxN fishing skills with better definitions for Fins we were able to fish these eliminations using NxN fish. I'm sure you remember the NoFish list!!!

daj95376 wrote:I also wait for someone to provide an example for this hypothesis. I'm not going to hold my breath.
I have no example to provide here. The person to provide this will be somebody who prefers Nx(N+k) to NxN fish.

tarek

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### Re: The Ultimate FISH Guide

David P Bird wrote:As a casual reader of this thread, I would also appreciate a reference to non-NxN fish and a link to their definition and proofs in your opening post.

I agree & will work on it

tarek

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### Re: re: 7\8 (Obi-Wahn's arithmetic)

Pat wrote:
daj95376 wrote:
Obi-Wahn wrote:Construction Rule: We have a Fish pattern if we can construct two sets of sectors, a Base set and a Cover set, in such a way that every candidate of a given digit belongs to at least as many Cover sectors as it belongs to Base sectors.

So forgive me here for not following this. Does this mean that in the Nx(N+k) construction rules as described by Obi-Wahn do not have Fins??!!!

My interpretation of his ideas was (using what was mentioned before) is: any PEs will have C/B = k+1 & any EE will be a PE that sees ALL fins.
Last edited by tarek on Sun Aug 04, 2013 9:29 pm, edited 1 time in total.

tarek

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### Re: The Ultimate FISH Guide

edit:
dannys post below is a much better explanation then my own failed attempts,

however i still don't agree with the word "Fin" being used, to me its a collection of cover sections.

since technically all cover sectors in a nx(n+k) could be the "fin sector".
Which is why there is no PE's and only eliminations when using it.
Last edited by StrmCkr on Mon Aug 05, 2013 4:19 am, edited 6 times in total.
Some do, some teach, the rest look it up.

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### Re: The Ultimate FISH Guide

[Deleted: The post was not needed anymore]
Last edited by tarek on Mon Aug 05, 2013 11:33 am, edited 1 time in total.

tarek

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### Re: re: 7\8 (Obi-Wahn's arithmetic)

tarek wrote:
daj95376 wrote:
Obi-Wahn wrote:Construction Rule: We have a Fish pattern if we can construct two sets of sectors, a Base set and a Cover set, in such a way that every candidate of a given digit belongs to at least as many Cover sectors as it belongs to Base sectors.

So forgive me here for not following this. Does this mean that in the Nx(N+k) construction rules as described by Obi-Wahn do not have Fins??!!!

My interpretation of his ideas was (using what was mentioned before) is: any PEs will have C/B = k+1 & any EE will be a PE that sees ALL fins.

My understanding is that finned NxN Fish have fin cells that are covered with fewer cover sectors than base sectors.

Obi-Wahn eliminated fin cells in his "arithmetic" for finned Nx(N+k) Fish by adding additional cover sectors until every candidate cell is as I quoted in his Construction Rule above. He refers to these extra cover sectors as fin sectors.

Obi-Wahn wrote:Number of fin sectors = Number of cover sectors - Number of base sectors

When talking of a finned fish, the distinction is in the dimensionality of the base and cover sets. In a finned NxN fish, I only consider fin cells being present. In a finned Nx(N+k) fish, I only consider (extra) fin sectors being present.

As far as the mathematics goes for Nx(N+k) fish, each cover sector's contribution is independent of any other cover sector's contribution -- even if it's the same house/unit being repeated.

In finned NxN fish, eliminations are determined using peers of the fin cells and PEs from the unfinned fish. In finned Nx(N+k) fish, eliminations occur in candidate cells where ( cover_sector_count - base_sector_count ) > k.
daj95376
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### Re: re: 7\8 (Obi-Wahn's arithmetic)

daj95376 wrote:In finned NxN fish, eliminations are determined using peers of the fin cells and PEs from the unfinned fish. In finned Nx(N+k) fish, eliminations occur in candidate cells where ( cover_sector_count - base_sector_count ) > k.

That is clear ....

So if you manage to construct through programming an Nx(N+K) fish according to the rules then the eliminations will not rely on knowing which sectors are the fin sectors just knowing the candidate cells where ( cover_sector_count - base_sector_count ) > k. That -from a programming point of view- shaves a few steps compared to NxN fish: Finding fin cells & then finding PEs that are peers of all fin cells.

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.

tarek

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### Re: re: 7\8 (Obi-Wahn's arithmetic)

tarek wrote: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.

Tarek, agree absolutely that "Fin Sectors" is a badly chosen term and should be replaced. It creates the impression that the are sectors that contain only fins which is completely wrong. I wouldn't want to introduce yet another term involving 'cover' though. To my mind there is a strong similarity with the ranking of truth and link sets, and so would suggest that readers would intuitively understand "k-rank" better.

Your definitions of cell types according the difference in the number of times they are covered by the base and cover sets remains as valid as ever and should be protected from being devalued or mutilated by the lop-sided fish advocates.

A lot of truly awful terms are now established in our vocabulary and I'd like to strangle this one as quickly as possible. See < this post > for an earlier rant of mine.

DPB
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### Re: re: 7\8 (Obi-Wahn's arithmetic)

tarek wrote: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.

The fin sectors for the N\( N + k ) fish would almost always be the same as for the corresponding finned N\N fish, so I don't see an issue here.

edit:
David P Bird wrote:Tarek, agree absolutely that "Fin Sectors" is a badly chosen term and should be replaced. It creates the impression that they are sectors that contain only fins which is completely wrong.

daj95376 has more experience on this point but I think every "fin sector", even for N\(N + k) fish, has at least one fin cell, so I see no issue here either.
ronk
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### Re: The Ultimate FISH Guide

ronk wrote:
David P Bird wrote:Tarek, agree absolutely that "Fin Sectors" is a badly chosen term and should be replaced. It creates the impression that they are sectors that contain only fins which is completely wrong.

daj95376 has more experience on this point but I think every "fin sector", even for N\(N + k) fish, has at least one fin cell, so I see no issue here either.

By Obi-Wahns construction rule every digit in the pattern "belongs to at least as many cover sectors as it belongs to base sectors". His fish patterns therefore contain no fin cells at all!

No individual sector can therefore be picked out as being a "fin sector" which is what the term implies.

Obi-Wahn's definition is "Number of fin sectors = Number of cover sectors - Number of base sectors". It therefore equals k in the Nx(N+k) description of the fish, and hence the suggested alternatives.

 Deleted a confused rant about Obi-Whan reversing the fin and PE terms
Last edited by David P Bird on Mon Aug 19, 2013 7:35 am, edited 1 time in total.
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