The Sudoku-X FISH appendix

Advanced methods and approaches for solving Sudoku puzzles

The Sudoku-X FISH appendix

Postby Ruud » Mon Nov 06, 2006 8:40 pm

The recent developments in fish variants is quite impressive. However, there is not much information about variants fish, despite the fact that several Sudoku variants could benefit from the recent discoveries.

In this thread, which you may consider an appendix to the Ultimate FISH guide, I would like to present some fish varieties which can be used in solving one of the popular Sudoku variants: Sudoku-X.

Sudoku-X has the same RCB constraints as vanilla Sudoku, with the addition of the 2 diagonals, usually referred to as the "main diagonal" and the "anti-diagonal".

Needless to say that all fish varieties for vanilla Sudoku can also be used in Sudoku-X. In the following examples, there are several types which form a combination with a vanilla fish type.

1-Fish - 1 diagonal * 1 box (Locked Candidates X)
Code: Select all
 *  .  .  | .  .  .  | .  .  .        /  .  .  | .  .  .  | .  .  . 
 .  *  .  | .  .  .  | .  .  .        .  /  .  | .  .  .  | .  .  . 
 .  .  *  | .  .  .  | .  .  .        .  .  /  | .  .  .  | .  .  . 
----------+----------+---------      ----------+----------+---------
 .  .  .  | X  /  /  | .  .  .        .  .  .  | X  *  *  | .  .  . 
 .  .  .  | /  X  /  | .  .  .        .  .  .  | *  X  *  | .  .  . 
 .  .  .  | /  /  X  | .  .  .        .  .  .  | *  *  X  | .  .  . 
----------+----------+---------      ----------+----------+---------
 .  .  .  | .  .  .  | *  .  .        .  .  .  | .  .  .  | /  .  . 
 .  .  .  | .  .  .  | .  *  .        .  .  .  | .  .  .  | .  /  . 
 .  .  .  | .  .  .  | .  .  *        .  .  .  | .  .  .  | .  .  /
Fig. 1A: b\d                         Fig 1A: d\b


2-Fish - 1 row + 1 diagonal * 2 columns (Skewed X-Wing)

Code: Select all
 /  *  .  | .  *  .  | .  .  .        *  /  .  | .  /  .  | .  .  . 
 .  X  .  | .  *  .  | .  .  .        .  X  .  | .  /  .  | .  .  . 
 .  *  /  | .  *  .  | .  .  .        .  /  *  | .  /  .  | .  .  . 
----------+----------+---------      ----------+----------+---------
 .  *  .  | /  *  .  | .  .  .        .  /  .  | *  /  .  | .  .  . 
 .  *  .  | .  X  .  | .  .  .        .  /  .  | .  X  .  | .  .  . 
 .  *  .  | .  *  /  | .  .  .        .  /  .  | .  /  *  | .  .  . 
----------+----------+---------      ----------+----------+---------
 .  *  .  | .  *  .  | /  .  .        .  /  .  | .  /  .  | *  .  . 
 .  *  .  | .  *  .  | .  /  .        .  /  .  | .  /  .  | .  *  . 
 /  X  /  | /  X  /  | /  /  /        *  X  *  | *  X  *  | *  *  *
Fig. 2A: rd\cc                       Fig 2A: cc\rd


Code: Select all
 /  .  .  | *  *  .  | .  .  .        *  .  .  | /  /  .  | .  .  . 
 .  /  .  | *  *  .  | .  .  .        .  *  .  | /  /  .  | .  .  . 
 .  .  /  | *  *  .  | .  .  .        .  .  *  | /  /  .  | .  .  . 
----------+----------+---------      ----------+----------+---------
 .  .  .  | X  *  *  | .  .  .        .  .  .  | X  /  *  | .  .  . 
 .  .  .  | *  X  *  | .  .  .        .  .  .  | /  X  *  | .  .  . 
 .  .  .  | *  *  /  | .  .  .        .  .  .  | /  /  *  | .  .  . 
----------+----------+---------      ----------+----------+---------
 .  .  .  | *  *  *  | /  .  .        .  .  .  | /  /  *  | *  .  . 
 .  .  .  | *  *  *  | .  /  .        .  .  .  | /  /  *  | .  *  . 
 /  /  /  | X  X  /  | /  /  /        *  *  *  | X  X  *  | *  *  *
Fig. 2B: rd\cc                       Fig 2B: cc\rd

Figure 2B is a combination of 2A + Vanilla type 2B (Locked Candidates)

2-Fish - 2 diagonals * 2 columns/rows (Hourglass X-Wing)

Code: Select all
 /  *  .  | .  .  .  | *  .  /        *  /  .  | .  .  .  | /  .  * 
 .  X  .  | .  .  .  | *  /  .        .  X  .  | .  .  .  | /  *  . 
 .  *  /  | .  .  .  | X  .  .        .  /  *  | .  .  .  | X  .  . 
----------+----------+---------      ----------+----------+---------
 .  *  .  | /  .  /  | *  .  .        .  /  .  | *  .  *  | /  .  . 
 .  *  .  | .  /  .  | *  .  .        .  /  .  | .  *  .  | /  .  . 
 .  *  .  | /  .  /  | *  .  .        .  /  .  | *  .  *  | /  .  . 
----------+----------+---------      ----------+----------+---------
 .  *  /  | .  .  .  | X  .  .        .  /  *  | .  .  .  | X  .  . 
 .  X  .  | .  .  .  | *  /  .        .  X  .  | .  .  .  | /  *  . 
 /  *  .  | .  .  .  | *  .  /        *  /  .  | .  .  .  | /  .  *
Fig. 2C: dd\cc                       Fig 2C: cc\dd

Variant (to be reclassified)
Code: Select all
 /  .  .  | .  .  .  | .  .  /
 .  X  .  | .  .  .  | .  /  .
 .  .  /  | .  .  .  | X  .  .
----------+----------+---------
 .  .  .  | /  .  /  | .  .  .
 .  *  .  | .  X  .  | *  .  .
 .  .  .  | /  .  /  | .  .  .
----------+----------+---------
 .  .  /  | .  .  .  | X  .  .
 .  X  .  | .  .  .  | .  /  .
 /  .  .  | .  .  .  | .  .  /
Fig. 2C?: dd\cc+center


2-Fish - 2 diagonals * 2 rows * 2 columns (Centered Square)
(invalid - removed)
Last edited by Ruud on Mon Nov 06, 2006 6:01 pm, edited 1 time in total.
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Postby tarek » Mon Nov 06, 2006 9:01 pm

Impressive Ruud.......

The N Sector * N Sector fish shoul;d hold for any variant based on Latin squares........ So Sudoku-X, Toroidals, DJs & Latin squares all benifit from this.....

It is easy to name them because basic, finned & sashimi are shared by all variants I think...

The word Box can be relaxed to Cover Sectors (so you could have part of a Franken fish in the diagonal) or just put everything in the Mutant fish part as that clearly says already N Sector * N Sector

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Postby ronk » Mon Nov 06, 2006 9:37 pm

Ruud, for those of us who haven't before taken interest in sudoku variants ... you might include the rule difference in your opening post.

I'm guessing each digit occurs exactly once in the diagonal (upper left to lower right) of a solution for a Sudoku-X puzzle. If so, is that simultaneously true for the anti-diagonal (lower left to upper right) also?
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Postby Ruud » Mon Nov 06, 2006 9:53 pm

Thanks tarek,

but I'm still learning...

Figure 2D is impossible. A sort of deadly square for a Sudoku-X. Both diagonals would eliminate all candidates from the other in a simple Crossover move. I will remove it from my original post.

Figure 2C has an interesting variation:
Code: Select all
 /  .  .  | .  .  .  | .  .  /
 .  X  .  | .  .  .  | .  /  .
 .  .  /  | .  .  .  | X  .  .
----------+----------+---------
 .  .  .  | /  .  /  | .  .  .
 .  *  .  | .  X  .  | *  .  .
 .  .  .  | /  .  /  | .  .  .
----------+----------+---------
 .  .  /  | .  .  .  | X  .  .
 .  X  .  | .  .  .  | .  /  .
 /  .  .  | .  .  .  | .  .  /

When the center cell also has a candidate, either this center candidate or the hourglass pattern is true. I'm just not sure how to classify this one.

ronk

You are correct. I will add it to the opening post.
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Postby Myth Jellies » Mon Nov 06, 2006 11:03 pm

For your 2C variant, it seems the center X could be a fin for each diagonal. Thus, either the hourglass is true or the fin is true.
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Postby tarek » Tue Jan 09, 2007 1:40 pm

As Myth said about this one........

It is a 2 diagonal+2 Column finned fish with the fin being the centre X.........

the safest name would be Mutant finned (where all unclassified finned fish regardless of variant should be)........

however there is no mixing of row & columns in the base or cover (actually there is no mixing of any sectors) ....... so franken finned -to me- seems to be best for this specific example .........

with latin squares & 3-sector-constraint variants of latin square(Toridals, vanillas), the classification still holds......

the tricky bit comes in when u have extra sector constraints like in Sudoku X or DJ....maybe in these cases a low threshold for mixing different sectors should exist (generating more mutant fish)

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Postby Pyrrhon » Sat Jan 12, 2008 12:26 pm

A further overview about Fishs in Sudoku-X is the Skewed Fish article and in the articles linked there.

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Postby Pyrrhon » Sat Jan 12, 2008 6:46 pm

I would call Ruuds last example a Mutant Skewed X-Wing. The defining set consist of both diagonals and the secondary set contains the Weak Link Squares {R2C2, R5C2, R8C2,R5C5} and {R3C7, R5C7, R7C7, R5C5}. So we have no fins.

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Postby tarek » Sun Jan 13, 2008 5:20 pm

basic fish:
1. where cover or base sectors cannot intersect
2. it seemed that everybody was going with having also these sectors as Rows & columns only (so rr/bb or bbb/ccc are not basic fish...they were added to the franken fish to the types of rb/cc & others)
This means that regardless of variant, anything not r1r2..rn/c1c2..cn or reverse is not a basic fish)


Sudoku x diagonals will intersect with any other sector, so any fish that involves a diagonal even if point 2 is relaxed cannot be a basic fish.

if we have data supporting the fish oputside the constraints then it is a kraken fish....

There is no need for the term skewed unless it descibes a special case where intersection of base sectors or cover sectors happens only between a diagonal & some other sector (even the other diagonal ), giving it a different name other than franken or mutant.

so we have a:
basic fish
a fanken fish
a skewed fish
a mutant fish

---just a suggestion.

if the time would roll back, I would suggest that fish would be named according to the number of different constraints in cover os base sectors:
1. basic if 1 (ex rr/bb rrr/ccc diagonal/box)
2. franken if 2 (RowRowDiagonal/BoxBoxBox)
3. mutant if 3 (DiagonalDiagonalRow/BoxBoxBox)
4. .........

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Postby Pyrrhon » Sun Jan 13, 2008 6:43 pm

Okay Tarek, you are right, its a Mutant X-Wing.

If we look at the fish guide we see with respect to the shape (fins, kraken and sashimi excluded here):

Basic: N rows * N columns or vv.
Franken: N (rows+boxes) * N (columns+boxes) or vv.
Mutant: N Sectors * N Sectors

We have now two different understandings of skewed fishs.

1) Ruud in this thread (if I'm right):

Skewed: N (rows+diagonals) * N (columns+diagonals) or vv.

2) Unkx80 in the sudopedia:

Skewed: N (rows+diagonals) * N columns
or N (columns+diagonals) * N rows

I think it's only a question of agreement which definition to follow. But in Ruuds example we have:

2 diagonals * 2 weak link squares

and so both definitions are not fullfilled and it is a Mutant X-Wing.

From another perspective we have defining constraints in the diagonals and secondary constraints C2 and C7. The fin organizing constraint would be R5. (That is the constraint that is in a Finned Basic X-Wing the box.) If we consider the example this way we would get a Finned Skewed X-Wing with Ruuds use as well as with Unkx80 use.

This matches also a generalization of Ruuds pattern:

Code: Select all
 
/  .  .  | .  .  .  | .  .  /
 .  X  .  | .  .  .  | .  /  .
 .  .  /  | .  .  .  | X  .  .
----------+----------+---------
 .  *  .  | X  .  X  | *  .  .
 .  .  .  | .  /  .  | .  .  .
 .  .  .  | /  .  /  | .  .  .
----------+----------+---------
 .  .  /  | .  .  .  | X  .  .
 .  X  .  | .  .  .  | .  /  .
 /  .  .  | .  .  .  | .  .  /
 


Here we would have also a Finned Skewed X-Wing with the Fin constraint in R4 and fins in R4C46.
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Postby ronk » Sun Jan 13, 2008 11:28 pm

Ruud wrote:Figure 2C has an interesting variation:
Code: Select all
 /  .  .  | .  .  .  | .  .  /
 .  X  .  | .  .  .  | .  /  .
 .  .  /  | .  .  .  | X  .  .
----------+----------+---------
 .  .  .  | /  .  /  | .  .  .
 .  *  .  | .  X  .  | *  .  .
 .  .  .  | /  .  /  | .  .  .
----------+----------+---------
 .  .  /  | .  .  .  | X  .  .
 .  X  .  | .  .  .  | .  /  .
 /  .  .  | .  .  .  | .  .  /

When the center cell also has a candidate, either this center candidate or the hourglass pattern is true. I'm just not sure how to classify this one.

I see no need for a separate classification or even a separate illustration. As Myth Jellies, tarek, and Pyrrhon have said ... it is a finned fish. In the style of exemplars on The Ultimate Fish Guide (TUFG), it would look like ...
Code: Select all
 /  *  . |  .  .  . |  *  .  /
 .  X  . |  .  .  . |  *  /  .
 .  *  / |  .  .  . |  X  .  .
---------+----------+----------
 .  *  . |  /  .  / |  *  .  .
 . **  . |  .  @  . | **  .  .
 .  *  . |  /  .  / |  *  .  .
---------+----------+----------
 .  *  / |  .  .  . |  X  .  .
 .  X  . |  .  .  . |  *  /  .
 /  *  . |  .  .  . |  *  .  /

Any one of the 13 empty cells could be a lone fin cell and there would still be valid eliminations. Besides these 13 single-finned fish, there are others with two and three fin cells ... still with valid elimination possibilities. It's just not practical to illustrate them all.

This unfinned fish is stable (viable), i.e., it does not degenerate to a smaller pattern. Illustrating a stable unfinned fish should be adequate, leaving the reader to mentally construct the finned possibilities.

The "unusual" thing about this finned fish is that the fin cell is common to two base sectors -- termed an endo-fin on the TUFG.
Last edited by ronk on Sun Jan 13, 2008 7:32 pm, edited 2 times in total.
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Postby tarek » Sun Jan 13, 2008 11:31 pm

Pyrrhon wrote:Okay Tarek, you are right, its a Mutant X-Wing.

Or is it 2 simultaneous skewed x-wing:?::D

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Postby Pyrrhon » Mon Jan 14, 2008 8:06 am

No it isn't two Skewed X-Wings. Or what would you call the defining and secondary constraints?

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Postby tarek » Mon Jan 14, 2008 12:19 pm

if the cover sectors are (r2r8c2c8)(r3r5r7c7+r5c5)....then you have a fish & 1 elimination & do the opposite on the other side another 1 elimination then you would achive the same with 2 fish without having 2 cover sets intersecting at r5c5

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Postby Pyrrhon » Mon Jan 14, 2008 1:14 pm

If I'm right then your cover sectors are neither lines nor diagonals but weak link squares. So what you have are by definition two Mutant 1-fishs in this grid. Isn't it?
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