The New Sudoku Players' Forum

[Yogi]

9..8..4.2.18....6.6...2..9....7.8..6..7...5..3..6.2....8..3...9.3....61.7.6..9..3
=> 9..8614.2218....6.6...2..9....7.8..6867...52.3..6.2....8..36..9.392..61.7.6..9..3
1r5c9 => 3r4c8
║
4r5c9 => 3r4c8, so to:
9.38614.2218...36.6...2..9....7.8.36867...52.3..6.2....8..36..9.392..61.7.6..9..3
Now what?

[Cenoman]

After your first move +3r4c8:

Code: Select all

 +----------------------+---------------------+----------------------+
 |  9     57     3      |  8      6     1     |  4      57    2      |
 |  2     1      8      |  459    479   457   |  3      6     57     |
 |  6     457    45     |  35     2     357   |  18     9     18     |
 +----------------------+---------------------+----------------------+
 |  145   2459   1245   |  7      145   8     |  19     3     6      |
 |  8     6      7      |  1349   149   34    |  5      2     14     |
 |  3     459    145    |  6      145   2     |  1789   478   1478   |
 +----------------------+---------------------+----------------------+
 |  145   8      1245   |  45     3     6     |  27     457   9      |
 |  45    3      9      |  2      478   457   |  6      1     458    |
 |  7     245    6      |  145    148   9     |  28     458   3      |
 +----------------------+---------------------+----------------------+

2. L3-Wing: (7)r2c5 = (7-8)r8c5 = (8-5)r8c9 = (5)r2c9 => -7 r2c9; 19 placements

or equivalently: 2. M3-Wing (5=7)r2c9 - r2c5 = (7-8)r8c5 = (8)r8c9 => -5 r8c9; 19 placements

Then, finish with a UR (type 1) or an ALS XZ rule:

Code: Select all

 +-----------------+-------------------+-----------------+
 |  9    5    3    |  8     6     1    |  4    7    2    |
 |  2    1    8    |  49    479   47   |  3    6    5    |
 |  6    7    4    |  3     2     5    |  18   9    18   |
 +-----------------+-------------------+-----------------+
 |  4    2    15   |  7     15    8    |  9    3    6    |
 |  8    6    7    |  149   149   3    |  5    2    14   |
 |  3    9    15   |  6     145   2    |  18   48   7    |
 +-----------------+-------------------+-----------------+
 |  1    8    2    |  45    3     6    |  7    45   9    |
 |  5    3    9    |  2     478   47   |  6    1    48   |
 |  7    4    6    |  15    18    9    |  2    58   3    |
 +-----------------+-------------------+-----------------+

3. UR(15)r46c35 using single internal => +4 r6c5; ste

or w/o uniqueness: 3. (8=4)r6c8 - (4=158)r469c5 => -8 r9c8; ste

[Yogi]

Thank you. Very interesting, although admittedly out of my depth. I think you have finally pushed me into asking about all these Wing things.
I do understand XY Wings (the simplest form of XY Chains) and of course the so-called XWings are in fact fish. If there is a UFG for Fish, what is the learning source for these things called L3 Wings, M Wings, H3 Wings, etc. Have they been gathered together in a body of work?
I can follow the logic of the chains you described to -7 r2c9 or -5 r8c9 but what makes these chains a pattern which is worthy of being given a name?

[Leren]

You'll find M Wings described here.

L3 and other related wings were described in a post by daj95376 in about 2013, but I can't seem to find it using the Advanced search feature. Maybe you or someone else may have more luck.

Leren

[yzfwsf]

Hi Leren,are you talking about this one?
Local - Wing: http://forum.enjoysudoku.com/local-wing-t34685.html

[Leren]

No but I eventually found it here at the bottom of the page.

Leren

[Cenoman]

Hi Yogi, Leren, yzfwsf,

Your links are "old stuff".
Since 2020, I use SpAce's work on wings' naming, which brought a neat improvement by linking the names and the link arrangements.
I guess it was exposed first in that post

For Yogi and maybe other players, I recall its content hereafter (with minor changes in Italic):

What's common to all one-letter wings is exactly three strong links in the unpacked chain, and all such three-link chains have a corresponding wing name. So, that's how you know you have some kind of wing, even if you don't know or care about the exact type.

The wing letter, i.e. the top level type, is determined by the configuration of bivalue (V) and bilocal (L) strong links. There are six distinct configurations: {VVV, VVL, VLV, LVL, VLL, LLL} corresponding with six different letters {Y, H, W, S, M, L}. (Y-Wing is of course the same as XY-Wing, but in this context, just one-letter is preferred.)

M-Wing and L-Wing have also more exact subtypes, because both have adjacent L-links allowing varying numbers of digits with distinct patterns. The subtype indicator is simply the number of digits used. The other wings don't have or need any subtypes.

Thus we have a total of nine distinct one-letter wings:

Code: Select all

VVV (3 digits) : Y-Wing  : (a=b) - (b=c) - (c=a) => -a (common peers)
VVL (3 digits) : H-Wing  : (a=b) - (b=c) - c = c => -a (last cell)
VLV (2 digits) : W-Wing  : (a=b) - b = b - (b=a) => -a (common peers)
LVL (2 digits) : S-Wing  : a = a - (a=b) - b = b => -a (last), -b (first)
VLL (2 digits) : M2-Wing : (a=b) - b = (b-a) = a => -a (common peers)
VLL (3 digits) : M3-Wing : (a=b) - b = (b-c) = c => -a (last cell)
LLL (1 digit)  : L1-Wing : a = a - a = a - a = a => -a (common peers)
LLL (2 digits) : L2-Wing : a = (a-b) = b - b = b => -a (last), -b (first)
LLL (3 digits) : L3-Wing : a = (a-b) = (b-c) = c => -a (last), -c (first)

Grouped logically like that, there is no problem remembering them. Noticing pairwise symmetries also helps. Y-Wing and L-Wing are polar opposites (VVV vs LLL), so are W-Wing and S-Wing (VLV vs LVL), and the same with H-Wing and M-Wing (VVL vs VLL).

Mostly the letters make some sense too, except 'M'. (Both W-Wing and M-Wing were named after persons, but the symmetry of 'W' also fits its pattern. In fact, only W-Wing has symmetry in both links and digits. Since 'M' is 'W' upside down it would make more sense for the 'S-Wing' pattern.)

Only H-Wing and M-Wing have asymmetric link configurations. On the subtype level L2-Wing is also asymmetric. Technically L2-Wing has a possible symmetric variant too: a = (a-b) = (b-a) = a. However, it degenerates into a Turbot Fish: a = a - a = a. So, not much point to worry about it.

Only M2-Wing and S-Wing can loop, and both result in the same pattern (typically called M-Ring, though S-Ring would be just as correct).

Complex variants. If a multi-cell ALS is used for any V-link, the wing name should be prefixed with "ALS". If group nodes or AHS are used in L-links, the name should be prefixed with "Grouped" or "AHS" correspondingly. Subtype numbers should not be used with ALS or AHS extensions because they only make sense with the simple types. The elimination logic may change, and also more loops become possible with the complex variants, so the only common thing with the corresponding simple type might be the link configuration.

[...] Note the changes in the H-Wing and M-Wing families, but it's nevertheless the only logical way to categorize them with the link-based system. [...] It translates to the old names like this:

Code: Select all

  new   |   old
--------+--------
H-Wing  | H3-Wing
M3-Wing | H2-Wing
M2-Wing | M-Wing

(Also, H1-Wing was sometimes used as a synonym for L2-Wing. The old H-Wing family was based on different logic, namely patterns of four cells, so it never played well with the other link-based one-letter wings.)

[Yogi]

Cenoman's last post, quoting and updating SpAce's thoughts, is a great help, providing a language we can understand and communicate with.
Thanks again.

[m_b_metcalf]

This is a nice pattern. Here's another version, minimal and with symmetry of the values of the givens:

Code: Select all

 1 . . 2 . . 3 . 4
 . 6 5 . . . . 2 .
 4 . . . 7 . . 9 .
 . . . 7 . 1 . . 2
 . . 1 . . . 9 . .
 8 . . 9 . 3 . . .
 . 1 . . 3 . . . 6
 . 8 . . . . 5 4 .
 6 . 7 . . 8 . . 9

1..2..3.4.65....2.4...7..9....7.1..2..1...9..8..9.3....1..3...6.8....54.6.7..8..9

Mike

[eleven]

Code: Select all

+----------------------+----------------------+----------------------+
| 1      79     89     | 2      5689   569    | 3      5678   4      |
| 379    6      5      | 1348   1489   49     | 178    2      178    |
| 4      23     238    | 13568  7      56     | 168    9      158    |
+----------------------+----------------------+----------------------+
| 359    3459   3469   | 7      4568   1      | 468    3568   2      |
| 2357   23457  1      | 4568   24568  2456   | 9      35678  3578   |
| 8      2457   246    | 9      2456   3      | 1467   1567   157    |
+----------------------+----------------------+----------------------+
| 259    1      249    | 45     3      24579  | 278    78     6      |
| 239    8      239    | 16     1269   2679   | 5      4      137    |
| 6      2345   7      | 145    1245   8      | 12     13     9      |
+----------------------+----------------------+----------------------+

Central (180° rotational) symmetry (19,28,37,46,55): 5r5c5

Code: Select all

+----------------------+----------------------+----------------------+
| 1      79     89     | 2     *689   *569    | 3     #56-78  4      |
| 379    6      5      | 1348   1489   49     | 178    2      178    |
| 4      23     238    | 13568  7     #56     | 168    9      158    |
+----------------------+----------------------+----------------------+
| 359    3459   3469   | 7      468    1      | 468    3568   2      |
| 237    2347   1      | 468    5      246    | 9      3678   378    |
| 8      2457   246    | 9      246    3      | 1467   1567   157    |
+----------------------+----------------------+----------------------+
| 259    1      249    | 45     3      24579  | 278    78     6      |
| 239    8      239    | 16     1269   2679   | 5      4      137    |
| 6      45-23  7      | 145    124    8      | 12     13     9      |
+----------------------+----------------------+----------------------+

Both 5 or 6 in r3c6 have to go to r1c8 => -78r1c8, (symm) -23r9c2

Code: Select all

+----------------+----------------+----------------+
| 1    7    8    | 2    69   569  | 3    5-6  4    |
| 9    6    5    | 3    18   4    | 17   2    78   |
| 4    3    2    | 18   7    56   | 16   9    58   |
+----------------+----------------+----------------+
| 35   9    36-4 | 7    468  1    | 4-6  568  2    |
| 7   #24   1    |#48   5    26   | 9    68   3    |
| 8    245  6-4  | 9    246  3    | 47-6 1    57   |
+----------------+----------------+----------------+
| 25   1   #49   |#45   3    29   | 8    7    6    |
| 23   8    39   | 6    29   7    | 5    4    1    |
| 6    5-4  7    | 145  14   8    | 2    3    9    |
+----------------+----------------+----------------+

skyscraper 4r5c24,r7c34 => -4r46c3,r9c2, -6r1c8,r46c7; stte

[yzfwsf]

Gurth's symmetry placement: => r5c5<>2468
Axisymmetric Conjugate Pair: r46c3<>4,r46c7<>6
Candidate's mapping in Central: 1<=>9 2<=>8 3<=>7 4<=>6 5<=>5
stte

[StrmCkr]

Dont take this the wrong way: i presently agree with renaming to some degree as I've been discussing this with a contributor to my own solver rebuild as a Gui

as consistency with a more fluid synapse does make transparency clearer and easier to follow.

i am looking for people that want to build a Hodoku esq/inspired upgraded solver { Its uses AIC over nice-loops and slowly incorporates all modern tech + adds the projects he never got to finish {like the named aic/als wings in Java [ its built using maven/eclipse ]

come on board and help code techniques into it our end goals is to release it freeware just dm me and i i'll add you via Github.
its a ground up rebuild of my pascal code converted to java with a lot more readability into it.

Also, H1-Wing was sometimes used as a synonym for L2-Wing. The old H-Wing family was based on different logic, namely patterns of four cells, so it never played well with the other link-based one-letter wings.)

---------------
wasn't just that,
daily Sudoku forum developed the l2-l3 wing wing independently, and had 3 separate equations for it for 2 and 3 digits, that part that's missing is the "Local" these wings where fixed to Bands or stacks at one point in time. these are the ones that really didn't play well with others, as none of the moves here had restrictions to stack or band. L1 wing never really seen anything formally documented/written about them but are surmised to be an x-chain of three nodes.

where the h-wing was here for types 1 - 6: it was based on linkage types of 3 Digits.
i dropped types 4-6 as they are ALS based over simpler strong link types

spaces list is also missing the iW wing {inverted w wing}

all the ring classes are never mentioned. {probably as the number of strong links change to close the loop +1 or +2 in w wing and iW wing case}
W-ring, iW- ring, XY - ring, or the ring classes within L123, and h123 wings

as these under the ordinal definitions have Ring classes

the unnamed types identified by space

Note 2: At least these unnamed configurations are also technically possible:

Code: Select all
VLL: (a=b) - b = b - b = b => -a (last), -b (first)
LLL: a = (a-b) = (b-a) = a => -a (common peers)

which belong to the lost wing types " Strong wing/ strong Ring"
http://forum.enjoysudoku.com/post22386.html?hilit=strong%20wing#p22386
http://forum.enjoysudoku.com/post276416.html#p276416
and one of them belongs to the over defined w-wing where the bivalves are in the middle as an over defined Skyscraper.http://forum.enjoysudoku.com/post261781.html#p261781 [ xsudo has these in it]


space tried to clean up the naming a bit via how the links operate with digits instead of equation based on structures viable for the digits count for the class {see the definitions below}

and tried to clean up names with the range of variability in Empty Rectangles as tower crane, loader crane instead of min /max then to my disagreement stuffed all named single digit stuff under "turbots subheading" when turbots are Nice-loops. not A.I.C where these are based out of.

Code: Select all

H-Wings and L3-Wing use three candidate values:
    L3-Wing:  (X)a = (X   -  Y)b = (Y-Z)c = (Z)d     "a" and "d" in same unit; a<>Z, d<>X

    H1-Wing:  same as L3-Wing
    H2-Wing:  (X=Y)a - (Y)b = (Y-Z)c = (Z)d          "a" and "d" in same unit; a<>Z, d<>X
    H3-Wing:  (X=Y)a - (Y=Z)b - (Z)c = (Z)d          "a" and "d" in same unit; a<>Z, d<>X


Code: Select all

L2-Wing, M-Wing, S-Wing, and W-Wing use two candidate values:

     M-Wing:  (X=Y)a - (Y)b ... = (Y-X)c = (X)d      strong link   at weak inferences
    gM-Wing:  (X=Y)a - (Y)b ... = (Y-X)c = (X)d      no constraint at weak inferences
                                                 =>  elims for (X) in peers common to "a","d"
     M-Ring:                                     =>  continuous loop if "a","d" in same unit

     W-Wing:  (X=Y)a - (Y)b     = (Y)c - (Y=X)d      strong link   at weak inferences
    eW-Wing:  (X=Y)a - (Y)b ... = (Y)c - (Y=X)d      no constraint at weak inferences
                                                 =>  elims for (X) in peers common to "a","d"

    iW-Wing:  (X)s = (X-Y)a = (Y)b - (Y)c = (Y-X)d = (X)t   Inverted W-Wing (courtesy of Norm)

     S-Wing:  (X)a = (X)b - (X=Y)c - (Y)d = (Y)e     "a" and "e" in same unit; a<>Y, e<>X
                                                     but not in the same cell -- else M-Ring

    L2-Wing:  (X)a = (X)b - (X)c = (X-Y)d = (Y)e     "a" and "e" in same unit; a<>Y, e<>X


Have they been gathered together in a body of work?

yes
quick link to the wing types

S-Wing can loop, and both result in the same pattern (typically called M-Ring, though S-Ring would be just as correct).

S - wings
do not connect at the end points else its a M - RING : I've made that distinction in my posts on them.

hence the name "Split" wing.
where i chose a S name that characterized how to find them where S is for strmckr/ { 2 strong links splitting away from the bivalve}

I do get that they can technically occupy the same sector and share an overlap at the end points ie the end points lock to the overlap digits for a M ring type c elimination

besides when you go from Wing to Ring your adding 1-2 strong link which changes the (LLL) types to having extra V'Ls to form the loop.
which makes the S wing never be identified as a Ring as its can only have 3 strong-links and 2 weak-inferences

[Hajime]

I found some more ?-Wings where ?=M,L,S,etc
In total there are 13 types of =-=-=

Code: Select all

(=)-(=)-(=) Y-Wing 2,3,4 digits
(=)-(=)-=   H-Wing 2,3 digits
(=)-=(-)=   M-Wing 2,3 digits
(=)-=-(=)   W-Wing 2,3 digits
(=)-=-=     ?-WIng 2 digits    (M-Wing?)
=(-)=(-)=   L-Wing 2,3 digits
=(-)=-(=)   ?-Wing 2,3 digits  (M-wing?)
=(-)=-=     L-Wing 2 digits    
=-(=)-(=)   ?-Wing 2,3 digits  (M-Wing?)
=-(=)-=     S-Wing 2 digits
=-=(-)=     ?-Wing 2 digits    (L-Wing?)
=-=-(=)     ?-Wing 2 digits    (S-Wing?)
=-=-=       L1-Wing 1 digit  or   X-chain of length 5 links (3 strong and 2 weak)

Deliberately the a,b,c digits are not visible because you can have 1,2,3,4 per same type of chain,
like (=)-(=)-(=) filled with a,b,c,d:

Code: Select all

VVV 3 digits : (a=b)-(b=c)-(c=b) => -a (last), -b (first) : Y3-Wing? : AIC Type 2
VVV 3 digits : (a=b)-(b=c)-(c=a) => -a (common peers)     : Y3-Wing  : AIC Type 1
VVV 2 digits : (a=b)-(b=a)-(a=b) => -a (last), -b (first) : Y2-Wing  : AIC Type 2
VVV 3 digits : (a=b)-(b=c)-(c=d) => -a (last), -d (first) : Y4-Wing? : AIC Type 2

AIC Type 1 and 2 see Hodoku

In general:

• if the first and last char are same, say "a",
  then common pears cannot hold "a" ; this is AIC Type 1.
• if the first and last char differs AND are both in a same row,col or box
  then b cannot be in first cell and a cannot be in last cell ; this is AIC Type 2.

spAce canceled =(-)=(-)=   
a=(a-b)=(b-a)=a for being a Turbot , however
a=(a-b)=(b-c)=c is not a Turbot but a Type 2 AIC

To Do:

• Please confirm above logic
• Determine proper names for above ?-Wings
• What to do with all those digits in the name

[StrmCkr]

the following exemplar have no names as it has a fish embedded in it and shouldn't come up in a hierarchy based solver, but they'd fall under hybrid wings as they use both cells and locations. {and only 2 digits} where the last reaming Hybrid wing uses 3 digits, we could slip it in there as a H -2 wing.

Code: Select all

 2 string kite with a bivalve in its elim spot.
+--------------+-----------------+---------+
| .  (+2-1)  . | .    -2(1)  .   | .  .  . |
| .  .       . | .    /      .   | .  .  . |
| .  .       . | .    /      .   | .  .  . |
+--------------+-----------------+---------+
| .  .       . | .    (1)    .   | .  .  . |
| /  (1)     / | (1)  /      (1) | /  /  / |
| .  .       . | .    (1)    .   | .  .  . |
+--------------+-----------------+---------+
| .  .       . | .    /      .   | .  .  . |
| .  .       . | .    /      .   | .  .  . |
| .  .       . | .    /      .   | .  .  . |
+--------------+-----------------+---------+

(1=2) r1c2 - (1) r5c2 = r5c46 -(1)r46c5 = r1c5 => r1c2 <>1 , r1c5 <> 2

Code: Select all

 empty rectangle with a bivalve in its elim spot.
+---------------+--------------+---------+
| /    (1)  /   | /  -2(1)   / | /  /  / |
| .    .    .   | .  .       . | .  .  . |
| .    .    .   | .  .       . | .  .  . |
+---------------+--------------+---------+
| /    (1)  /   | .  .       . | .  .  . |
| (1)  (1)  (1) | .  (+2-1)  . | .  .  . |
| /    (1)  /   | .  .       . | .  .  . |
+---------------+--------------+---------+
| .    .    .   | .  .       . | .  .  . |
| .    .    .   | .  .       . | .  .  . |
| .    .    .   | .  .       . | .  .  . |
+---------------+--------------+---------+


(2=1)r5c5 - (1) r5c123 = r46c2 -(1) r1c2 = r1c5 => r1c5 <>2,r5c5 <> 1

i have this one classed under W-wings as it shows up with this tag in xsudo i call it the unconventional W-wing as the bivalves and strong links swap spots.

Code: Select all

 skyscraper with two bivalves over top the single digit strong link
+------------+--------------+---------+
| /  (1)   / | (1)  /     / | /  /  / |
| .  .     . | .    -1    . | .  .  . |
| .  .     . | .    -1    . | .  .  . |
+------------+--------------+---------+
| .  .     . | -1   .     . | .  .  . |
| /  (12)  / | /   (12)   / | /  /  / |
| .  .     . | -1   .     . | .  .  . |
+------------+--------------+---------+
| .  .     . | .    .     . | .  .  . |
| .  .     . | .    .     . | .  .  . |
| .  .     . | .    .     . | .  .  . |
+------------+--------------+---------+


(1)r1c4 = r1c2 -(1=2)r5c2 - (2=1) r5c3 => r23c5,r46c4 <> 1


as for this list of questions:

VVV 3 digits : (a=b)-(b=c)-(c=b) => -a (last), -b (first) : Y3-Wing? : AIC Type 2
- this would be a naked pair elimination but is also valid as an xy - wing

VVV 3 digits : (a=b)-(b=c)-(c=a) => -a (common peers) : Y3-Wing : AIC Type 1
- valid XY - Wing

VVV 2 digits : (a=b)-(b=a)-(a=b) => -a (last), -b (first) : Y2-Wing : AIC Type 2
- not possible on any grid.

VVV 3 digits : (a=b)-(b=c)-(c=d) => -a (last), -d (first) : Y4-Wing? : AIC Type 2
- no elms possible since first and last are bivalves, and cannot have these values.

for generating Names in a singular output it helps to have both forward and reverse direction listing of links
that way you can categorize them in "smallest" listing first to catch the forward/back ward ordering of the same chains.
for the 5 links used for Cell or Location { Value Location } which every u wish to use.

then when you get to rings class you add on extra links for it to id properly.

What to do with all those digits in the name

digits represent the number of values needed for the wing to operate: and isn't listed if the class only has 1 type.
xy-wing only work with 3 digits,
W- wings only use 2
M - wings use 2,3 digits
Split wings only use 2

Hybrid wings use 3 { or now 2 thanks to the add on class here in}
Local wings use 1,2,3 digits in linkage.

Strong Rings/wings use 2-3 { the class your missing }

[Hajime]

Thanks StrmCkr, I will dive into it.