The New Sudoku Players' Forum

[daj95376]

N*(N-1) Finned Fish Triads

For a fish elimination to occur, the cover set must include at least one unit from each elimination cell. However, what about a fish pattern for an elimination cell that doesn't include the cell in the cover set, i.e. N*(N-1), but adding any one of the three units for that cell to the cover set results in a valid fish elimination for that cell.

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Lucky Puzzle #13
 5.3....81..8...32..1..8.5.4.51..8....7..5..4....1..25.2.7.1..3..85...6..34....8.2

# partial (5x4) mutant Starfish r38c267\r17c8b1 for <9>
# there are now three uncovered cells that 'see' [r6c9]
# add r6|c9|b6 to the cover set and [r6c9] is covered w/two fin cells remaining
 *-----------------------------------------------------*
 |  5    2    3    |  679  4    679  |  79   8    1    |
 |  4    69   8    |  5    79   1    |  3    2    679  |
 |  7    1    69   |  3    8    2    |  5    69   4    |
 |-----------------+-----------------+-----------------|
 |  69   5    1    |  4    2    8    |  79   679  3    |
 |  689  7    2    |  69   5    3    |  1    4    689  |
 |  689  3    4    |  1    679  679  |  2    5    8-9  |
 |-----------------+-----------------+-----------------|
 |  2    69   7    |  8    1    69   |  4    3    5    |
 |  1    8    5    |  2    3    4    |  6    79   79   |
 |  3    4    69   |  679  679  5    |  8    1    2    |
 *-----------------------------------------------------*


There are six triads for this puzzle. All start with finned mutant Starfish r3c67\r1c8.

[Pat]

N*(N-1) Finned Fish Triads

For a fish elimination to occur, the cover set must include at least one unit from each elimination cell. However, what about a fish pattern for an elimination cell that doesn't include the cell in the cover set, i.e. N*(N-1), but adding any one of the three units for that cell to the cover set results in a valid fish elimination for that cell.

hi daj95376,

could you please clarify if this is perhaps related to

About the arithmetic of Ultimate Fish
by Obi-Wahn (2007.Feb.1)

[daj95376]

could you please clarify if this is perhaps related to ...

Hello Pat,

I've read Obi-Wahn's posts, often several times, and must admit that there are some that I don't fully understand. The post you cite is no exception -- especially the lower half. Specifically in this citation, Obi-Wahn refers to extending the Fish definition past N*N and using a count to test a pattern for elimination cells and mandatory empty cells.

My discussion is strictly limited to N*N fish and a single elimination cell. I've found a lot of N*(N-1) fish patterns where (finally) adding any one of the units for an elimination cell to the cover set results in a valid N*N fish. This Triad, as I call it, has occurred in every solution that I've generated with my Generic Fish Finder (GFF) program. I'm hoping that this property isn't restricted to how GFF finds fish. In any event, the existence of Triads is still valid.

Addendum:

Is there anything in the definition of FISH that says every cover set unit must contain at least one cell from the base set?

[Pat]

My discussion is strictly limited to N*N fish and a single elimination cell. I've found a lot of N*(N-1) fish patterns where (finally) adding any one of the units for an elimination cell to the cover set results in a valid N*N fish.

hi daj95376,

i apologize for creating any confusion between your idea and Obi-Wahn's -- i saw your N*(N-1) and it reminded me of his stuff --

Obi-Wahn considers order j\J,
where the "cover" is increased to J units ( j < J )
e.g. j\j+1

-- but i now realize i should've read your post more carefully.

Is there anything in the definition of FISH that says every cover set unit must contain at least one cell from the base set?

the fish observation is --
A. each unit of the "base" can only have the digit somewhere in the "cover", and
B. the digit cannot occur in the overlap of units of the "base" [ new for Franken and Mutant ]

-- thus we know that the "base" (of j units) will provide the digit j times in the "cover" -- so surely the "cover" must consist of at least j units which do share cells with the "base"

but i can't comment on finned fish

[ronk]

Is there anything in the definition of FISH that says every cover set unit must contain at least one cell from the base set?

Weeell, that's the safe way to view it. However, if one doesn't see the franken swordfish, here is an apparent exception:

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

sashimi swordfish c578\r258 plus fin r13c78, implies r2c9<>X

I think this is just a franken masquerading as a basic fish. In any case, I don't know how to write a rule that encompasses such an exception.

[daj95376]

Pat,

I've encountered numerous N*(N-1) pre-fish patterns where all of the fin cells in the Base Set see an elimination cell, but the elimination cell isn't covered ... yet. Simply adding the row/column/box containing the elimination cell to the Cover Set results in a N*N pattern. However, sometimes the added row/column never intersects any cells in the Base Set. Is this a valid N*N Fish pattern under these circumstances?

ronk,

A good example of my scenario! Yes, the Franken alternative should always work. However, that still leaves the question of whether or not my Lucky Puzzle #13 is a valid fish if [r6] is added to the cover set instead of [b6].

[daj95376]

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Obi-Wahn's NoFish5:
 ..4..8......2..1.....5.134.9.....8.64.8...71.751.....383...4........6..75.7......

 *--------------------------------------------------------------*
 |  1236  179   4     |  3679  3679  8     |  29    67    5     |
 |  36    789   5     |  2     4     39    |  1     67    89    |
 |  26    789   29    |  5     679   1     |  3     4     289   |
 |--------------------+--------------------+--------------------|
 |  9     2     3     |  4     1     7     |  8     5     6     |
 |  4     6     8     |  39    239   5     |  7     1     29    |
 |  7     5     1     |  68    68    29    |  4     29    3     |
 |--------------------+--------------------+--------------------|
 |  8     3     6     |  79    579   4     |  259   29    1     |
 |  12    4     29    |  1389  3589  6     |  59    38    7     |
 |  5     19    7     |  1389  2389  239   |  6     38    4     |
 *--------------------------------------------------------------*

# finned Franken Starfish r157b37\c24579 w/endo-fin [r1c7] and fins [r7c8],[r8c3]
*-----------------------------------*
|  . *9  .  | *9 *9  .  | @9  .  .  |
|  .  9  .  |  .  .  9  |  .  . *9  |
|  .  9  9  |  .  9  .  |  .  . *9  |
|-----------+-----------+-----------|
|  9  .  .  |  .  .  .  |  .  .  .  |
|  .  .  .  | *9 *9  .  |  .  . *9  |
|  .  .  .  |  .  .  9  |  .  9  .  |
|-----------+-----------+-----------|
|  .  .  .  | *9 *9  .  | *9 #9  .  |
|  .  . #9  |  9  9  .  | -9  .  .  |
|  . *9  .  |  9  9  9  |  .  .  .  |
*-----------------------------------*


ronk: Sorry, but [r8c3]=9 is true!

[tarek]

Nice catch daj95376,

Why is r1c7 an endofin & not a vertix (it is a cover set base set intersection cell)?

tarek

[daj95376]

Why is r1c7 an endofin & not a vertix (it is a cover set base set intersection cell)?

It's simply how you define endo-fin cells. I'm currently using ronk's (old?) definitions, and [r1c7] is in two base sectors.

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endo-  fin-cell  <-->   A cell inside  the intersection of two or more base sectors
exo-   fin-cell  <-->   A cell outside the intersection of the base and cover sectors
remote fin cell  <-->   A cell that only sees an elimination cell indirectly -- sd-Kraken


Note: Discussions as recent as today between ronk and myself are leading (at least me) to drop the endo-fin designation. I've switched over to a variant of Obi-Wahn's counting technique and endo-fin cells lack distinction from other fin cells. I believe that ronk's current effort is having similar results. But, I'm dense and don't always catch the nuances in his explanations. Fortunately, he's very patient!

[ronk]

It's simply how you define endo-fin cells. I'm currently using ronk's (old?) definitions [...]

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endo-  fin-cell  <-->   A cell inside  the intersection of two or more base sectors
exo-   fin-cell  <-->   A cell outside the intersection of the base and cover sectors
remote fin cell  <-->   A cell that only sees an elimination cell indirectly -- sd-Kraken


Those are the current definitions. As I've recently been incorrectly calling remote fins as exo-fins instead, I should read them once in a while.

[tarek]

I understand that the "vertix" & "fin" definitions need to be revised based on the Remote ExoEndo theory. The recent catch by daj95376 of that finned franken starfish still shows that we need to wait before updating them.

your last endofin was not just an intersection of 2 base sets, it was the intersection of a base/cover set so makes it a bit confusing....No more possible eliminations are created if consider r1c7 a vertix.

I suggest fishing again in the NoFish puzzles.

tarek

[ronk]

your last endofin was not just an intersection of 2 base sets, it was the intersection of a base/cover set so makes it a bit confusing....No more possible eliminations are created if consider r1c7 a vertix.

In order for the constraint set elimination rule (do we have one?) to apply, there should not be a candidate in the intersection of two base sectors (units). If one does exist, it is an extra candidate -- which we call an endo-fin.

This means the endo-fin, just like any other fin, must "see" the elimination cell. Whether or not the endo-fin is also included in the cover set matters not.

[tarek]

This means the endo-fin, just like any other fin, must "see" the elimination cell. Whether or not the endo-fin is also included in the cover set matters not.

It matters, because the fin definitions need to be changed........

The rise of mutant and franken fish created this "endo-fin". I'm assuming that you have several examples of these endofins that are not part of the cover set.

tarek

[ronk]

This means the endo-fin, just like any other fin, must "see" the elimination cell. Whether or not the endo-fin is also included in the cover set matters not.

[...] I'm assuming that you have several examples of these endofins that are not part of the cover set.

I would be surprised if an endofin wasn't a member of a useful cover set. I only meant that an endofin cell must "see" the elimination cell, even though that endofin cell is a member of the cover set,

[daj95376]

If you use Obi-Wahn's counting technique, then a cell that's present in more base sectors than cover sectors must be empty ... or ... (daj95376's corollary) treated as a fin cell. It doesn't matter if the number of cover sectors is zero or one!