## Als- W- Wings & Rings

Advanced methods and approaches for solving Sudoku puzzles

### Als- W- Wings & Rings

This is an advancement of the classic w wing move set utilizing the bivalve cell as a group of n cells consisting of an almost locked set of n+1 digits

Theorem:

Als set A of n cells with n+1 digits
Als set B of n cells with n+1 digits
Where A & B share 2 or more candidates.

Part C: Candidate X or group of X candidates visible to all X digits of set A
Part D: Candidate X or group of X candidates visible to all X digits of set B
Where C and D form a strong link

Placement of X as C or D restricts A or B by -1 candidate reducing A or B to a locked set {n cells with n digits} there by
Eliminating common z digits of sets A & B in all peer cells visible to all cells holding z in both A and B

Ring eliminations:
Type 1: A and B have a 2ndary C & D strong link using a different X2 candidate
Because A or B is restricted by (x1 or x2) it forms a loop and restrict the placements of A and b both by -1 candidate ( A & B are n cells with n digits}.

Therefore
All peers of als A for x and peers of CD x cells may be eliminated
All peers of als B for x1 and peers of CD x cells are eliminated
All digits - x in als B form a locked set eliminate peers of these candidates
All digits - x in als B form a locked set eliminate peers of these candidates
All common shared digits of A&B eliminate peers cells of these candidates.

Type 2:
Als A and als B house a restricted common that is not equal to x.
the CD link for x restricts both A and B by 1 digit and further restricting A and B by the restricted common which reduces both als A and als B as a locked set.

Therefor eliminate
All digits - x in als A form a locked set eliminate peers of these candidates
All digits - x in als B form a locked set eliminate peers of these candidates
All common shared digits of a&b eliminate peers cells of these candidates.

Als w wing
Code: Select all
`     +--------------------+------------------+---------------------+    | 1       23   236   | 4      3567  367 | 5678   568(3)  9    |    | (3467)  5    (346) | (367)  8     9   | (167)  2       1(3) |    | 367     9    8     | 1      3567  2   | 567    56(3)   4    |    +--------------------+------------------+---------------------+    | 2346    234  1     | 367    367   5   | 468    9       38   |    | 5       8    346   | 9      2     36  | 46-1   (136)   7    |    | 36      7    9     | 8      4     1   | 2      (36)    5    |    +--------------------+------------------+---------------------+    | 39      6    7     | 5      1     8   | 39     4       2    |    | 8       1    35    | 2      9     4   | 35     7       6    |    | 249     24   245   | 367    367   367 | 1589   158     18   |    +--------------------+------------------+---------------------+`

ALS - W - RING type 2
Code: Select all
`     +-------------------+----------------------+-------------------+    | 1     (478)  26-8 | (278)   9     (2468) | 27   5     3      |    | 9     (78)   5    | (278)   1     3      | 4    6     27     |    | 2346  34(7)  236  | 5-2(7)  (26)  45-26  | 9    1     8      |    +-------------------+----------------------+-------------------+    | 7     1      2389 | 6       238   289    | 358  4     59     |    | 23    39-8   4    | 12389   5     1289   | 6    3789  79     |    | 5     6      389  | 389     4     7      | 38   2     1      |    +-------------------+----------------------+-------------------+    | 8     349    1    | 2359    236   2569   | 27   379   245679 |    | 36    5      7    | 4       2368  2689   | 1    389   269    |    | 346   2      369  | 13589   7     15689  | 358  389   4569   |    +-------------------+----------------------+-------------------+`

Code: Select all
`         -bce abce  a |  .  .  . |  .  .  .       -bce abce  . |  .  .  . |  .  .  .        -bce abce  a |  .  .  . |  .  .  .       -bce abce  . |  .  .  . |  .  .  .        -bce abce  a |  .  .  . |  .  .  .       -bce abce  . |  .  .  . |  .  .  .        ---------+----------+----------      ---------+----------+----------        abce -bce  a |  .  .  . |  .  .  .       abce -bce  . |  .  .  . |  .  .  .        abce -bce  a |  .  .  . |  .  .  .       abce -bce  . |  .  .  . |  .  .  .        abce -bce  a |  .  .  . |  .  .  .       abce -bce  . |  .  .  . |  .  .  .        ---------+----------+----------      ---------+----------+----------         .  .  / |  .  .  . |  .  .  .        a  a  / |  .  .  . |  .  .  .         .  .  / |  .  .  . |  .  .  .        a  a  / |  .  .  . |  .  .  .         .  .  / |  .  .  . |  .  .  .        a  a  / |  .  .  . |  .  .  .`

Als w wing
Code: Select all
`     +----------------+------------------------+---------+    | .  (1234)  .   | .       .       .      | .  .  . |    | .  (1234)  (1) | .       .       .      | .  .  . |    | .  (1234)  .   | .       .       .      | .  .  . |    +----------------+------------------------+---------+    | .  .       .   | .       .       .      | .  .  . |    | .  -234    (1) | (1234)  (1234)  (1234) | .  .  . |    | .  .       .   | .       .       .      | .  .  . |    +----------------+------------------------+---------+    | .  .       .   | .       .       .      | .  .  . |    | .  .       .   | .       .       .      | .  .  . |    | .  .       .   | .       .       .      | .  .  . |    +----------------+------------------------+---------+`

Code: Select all
`+------------------+-----------------+-----------------+| 7     1      2   | 8    56     9   | 3     4    56   || 6     8      9   | 4    137    135 | 2     157  157  || 3     4      5   | 2    17     16  | 8     179  1679 |+------------------+-----------------+-----------------+| 2589  2359   38  | 1    4      7   | 6     259  589  || 1258  25     7   | 36   9      36  | 4     125  158  || 1(9)  6      4   | 5    8      2   | 1(9)  37   37   |+------------------+-----------------+-----------------+| 4     57     13  | 67   56     8   | (19)  139  2    || (29)  279-3  6   | 7-3  (123)  4   | 5     8    (13) || 258   235    138 | 9    123    135 | 7     6    4    |+------------------+-----------------+-----------------+`

ALS -W-Wing
Als A) 1239 @R8C159
Als B) 139 @ R7C7,R8C9
(9) R6C1 = R6C7 (9)
=> R8C24 <> 3
stte

NOTES:
the strong link featured in the W-wing thread is used identically for these, the only differences is that i have expanded both of the bivalves from als size 1 to variable size range 1->8
the strong link can also be expanded for an extended variation.
Last edited by StrmCkr on Thu Nov 21, 2019 10:22 am, edited 4 times in total.
Some do, some teach, the rest look it up.

StrmCkr

Posts: 1283
Joined: 05 September 2006

### Re: Als- W- wing/ring

Here my summary:
ALS XZ: ALS+ALS (Rank 1)
ALS W-Wing: ALS+ALS+strong link from digit (Rank 0)
ALS XY: ALS+ALS+ALS (Rank 1)
Death Blossom: cell > multiple ALS (Rank 1)

Rank 0 in those cases: begin AND end of chain are true.
Rank 1 in those cases: begin OR end of chain must be true.
Exclusions will follow the AND and OR rules.

So many names introduced lately (empty rectangle). Generalizing strategies could help improve software solvers.
creint

Posts: 340
Joined: 20 January 2018

### Re: Als- W- wing/ring

creint wrote:Death Blossom: cell > multiple ALS (Rank 1)

Rank N-1, where N is the ALS count -- like a "Kraken (something)" with N branches.

creint wrote:ALS W-Wing: ALS+ALS+strong link from digit (Rank 0)

Rank 1, like ALS-XY.

creint wrote:Rank 0 in those cases: begin AND end of chain are true.

Chain is not a chain, per se, but a loop.
One end is weakly linked to the other, and only one can be true.
The strong link from the chain establishes that it's exactly one that's true.
blue

Posts: 909
Joined: 11 March 2013

### Re: Als- W- wing/ring

Hi creint,

I second blue's remarks.

Just in case, here's a little recap on the Rank stuff. Any chain, kraken, or loop can be represented as a fish, with the strong links as the base sets (truths) and the weak links as the cover sets (links). Rank is simply the excess of covers vs bases, forgetting any triplet complications. It tells how many cover sets are unoccupied by a base candidate, and consequently how many overlapping covers are needed to guarantee an elimination. Any non-base candidate that is covered by at least Rank+1 links can be safely eliminated, because at least one of those links must be occupied by a base candidate. Cannibal eliminations, i.e. base candidates, need additionally one extra cover for each base they're part of.

Loops (such as rings and SDC) are Rank 0, because there are just as many weak links (covers) as strong links (bases). It means that every weak link has a base candidate at one or the other end (not both!), so any other candidates along those weak links can be eliminated. Non-looping chains (such as wings) are Rank 1, so you need two overlapping cover sets (e.g. a row and a column, or a box and a cell, etc.) to get an elimination. In other words, both chain ends need to see the victim. Three-way krakens, such as a Death Blossom with three branches, are Rank 2, so you need three overlapping cover sets.

Rank 3 is the highest possible without triplets because there are only four ways (row, column, box, cell) to cover something in vanilla sudoku. An even higher global rank is possible with triplets (base candidates belonging to multiple base or cover sets), because they lower the local rank in some regions to allow eliminations. In obi-fishes the triplet logic is replaced with duplicated sets, which I find simpler.

Generalizing strategies could help improve software solvers.

Tactics. Or techniques. Not strategies. Thanks! (My pet peeve )

SpAce

Posts: 2671
Joined: 22 May 2017

### Re: Als- W- wing/ring

With Rank 0 I meant there are enough links to reduce every ALS to a Locked set. And the links is also locked inside those sets.

And with Rank 1 I meant there is missing one link from every ALS, so it stays ALS. The link is optional but every other candidate that is shared between (begin and end of chain, or something more complex) you could remove.

I just want a variant solver like Xsudo, where you could have an find all tactics.
Chains/loops/nets with single links are easy to program. But nets with overlapping truths/links is much harder.
Any idea on how to implement it?
creint

Posts: 340
Joined: 20 January 2018

### Re: Als- W- wing/ring

Yes, but its a question of depth and lenght that you want to explore.

Then its a matter of having a depth/bredth chain engine that can cycle through all linkable objects correctly.

Strong/weak/group on a single digits
Bivavle
grouped digit sets (locked)
(almost locked set)
N+1 digits in n cells
N+x digits in n cells
(hidden almost locked set)
N + 1 cells with n digits
N+x cells with n digits.

There really isnt any solver to my knowledge released that uses the full data together. Some cover partial sections.
A find all for this size of data is prbably in the hour range
Like my als xy early code without redundance shortcuts for class size 1-8 x27 sectors searching for size 1-8 x 27 sectors for a and b
On some grids took 45mins alone to run once...

Small coded techniques are sometimes easier to work with to build fundaments for a catch all
To catch all for these would be als chain code and aic chain code combined.

I do have thes als w wing coded into my solver presently, a few more tweaks befor i release an updated build.

Anway this thread is for the aforementioned als-w-wing for documentation purposes and coding it.
xsudoku can list these and finds them manuly as mixed type als
Some do, some teach, the rest look it up.

StrmCkr

Posts: 1283
Joined: 05 September 2006

### Re: Als- W- wing/ring

From space in todays puzzel
Code: Select all
` .-----------------.----------.-------------------.    | 8   467  b(627) | 1  45  9 | b45(7)  b(27)  3  |    | 5   14     3    | 8  7   2 |  14       9    6  |    | 29  147    279  | 6  45  3 |  8        127  25 |    :-----------------+----------+-------------------:    | 4   3      8    | 2  1   6 |  9        5    7  |    | 7   2      5    | 9  8   4 |  6        3    1  |    | 6   9      1    | 5  3   7 |  2        4    8  |    :-----------------+----------+-------------------:    | 29  567    4    | 3  69  1 | a5[7]     8    25 |    | 1   8      7-6  | 4  2   5 |  3      a[67]  9  |    | 3   56     29   | 7  69  8 |  15       126  4  |    '-----------------'----------'-------------------'`

ALS-W-Wing (seems to be fashionable):

(67)b9p51 = (726)r1c783 => -6 r8c3; stte
Some do, some teach, the rest look it up.

StrmCkr

Posts: 1283
Joined: 05 September 2006

### Re: Als- W- Wings & Rings

StrmCkr wrote:Where C and D form a strong link

As I'm in the pre coding phase of W-wing in Sukaku Explainer I had some thoughts.
The Ring would happen when C <----> D are linked using the other candidate as well (and the chain can be of different length to the 1st one)
With the ring you can eliminate those single values of the Weak links that are not part of the strong links (Obviously there could be fin(s) but I'll skip that one for now)

The bivalue cell that we have in the w-wing is an ALS so there is no reason why that can't be extended into a full ALS concept as you outlined

I know I'm just recapping what has already been said … I think of it as an exercise to make sure I got all that has been said

tarek

Posts: 3759
Joined: 05 January 2006

### Re: Als- W- Wings & Rings

Correct, i used w wings to formulate it and test code as its the simplest als.
then expanded the als size and analyzed for eliminations

The c d chain strong could be replaced with a chain of x length as long as it links back up.
Or
an muti-digit chain structures, that could also using other als in sections but that goes beyond what i was trying to do here.
Some do, some teach, the rest look it up.

StrmCkr

Posts: 1283
Joined: 05 September 2006