The New Sudoku Players' Forum

[Pat]

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 *-----------------------------------------------------------------------*
 |  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      |
 *-----------------------------------------------------------------------*



to exclude 1 at r1c4 --

• finned Franken X-wing cb\rr: c7b5\r16, with fin = r45c4
• finned Mutant X-wing cb\rc: c7b5\r6c4, with fin = r1c7
• cb\rrc: c7b5\r16c4

the last one uses Obi-Wahn's idea -- a "base" of 2 units, but 3 units in the "cover" -- and thus has no fin

do we still call it a fish?
(Obi-Wahn gave no new name for these creatures)

[daj95376]

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# wapati puzzle PM
 *-----------------------------------------------------------------------*
 |  9      4      3      |  5      16     7      |  16     2      8      |
 |  26     7      5      |  12     8      13     |  169    4      1369   |
 |  8      1      26     |  24     346    9      |  67     37     5      |
 |-----------------------+-----------------------+-----------------------|
 |  7      5      12     |  6      1249   24     |  3      8      149    |
 |  126    9      4      |  3      127    8      |  5      17     16     |
 |  136    36     8      |  149    1479   5      |  14679  179    2      |
 |-----------------------+-----------------------+-----------------------|
 |  4      368    169    |  7      129    26     |  189    5      139    |
 |  13     2      7      |  89     5      13     |  489    6      49     |
 |  5      68     169    |  1489   134    46     |  2      139    7      |
 *-----------------------------------------------------------------------*


Siamese fish or Remoras? If Remoras, what's the larger fish?

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Sashimi Swordfish r248\c479 w/fin cell B=[r4c5] => [r6c4]<>9
Sashimi Swordfish r248\c579 w/fin cell A=[r8c4] => [r7c5]<>9
*-----------------------------------*
|  9  .  .  |  .  .  .  |  .  .  .  |
|  .  .  .  |  .  .  .  | *9  . *9  |
|  .  .  .  |  .  .  9  |  .  .  .  |
|-----------+-----------+-----------|
|  .  .  .  |  . B9  .  |  .  . *9  |
|  .  9  .  |  .  .  .  |  .  .  .  |
|  .  .  .  | -9  9  .  |  9  9  .  |
|-----------+-----------+-----------|
|  .  .  9  |  . -9  .  |  9  .  9  |
|  .  .  .  | A9  .  .  | *9  . *9  |
|  .  .  9  |  9  .  .  |  .  9  .  |
*-----------------------------------*


[ronk]

Siamese fish or Remoras? If Remoras, what's the larger fish?

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Sashimi Swordfish r248\c479 w/fin cell B=[r4c5] => [r6c4]<>9
Sashimi Swordfish r248\c579 w/fin cell A=[r8c4] => [r7c5]<>9
*-----------------------------------*
|  9  .  .  |  .  .  .  |  .  .  .  |
|  .  .  .  |  .  .  .  | *9  . *9  |
|  .  .  .  |  .  .  9  |  .  .  .  |
|-----------+-----------+-----------|
|  .  .  .  |  . B9  .  |  .  . *9  |
|  .  9  .  |  .  .  .  |  .  .  .  |
|  .  .  .  | -9  9  .  |  9  9  .  |
|-----------+-----------+-----------|
|  .  .  9  |  . -9  .  |  9  .  9  |
|  .  .  .  | A9  .  .  | *9  . *9  |
|  .  .  9  |  9  .  .  |  .  9  .  |
*-----------------------------------*


One could also ask ... what's the smaller fish?

[daj95376]

Do we have an example where the Base and Cover sets exist in every box and there aren't any remoras?

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 *-----------------------*
 | 5 . . | . . . | . 7 . |
 | 3 6 . | . . 7 | . 9 5 |
 | . . 7 | . 5 . | 4 . . |
 |-------+-------+-------|
 | . . 9 | 6 . . | . 8 . |
 | . . 6 | . 7 . | 3 . . |
 | . 1 . | . . 9 | 7 . . |
 |-------+-------+-------|
 | . . 8 | . 6 . | 9 . . |
 | 1 9 . | 3 . . | . 6 7 |
 | . 5 . | . . . | . . 4 |
 *-----------------------*


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Franken Whale c2569b27\r13457b8
 *--------------------------------------------------------------*
 |  5     248   14-2  |  9     12    3     |  6     7     128   |
 |  3     6     12    |  48    48    7     |  12    9     5     |
 |  9     28    7     |  12    5     6     |  4     123   1238  |
 |--------------------+--------------------+--------------------|
 |  7     3     9     |  6     124   1245  |  125   8     12    |
 |  248   24    6     |  1258  7     128   |  3     1245  9     |
 |  248   1     5     |  28    3     9     |  7     24    6     |
 |--------------------+--------------------+--------------------|
 |  24    7     8     |  1245  6     124   |  9     1235  123   |
 |  1     9     24    |  3     248   2458  |  258   6     7     |
 |  6     5     3     |  7     9     128   |  128   12    4     |
 *--------------------------------------------------------------*


[ronk]

Do we have an example where the Base and Cover sets exist in every box and there aren't any remoras?
[...]
Franken Whale c2569b27\r13457b8

That's not a stat I track , but I think a fish that large without a single remora would be rare. In any case, a ccccbb\rrrrrb whale -- basic or sashimi -- is not currently in the exemplar catalog. Great fishin'.

BTW how many fin sectors did you allow for the potential remora?

[daj95376]

BTW how many fin sectors did you allow for the potential remora?

I must admit that I'm working with questionable definitions for Siamese fish and Remora.

Siamese fish: Two N*N fish that share N*(N-1) sectors. They use a different N-th Cover sector to complete the fish pattern and perform different eliminations. The Sashimi Swordfish a couple of posts back are an (imperfect) example.

Remora fish: An N*N fish where most of its eliminations are common to an (N+M)*(N+M) fish that performs even more eliminations. More of a subset relationship to the larger fish than a conjugate/conjoined relationship between two fish.

===== ===== ===== Addendum

FWIW: The smallest fish that I found were either Franken or mutant Whales for [r1c3]<>2. Since there is only this one Template elimination present, it doesn't make sense to look for a larger fish to encompass these fish.

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6-Fish c2569b27\r3457b8  +r1|c3|b1  <> 2  [r1c3]
6-Fish r2689b27\c1478b8  +r1|c3|b1  <> 2  [r1c3]
6-Fish r2689c2b2\c478b48 +r1|c3|b1  <> 2  [r1c3]
6-Fish r26c29b27\r7c4b346+r1|c3|b1  <> 2  [r1c3]
6-Fish r26c2b279\r7c478b4+r1|c3|b1  <> 2  [r1c3]
6-Fish r2c2569b7\r457b38 +r1|c3|b1  <> 2  [r1c3]
6-Fish r2c29b257\r457c4b3+r1|c3|b1  <> 2  [r1c3]


[ronk]

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

Without proof and with little comment, here is something along the same line. A specific case from the exemplar catalog:

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 /  X  / |  .  .  . |  .  /  .
 X *X  X |  *  *  * |  *  X  *
 /  X  / |  .  .  . |  .  /  .
---------+----------+----------
 .  *  . |  .  .  . |  .  /  .
 .  *  . |  .  .  . |  .  /  .
 .  *  . |  .  .  . |  .  /  .
---------+----------+----------
 .  *  . |  .  .  . |  *  X  *
 /  X  / |  /  /  / |  X  /  X
 .  *  . |  .  .  . |  *  X  *
 finless mutant swordfish r8c8b1\r2c2b9+[r8c8]

Where:
 r8c8b1 is the base set, a union of units r8, c8 and b1
 r2c2b9 is the cover set, a union of units r2, c2 and b9
 r8c8b1\r2c2b9 is the base set minus the cover set, all the empty cells shown EXCEPT for cell r8c8
 [r8c8] is cell r8c8, as opposed to the union of r8 and c8 (equivalent to r8*c8)
 r8c8b1\r2c2b9+[r8c8] is the union of r8c8b1\r2c2b9 and [r8c8], ALL the empty cells shown


Note that most of the union operators are omitted, and the difference operator '\' takes precedence over the union operator '+'.

In general:

Code: Select all

Sets:
  Base (size N):   B  = B1 + B2 ... + BN
  Base Intersect:  BI = B1*B2 + B1*B3 ... + B1*BN + B2*B3 ... + (BN-1)*BN
  Cover (size N):  C  = C1 + C2 ... + CN
  Cover Intersect: CI = C1*C2 + C1*C3 ... + C1*CN + C2*C3 ... + (CN-1)*CN
  Hidden:          H  = (B \ C) + BI
  Exclusion:       E  = (C \ B) + CI

Symbol Key:
    Bn    <=> units (sectors) of the base set
    Cn    <=> units (sectors) of the cover set
    '+'   <=> union operator ('|' in C++)
    '*'   <=> intersection operator ('&' in C++)
    '\'   <=> difference operator (substraction) ("X & ~Y" in C++)
  "(...)" <=> operator precedence


Conjecture: If the "hidden set" is empty, then the "exclusion set" is empty also.

The next step is to generalize this even further with a fin or two. As I'm a novice on set mathematics and set notation, comments are invited.

[daj95376]

[Withdrawn] I misunderstood.

[ronk]

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Symbol Key:

    Bi  : occupied cells of i-th base  sector/unit
    Ci  : occupied cells of i-th cover sector/unit
    '|' : set-wise OR
    '&' : set-wise AND
    '\' : set-wise NOT


Thanks for the feedback, for which I've a few questions.

1) [edit: Dumb question deleted.]
2) What precisely does "set-wise" mean?

Code: Select all

Sets:
  Base  Set       : B  = |( Bi        i=1..N             )


I'm assuming that's equivalent to the C-language statement ...

for (B = 0, i =1; i <= N; ++i) B |= Bi; (assuming a large enough int)

... but 3)would you please translate that into English? (This question refers to the " = |( .... )" part, not the portion between the parentheses.)

Code: Select all

 r8c8b1 is B             : an OR of occupied cells in units r8, c8, and b1
 r2c2b9 is C             : an OR of occupied cells in units r2, c2, and b9
 r8c8b1\r2c2b9 is Exclude: ( ( C \ B ) | CI ) -or-
                           ( ( C \ B ) | CI ) & BB (if BI not empty )


At first glance, the BB part doesn't seem correct ... unless you're considering only endo-fins, and not exofins.

[tarek]

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  Hidden:          H  = (B \ C) + BI
  Exclusion:       E  = (C \ B) + CI

Very nice indeed, where is the line regarding the fin

daj95376: I have to digest what you've written before commenting

tarek

[Pat]

Code: Select all

Sets:
                   Bn = units of the base set
  Base (size N):   B  = B1 + B2 ... + BN
  Base Intersect:  BI = B1*B2 + B1*B3 ... + B1*BN + B2*B3 ... + (BN-1)*BN

                   Cn = units of the cover set
  Cover (size N):  C  = C1 + C2 ... + CN
  Cover Intersect: CI = C1*C2 + C1*C3 ... + C1*CN + C2*C3 ... + (CN-1)*CN

  Hidden Pattern:  H  = (B \ C) + BI
  Exclusion:       E  = (C \ B) + CI


Symbol Key:
    '+'   <=> union         ('|'       in C++)
    '*'   <=> intersection  ('&'       in C++)
    '\'   <=> substraction  ('X & ~Y'  in C++)



Conjecture:

If the "hidden set" is empty,
then the "exclusion set" is empty also.

yes

1. exclusions in C\B

B will supply the digit N times (not less since none in BI);
all N will be in C (since none in B\C);
this supplies the full quota of the digit for the N units of C,
therefore, exclude it elswhere in C (i.e. in C\B).

2. exclusions in CI

placing the digit in CI would satisfy 2 (or more) units of C,
so C could only take N-1 of the total N supplied by B.

anyone wants to phrase it more formally?

[ronk]

daj95376, I'm disappointed to see that you withdrew your post. It was mostly, if not all, correct and exactly the type of feedback I'd hoped for. I intend to adopt some of your notation, and now I'll have to go by memory. You know how bad that can be.

Hidden Pattern: H = (B \ C) + BI
Exclusion: E = (C \ B) + CI

Conjecture:

If the "hidden set" is empty,
then the "exclusion set" is empty also.

1. exclusions in C\B

B will supply the digit N times (not less since none in BI);
all N will be in C (since none in B\C);
this supplies the full quota of the digit for the N units of C,
therefore, exclude it elswhere in C (i.e. in C\B).

2. exclusions in CI

placing the digit in CI would satisfy 2 (or more) units of C,
so C could only take N-1 of the total N supplied by B.

anyone wants to phrase it more formally?

Pat, that's a fine enough proof for me. Thanks.

[daj95376]

I'm not up on degenerate finned fish, so my question may have been addressed elsewhere. Please be patient if you consider the topic old information.

Does anyone have an example of a degenerate finned fish where all of the fin cells are false? TIA!!!

[ronk]

Does anyone have an example of a degenerate finned fish where all of the fin cells are false? TIA!!!

Here is a sashimi x-wing where both fin cells are ultimately false.

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top1465 #1046
..8.1..2...4689..1..7........9..8.5.8....2.......7.9..3..2...7......74......3...2

After SSTS:
 69    369   8     | 7     1     345   | 56    2     4569
 25    25    4     | 6     8     9     | 7     3     1
 1     369   7     | 45    2     35    | 568   4689  5689
-------------------+-------------------+------------------
 47    1     9     | 3     46    8     | 2     5     467
 8     47   *56    | 9    *456   2     | 3     1     467
 256   256   3     | 45    7     1     | 9     468   68
-------------------+-------------------+------------------
 3     45689 16-5  | 2     459   456   | 1568  7     5689
#569  #5689 *2     | 1    *59    7     | 4     689   3
 47    45679 16-5  | 8     3     456   | 156   69    2

 sashimi x-wing r58\c35 plus fin r8c12, implies r79c3<>5

In the solution, fin r8c12<>5.

[daj95376]

Thanks ronk! There went a crazy hypothesis.