2 fish in 1 ?Siamese

Advanced methods and approaches for solving Sudoku puzzles

2 fish in 1 ?Siamese

Postby tarek » Sun Jan 05, 2020 9:23 pm

I caught this nice specimen recently which is a sudoku X:
Code: Select all
+--------+--------------------------------------------------------------+--------+
| 1      | 68       2       *67       9       *5678     3        4      |*5678   |
+--------+--------+                                            +--------+--------+
| 578    | 9      | 3456     1       *3478    3568-7    25678  | 258-7  | 5678   |
|        +--------+--------+                          +--------+--------+        |
| 578      3468   | 3456   | 2       *3478    3568-7  | 568-7  | 1        9      |
|                 +--------+--------+        +--------+--------+                 |
| 34       1        7      | 8      | 35     | 9      | 45       6        2      |
|                          +--------+--------+--------+                          |
| 6        2        8        347    |*357    | 1        4579     3579     345-7  |
|                          +--------+--------+--------+                          |
| 34       5        9      | 34-7   | 6      | 2      | 478      378      1      |
|                 +--------+--------+        +--------+--------+                 |
| 258      7      | 1      | 369      38       368    | 456    | 23589    34568  |
|        +--------+--------+                          +--------+--------+        |
| 9      | 3468   | 346      5        2        3678     1      | 37     | 3678   |
+--------+--------+                                            +--------+--------+
| 258    | 368      356      3679     1        4        256789   35789  | 356-7  |
+--------+--------------------------------------------------------------+--------+

Note that we have here what appears to be a Grouped 2-String Kite but it is so much more.
As it is a grouped L1-ring Then it is technically an X-Wing (a grouped X-Wing).
7r1c5\b5d/ explains most of the eliminations ... The eliminations in r59c9 require
7r1c5\c9d/

So you may have here a Siamese Grouped X-Wing

tarek
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Re: 2 fish in 1 ?Siamese

Postby SpAce » Sun Jan 05, 2020 11:09 pm

Hi tarek,

tarek wrote:I caught this nice specimen recently which is a sudoku X

Yes, a very interesting species!

diagram: Show
Code: Select all
+--------+--------------------------------------------------------------+--------+
| 1      | 68       2       *67       9       *5678     3        4      |*5678   |
+--------+--------+                                            +--------+--------+
| 578    | 9      | 3456     1       *3478    3568-7    25678  | 258-7  | 5678   |
|        +--------+--------+                          +--------+--------+        |
| 578      3468   | 3456   | 2       *3478    3568-7  | 568-7  | 1        9      |
|                 +--------+--------+        +--------+--------+                 |
| 34       1        7      | 8      | 35     | 9      | 45       6        2      |
|                          +--------+--------+--------+                          |
| 6        2        8        347    |*357    | 1        4579     3579     345-7  |
|                          +--------+--------+--------+                          |
| 34       5        9      | 34-7   | 6      | 2      | 478      378      1      |
|                 +--------+--------+        +--------+--------+                 |
| 258      7      | 1      | 369      38       368    | 456    | 23589    34568  |
|        +--------+--------+                          +--------+--------+        |
| 9      | 3468   | 346      5        2        3678     1      | 37     | 3678   |
+--------+--------+                                            +--------+--------+
| 258    | 368      356      3679     1        4        256789   35789  | 356-7  |
+--------+--------------------------------------------------------------+--------+

Note that we have here what appears to be a Grouped 2-String Kite but it is so much more.

Indeed.

As it is a grouped L1-ring Then it is technically an X-Wing (a grouped X-Wing).

Note that those are two different things, and only the latter is correct. An L1-Ring has exactly three strong links (like all one-letter wings/rings), but we only have two here. Thus it's a weird kind of X-Wing, but not an L1-Ring. (I guess a generalized name could be "Turbot Ring" since the definition of a Turbot Fish is a single digit pattern with exactly two strong links. The only looping turbot in vanilla sudoku is the normal X-Wing, but as we can see, others are possible in variants.)

No matter what it's called, all of the eliminations are caught with a single loop (made possible because of the diagonal link):

Code: Select all
.-----------------.------------------------.-------------------------.
| 1    68    2    | c67*    9      c5687*  | 3       4      d568(7)* |
| 578  9     3456 |  1     b3487*   3568-7 | 25678   258-7   5678    |
| 578  3468  3456 |  2     b3487*   3568-7 | 568-7   1       9       |
:-----------------+------------------------+-------------------------:
| 34   1     7    |  8      35      9      | 45      6       2       |
| 6    2     8    |  347   a35[7]*  1      | 4579    3579    345-7   |
| 34   5     9    |  34-7   6       2      | 478     378     1       |
:-----------------+------------------------+-------------------------:
| 258  7     1    |  369    38      368    | 456     23589   34568   |
| 9    3468  346  |  5      2       3678   | 1       37      3678    |
| 258  368   356  |  3679   1       4      | 256789  35789   356-7   |
'-----------------'------------------------'-------------------------'

Grouped X-Loop (or Grouped Turbot Ring ?):

(7)r5c5 = r23c5 - r1c46 = (7)r1c9 - loop => -7 r23c6,r2c8,r3c7,r6c4,r59c9

The fish POV is a bit more complicated, as you imagined. In fact, it's quite difficult to get all the eliminations with a single fish, Siamese or not.

7r1c5\b5d/ explains most of the eliminations ...

Yes, though it's b2, not b5.

D-Mutant X-Wing (Rank 0):

2x2: (7)R1C5\b2d/ => -7 r23c6,r2c8,r3c7,r6c4

The eliminations in r59c9 require
7r1c5\c9d/

That can't work, unless I'm missing something. As far as I see, you need these two finned fishes (Rank 1) for those eliminations:

2x3: (7)R1C5\b2[c9r5] => -7 r5c9
2x3: (7)R1C5\b2[c9d\] => -7 r9c9

...which can be combined into a single Siamese fish:

Siamese Finned D-Mutant X-Wing (Rank 1):

2x[2x3]: (7)R1C5\b2[c9[r5|d\]] => -7 r59c9

So you may have here a Siamese Grouped X-Wing

Yes, but not just that (and I wouldn't use the terms Siamese and Grouped together, as one is a fish term and the other a chain/loop term, which are two different POVs). As you can see, I need three fishes for all the eliminations, two of which can be combined into a Siamese one. It gets pretty hairy if all three are combined into a doubly siamese fish, because part of it is Rank 0 and other parts Rank 1. Here's my attempt to build a single omnipotent fish anyway:

Doubly Siamese Finned D-Mutant X-Wing (Mixed Rank (0|1|1)):

[2x2]+2x[2x3]: (7)R1C5\b2[d/|c9[r5|d\]] => -7 r23c6,r2c8,r3c7,r6c4,r59c9

Needless to say, I'd opt for the loop POV here.
Last edited by SpAce on Mon Jan 06, 2020 2:09 am, edited 1 time in total.
-SpAce-: Show
Code: Select all
   *             |    |               |    |    *
        *        |=()=|    /  _  \    |=()=|               *
            *    |    |   |-=( )=-|   |    |      *
     *                     \  ¯  /                   *   

"If one is to understand the great mystery, one must study all its aspects, not just the dogmatic narrow view of the Jedi."
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Re: 2 fish in 1 ?Siamese

Postby tarek » Mon Jan 06, 2020 2:06 am

SpAce wrote:
7r1c5\b5d/ explains most of the eliminations ...

Yes, though it's b2, not b5.

Thanks...
The loop can be made possible either by linking through d/ or b2 as r1 and c5 have grouped strong links that can weakly link through b2 leaving r1c9 and r5c5 seeing all the rest of the eliminations outside b2 … It actually looks like a 2-string kite!

I must have started to see things diagonally as indeed the r59c9 eliminations can't be explained by the 2 other fishes I posted

I'll keep using the X-loop from now on … mentioning L1 ring when it is 3 strong links only.

I'll keep posting these challenging examples when I find them

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Re: 2 fish in 1 ?Siamese

Postby SpAce » Mon Jan 06, 2020 2:52 am

tarek wrote:The loop can be made possible either by linking through d/ or b2

I'm not sure what you mean by that. I think you need both d/ AND b2, not either or, for the Rank 0 loop here. The Rank 1 eliminations (r59c9) don't use d/, but all of them link through b2.

as r1 and c5 have grouped strong links that can weakly link through b2 leaving r1c9 and r5c5 seeing all the rest of the eliminations outside b2 … It actually looks like a 2-string kite!

Well, it is a Grouped 2-String Kite, just like you said originally. The r5c9 elimination is a normal Kite elimination (Rank 1). In a vanilla sudoku it would be all we'd get, but here the diagonals give us much more. The r9c9 elimination is already an X-variant specific Kite elimination (Rank 1), but what makes this pattern really interesting is the other diagonal link between the end points closing the loop and giving us Rank 0 Kite eliminations too. Without that we'd just have the two Rank 1 eliminations (still one more than what's possible in vanilla):

Code: Select all
.---------.---------.---------.
| /  /  / | *  *  / | /  /  * |
| .  .  . | .  .  * | .  .  . |
| .  .  . | .  .  * | .  .  . |
:---------+---------+---------:
| .  .  . | .  .  / | .  .  . |
| .  .  . | .  .  / | .  .  . |
| .  .  . | .  .  * | .  .  x |
:---------+---------+---------:
| .  .  . | .  .  / | .  .  . |
| .  .  . | .  .  / | .  .  . |
| .  .  . | .  .  / | .  .  x |
'---------'---------'---------'

(*)r6c6 = r23c6 - r1c45 = (*)r1c9 => -* r69c9

I'll keep using the X-loop from now on … mentioning L1 ring when it is 3 strong links only.

Ok, good. What do you think of the "Turbot Ring" idea? Or perhaps "X-Ring"? I think it's a pity that we have specific term for the three-link version (L1-Ring), but not for the simpler two-linker. Of course we don't need it in vanilla sudoku, but this example demonstrated that it would be useful in general.

I'll keep posting these challenging examples when I find them

Nice! Thanks for this one too. It's very educational to look at how extra constraints can give new powers to familiar patterns. The X-variant is particularly interesting in that regard.
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Re: 2 fish in 1 ?Siamese

Postby tarek » Mon Jan 06, 2020 2:02 pm

SpAce wrote: both d/ AND b2
That is what I meant! there will be more of these variant creatures appearing every now and then with the Strong links algorithm support for variants in Sukaku explainer in the next release
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