The New Sudoku Players' Forum

[daj95376]

I suggest fishing again in the NoFish puzzles.

I hope you like finned Mutant Jellyfish/Starfish/Whales !!!

[denis_berthier]

I also posted this in the "fully supersymmetric chains" thread, but it is also relevant here.

Theorem: Finned and sashimi fish are subsumed by nrcz-chains.

Preliminaries: standard x-wing on rows is the following special case of nrc-chains: n1{r1c2 r1c1} - n1{r2c1 r2c2}, where:
- the two nrc-conjugate links are along rows r1 and r2,
- the mere central nrc-link is along column c1,
- the targets are considered as linked to both endpoints of this chain along column c2;
(Notice that, for the eliminations on column c1, chain n1{r1c1 r1c2} - n1{r2c2 r2c1} must be considered instead.)

Proof for finned fish: In the corresponding finned fish, additional candidate n1 may appear in the same row and block as the last candidate (n1r2c2) of this chain. But, any candidate n1rc such that rc is in the intersection of the column and the block containing the last candidate, makes the n1{r1c2 r1c1} - n1{r2c1 r2c2} chain into an nrcz-chain wrt target n1rc, and the additional candidates on row r2 are exactly what the z-extension allows to disregard.

Proof for sashimi fish: In the corresponding sashimi fish, additional candidate n1 may appear in the same row and block as the last candidate (n1r2c2) of this chain and candidate n1 may be absent from r2c2. But, any candidate n1rc such that rc is in the intersection of the column and the block containing the last candidate, still makes the n1{r1c2 r1c1} - n1{r2c1 r2c2} chain into an nrcz-chain wrt target n1rc, and the additional candidates on row r2 are exactly what the z-extension allows to disregard.

The same considerations apply to x-wing on columns, by row-column symmetry.
Their generalisation to swordfish and jellyfish is straightforward: if the last cell is still the one around which the modifications are made, consider respectively the associated nrc-chains (and their z-extension for the proof):
n1{r2c3 r2c2} - n1{r1c2 r1c1} - n1{r3c1 r3c4}
and
n1{r3c4 r3c3} - n1{r2c3 r2c2} - n1{r1c2 r1c1} - n1{r4c1 r4c4}

As a result, nrcz-chains are a way of integrating finned and sashimi fish into chains.

[denis_berthier]

erratum:
in the previous post, in the proof for sashimi, the last candidate in each of the 3 chains used:
n1{r1c2 r1c1} - n1{r2c1 r2c2}
n1{r2c3 r2c2} - n1{r1c2 r1c1} - n1{r3c1 r3c4}
and
n1{r3c4 r3c3} - n1{r2c3 r2c2} - n1{r1c2 r1c1} - n1{r4c1 r4c4}
does not exist.
It should be replaced by candidates in the fin, resp.: n1r2c+, n1r3c+, n1r4c+


Moreover, I've not been careful enough in the generalisations to Swordfish and Jellyfish. Here are the correct theorems.

1) x-wing:

Theorem: Any finned x-wing, with or without sashimi, is subsumed by an nrcz2-chain.


2) Swordfish

After re-ordering of the rows and columns, the most general Swordfish on rows for number n1 looks like:

..1a === 1b === (1c)
(1d) === 1e === 1f
..1g == (1h) === 1i
===========1x

where (1c), (1d) and (1h) are optional and 1x is a target in the same column as 1i

The most general Swordfish on rows for number n1, with fin (and sashimi), looks like:

..1a === 1b === (1c)
(1d) === 1e === 1f
..1g == (1h) === [1i] === 1fin
===========1x

1+ is a candidate in the fin
1fin is any candidate in the fin
[1i] is the potentially missing candidate in the sashimi
1x is a target in column and block of the last candidate: [1i]

Consider the chain 1f-1e-1b-1a-1g-(1i if present, 1fin otherwise). It is an nrczt-chain wrt 1x, If 1d is absent.

- additional candidates 1c and 1fin (if present) are justified by the z extension
- additional candidate 1h i(if present) s justified by the t extension

Theorem: Any 2x2x2 finned swordfish with or without sashimi is subsumed by an nrcz3-chain
Theorem: Any other finned swordfish with or without sashimi is subsumed by an nrczt3-chain, provided that it has a missing candidate in a row and a column different from that of the base for the fin/sashimi.

For the only remaining case, I need something I haven't yet spoken of. I haven't yet found time to write it down properly, so be patient.


3) Jellyfish

After re-ordering of the rows and columns, the most general Jellyfish on rows for number n1 looks like:

..1a === 1b === (1c) == (1d)
(1e) === 1f ==== 1g == (1h)
(1i) === (1j) === 1k === 1l
1m === (1n) == (1o) == 1p
==================1x
with similar conventions.


The most general Jellyfish on rows for number n1, with fin (and sashimi), looks like:
..1a === 1b === (1c) == (1d)
(1e) === 1f ==== 1g == (1h)
(1i) === (1j) === 1k === 1l
1m === (1n) == (1o) == [1p] === 1fin
==================1x

Consider the chain: 1l-1k-1g--1f-1b-1a-1m-(1p if present, 1fin otherwise). It is an nrczt-chain if 1e, 1i and 1j are absent:
- additional candidates 1d, 1h and 1fin (if present) are justified by the Z extension
- additional candidates 1c, 1n and 1o (if present) are justified by the t-extension

Theorem: Any 2x2x2x2 finned jellyfish with or without sashimi is subsumed by an nrcz4-chain
Theorem: Any other finned jellyfish with or without sashimi is subsumed by an nrczt4-chain, provided that it has:
- a row different from that of the fin/sashimi base with a missing candidate in a column different from that of the fin/sashimi base
- another row different from that of the fin/sashimi base with a missing candidate in the previous column and a missing candidate in another column also different from that of the fin/sashimi base

[Pat]

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 units.

If one does exist, it is an extra candidate

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

yes

the present text in the head-post still says --

*Fin*: A candidate that occupies a cell within a "Base sector" but not within any "Cover sector"---

starting with the (un-finned) fish,
and clearly defining the observation ("/" -- the "hidden pattern"),
it will become easy to define fin-cells --

a fin-cell ("#") is a "/" that isn't

the 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 will provide the digit N times in the cover,
and the conclusion is --
exclude the digit in the cover outside the base



beyond the fish exclusion,
if there's overlap in the cover (i.e. new for Franken and Mutant)
we have an extra type of exclusion --
exclude the digit in the overlap of units of the cover

[daj95376]

After recent pm's with ronk, and re-reading Obi-Wahn's Arithmetic post, it appears to me that finned fish is a misnomer. Since additional Cover sectors are now used to cover all Base set candidates, there aren't any fin cells remaining. There are only N*N Fish, N*(N+1) Fish, and N*(N+2) Fish. Right?

Maybe three new threads should be started.

TUFG for N*N Fish
TUFG for N*(N+1) Fish
TUFG for N*(N+2) Fish

My contribution to the N*(N+2) Fish thread.

Code: Select all

Puzzle #B026 8:
7..96.1....4..26.3...3.5.2.14......2..7.2.4..9......67.7.6.3...4.62..3....1.94..6

 Jellyfish   c147b3\r2579c9b 6  <> 8  [r5c9b6]   -or-
 Jellyfish r3c147  \r 579c9b16  <> 8  [r5c9b6]
 *-----------------------------------------------------------------------*
 |  7      23     23     |  9      6      8      |  1      45     45     |
 |  58     589    4      |  1      7      2      |  6      89     3      |
 |  6      1      89     |  3      4      5      |  7      2      89     |
 |-----------------------+-----------------------+-----------------------|
 |  1      4      358    |  7      358    6      |  9      358    2      |
 |  358    6      7      |  58     2      9      |  4      1358   158    |
 |  9      2358   2358   |  4      358    1      |  58     6      7      |
 |-----------------------+-----------------------+-----------------------|
 |  258    7      589    |  6      158    3      |  258    1458   14589  |
 |  4      589    6      |  2      158    7      |  3      1589   158    |
 |  2358   358    1      |  58     9      4      |  258    7      6      |
 *-----------------------------------------------------------------------*


[ronk]

After recent pm's with ronk, and re-reading Obi-Wahn's Arithmetic post, it appears to me that finned fish is a misnomer. Since additional Cover sectors are now used to cover all Base set candidates, there aren't any fin cells remaining. There are only N*N Fish, N*(N+1) Fish, and N*(N+2) Fish. Right?

Maybe three new threads should be started.

TUFG for N*N Fish
TUFG for N*(N+1) Fish
TUFG for N*(N+2) Fish

Right, but I think the fin concept is a useful, maybe even necessary, step in the learning process ... and keeping them together highlights the differences.

For example, consider the unfinned mutant jellyfish below, with the 21 possible eliminations illustrated. Merely add one fin cell at r4c5 -- covered by r4 -- and only three eliminations remain at double-covered cells r4c279.

Code: Select all

 .  *  . |  .  /  . |  .  /  .        .  *  . |  .  /  . |  .  /  .
 *  *  * |  *  X  * |  *  X  *        *  *  * |  *  X  * |  *  X  *
 .  *  . |  .  /  . |  .  /  .        .  *  . |  .  /  . |  .  /  .
---------+----------+----------      ---------+----------+----------
 .  *  . |  .  /  . |  *  X  *        . **  . |  .  #  . | **  X **
 /  X  / |  /  /  / |  X  /  X        /  X  / |  /  /  / |  X  /  X
 .  *  . |  .  /  . |  *  X  *        .  *  . |  .  /  . |  *  X  *
---------+----------+----------      ---------+----------+----------
 .  *  . |  *  X  * |  .  /  .        .  *  . |  *  X  * |  .  /  .
 /  X  / |  X  /  X |  /  /  /        /  X  / |  X  /  X |  /  /  /
 .  *  . |  *  X  * |  .  /  .        .  *  . |  *  X  * |  .  /  .
 "N\N" for 21 elims shown             "N\(N+1)" for r4c279<>X
 unfinned mutant jellyfish rrcc\rcbb  1-finned mutant jellyfish rrcc\rcbb+r

Add at least one of the fin cells r5c13 -- covered by b4 --- and the only remaining elimination is the triple-covered cell r4c2.

Code: Select all

 .  *  . |  .  /  . |  .  /  .
 *  *  * |  *  X  * |  *  X  *
 .  *  . |  .  /  . |  .  /  .
---------+----------+----------
 . *** . |  .  #  . | **  X **
 #  X  # |  /  /  / |  X  /  X
 . **  . |  .  /  . |  *  X  *
---------+----------+----------
 .  *  . |  *  X  * |  .  /  .
 /  X  / |  X  /  X |  /  /  /
 .  *  . |  *  X  * |  .  /  .
 "N\(N+2)" for r4c2<>X
 2-finned mutant jellyfish rrcc\rcbb+rb

[Ruud]

Maybe three new threads should be started.

TUFG for N*N Fish
TUFG for N*(N+1) Fish
TUFG for N*(N+2) Fish

I wonder if there is N*((N-1)+2) Fish not replicated by N*(N+1). Given the fact that N*(N+2) exists, could there also be N*((N-1)+3)? If we could formulate all theoretical possibilities in set definitions, at least we would know what to look for.

See also my recent post on the Eureka forum
{ broken link www.sudoku.org.uk/discus/messages/29/4672.html?1189931927 }

Ruud

[Mage]

[silly post, withdrawn]

Mage

[RW]

Staying at the surace, what I see here is an AIC loop :
r2c5=r2c8 - r46c8=r5c79 - r5c2=r8c2 - r8c46=r79c5

witch eliminates a lot more than the 21 illustrated:

Code: Select all

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

39 elimination cells !... and adding r4c5 or r5c13 doesn't change anything.

Mage

Here's a solution to the digit template that doesn't hit any of the forbidden cells and occupies 7 of your elimination cells...

Code: Select all

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

RW

[Mage]

Thanks RW for putting me back in my corner...

and sorry to have disturbed your fishing

[ronk]

I wonder if there is N*((N-1)+2) Fish not replicated by N*(N+1). Given the fact that N*(N+2) exists, could there also be N*((N-1)+3)?

I believe the N\((N-1)+M) notation helps us to eliminate equivalent fish, but will not uncover any new fish.

To illustrate:

Code: Select all

.7...3.91......5..3....6..7....3....2....9715.9..12.8..31..79......6.....57..1.62 #N014 4

After SSTS
 4  .  4 |  4  4  . |  4  .  .
 4  4  4 |  4  . #4 |  .  . -4
 .  4  4 |  4  4  . |  4 #4  .
---------+----------+----------
 4  4  4 |  4  . *4 |  4 *4  .
 .  4  . |  4  4  . |  .  .  .
 4  .  4 |  4  .  . |  4  .  4
---------+----------+----------
 .  .  . | *4 *4  . |  .  . #4
 4  4  4 |  4  . *4 |  .  .  4
 4  .  . |  4  4  . |  4  .  .

Since any two of the three cells tagged '#' can be considered as the fins of a 2-finned swordfish, it would appear there are three different N\(N+2) sashimi mutant swordfish with the identical r2c9<>4 elimination.

r7c68\r24b8+c9b3
r7c68\r4c9b8+r2b3
r7c68\r4b38+r2c9

But there's only one N\((N-1)+3) fish which can be written r7c68\r4b8+r2c9b3 ... or more compactly as r7c68\r24c9b38.

Of the two styles, I prefer the first ... but it's a bit harder to program, so my fish finder utility outputs the second for now. I have bigger fish (pun) to fry.

Question: Do we still consider a fish expressed as N\((N-1)+3) to be a 2-finned fish

BTW I think the N\(N-1)+M) POV makes sense for sashimi, but not for finned viable (non-degenerate) fish.

[daj95376]

FWIW: When there's only a single elimination to consider, then N*((N-1)+3) format may be overkill, but it covers all possible fin sector/cell combinations. In this case, I just compress the listing to where the +3 sectors are in the elimination cell:

Code: Select all

mutant Starfish r37c7b35\r6c458 <> 1 [r1c1b1]


(Yes ronk, I know you hate this format. Sorry!)

----- ----- ----- ----- ----- -----

Say, with the additional Cover sectors, what happens to the definition of a finned Franken fish?

[ronk]

Code: Select all

mutant Starfish r37c7b35\r6c458 <> 1 [r1c1b1]


(Yes ronk, I know you hate this format. Sorry!)

How does that notation apply when the fish exists, but there is no elimination

From that notation, how does one tell whether there are one or two fin sectors BTW I think Obi-Wahn showed that three actual fin sectors are possible too.

Say, with the additional Cover sectors, what happens to the definition of a finned Franken fish?

As I stated in my prior post, I think this N\((N-1)+M) discussion applies to sashimi only. Unfinned and finned fish -- basic, franken and mutant -- would remain N\N+M.

Non-sashimi fish would be notated with an "N cover sectors" and sashimi with an "N-1 cover sectors." This lack of consistency is actually a bit bothersome.

[daj95376]

How does that notation apply when the fish exists, but there is no elimination

How does any notation apply when there is a fish but no elimination?

From that notation, how does one tell whether there are one or two fin sectors BTW I think Obi-Wahn showed that three actual fin sectors are possible too.

In order to show the number of fin sectors, you are forced to write multiple, specific entries for the same elimination. With my format, the number of fin sectors can be derived, but aren't directly listed. Brevity comes at a price!

As I stated in my prior post, I think this N\((N-1)+M) discussion applies to sashimi only. Unfinned and finned fish -- basic, franken and mutant -- would remain N\N+M.

Do previous sashimi franken fish still qualify as franken fish in the N\((N-1)+M) notation?

Code: Select all

 *-----------------------------------------------------------------------*
 |  127    3      279    |  129    6      8      |  15     159    4      |
 |  128    5      4      |  1239   7      12     |  6      19     38     |
 |  6      18     19     |  1349   149    5      |  7      2      38     |
 |-----------------------+-----------------------+-----------------------|
 |  147    14     3      |  17     5      6      |  9      8      2      |
 |  5      6      17     |  8      2      9      |  4      3      17     |
 |  9      2      8      |  147    3      147    |  15     6      157    |
 |-----------------------+-----------------------+-----------------------|
 |  28     7      5      |  6      14     3      |  28     14     9      |
 |  124    149    6      |  12479  8      247    |  3      145    15     |
 |  3      1489   12     |  5      49     124    |  28     7      6      |
 *-----------------------------------------------------------------------*


Empty Rectangle c7b5 can be listed as either:

Code: Select all

sashimi mutant  X-Wing c7b5\r6+c4    <> 1  [r1c4]
sashimi franken X-Wing c7b5\r6+r1    <> 1  [r1c4]


With all fin sectors covered:

Code: Select all

sashimi ???     X-Wing c7b5\r6+r1c4  <> 1  [r1c4]


[ronk]

Do previous sashimi franken fish still qualify as franken fish in the N\((N-1)+M) notation?

Code: Select all

 *-----------------------------------------------------------------------*
 |  127    3      279    |  129    6      8      |  15     159    4      |
 |  128    5      4      |  1239   7      12     |  6      19     38     |
 |  6      18     19     |  1349   149    5      |  7      2      38     |
 |-----------------------+-----------------------+-----------------------|
 |  147    14     3      |  17     5      6      |  9      8      2      |
 |  5      6      17     |  8      2      9      |  4      3      17     |
 |  9      2      8      |  147    3      147    |  15     6      157    |
 |-----------------------+-----------------------+-----------------------|
 |  28     7      5      |  6      14     3      |  28     14     9      |
 |  124    149    6      |  12479  8      247    |  3      145    15     |
 |  3      1489   12     |  5      49     124    |  28     7      6      |
 *-----------------------------------------------------------------------*


Empty Rectangle c7b5 can be listed as either:

Code: Select all

sashimi mutant  X-Wing c7b5\r6+c4    <> 1  [r1c4]
sashimi franken X-Wing c7b5\r6+r1    <> 1  [r1c4]


With all fin sectors covered:

Code: Select all

sashimi ???     X-Wing c7b5\r6+r1c4  <> 1  [r1c4]


When one can interpret the same five cells two different ways, one time as a franken and another time as a mutant, it should be a red flag that something is amiss. It's actually a mutant in both interpretations, and the notation ...

sashimi mutant X-Wing c7b5\r6+r1c4 <> 1 [r1c4]

... with both rows and columns in the cover, makes that abundantly clear.

As one of the drum majors in this marching band, I take full responsibility for not recognizing this long ago.

And perhaps we've both overdone brevity here. For this fish, I would like to see fins r1c7 and r4c4 identified for the reader.