## WXYZ Wing questions

Post the puzzle or solving technique that's causing you trouble and someone will help

### WXYZ Wing questions

Hi,

This is my first post, so I hope I don't make a total fool of myself, but I think I have a legit novice level question.

I'm just learning the more advanced techniques, while programming my own Sudoku helper.

While programming the WXYZ Wing technique I came across the attached puzzle.

The WXYZ pivot cell is C7, and the outlier cell is B7. As I understand the technique, candidate 1 is the non-restrictive common candidate because the 1 in the outlier cell cannot see all of the 1's in the base cells (i.e. the B7 1 can't see the C1 and C2 1's) and therefore the 1 in C8, which can see all of the 1's in all of the cells of the WXYZ can be eliminated.

Two questions:

(1). All the examples I've seen of this technique have had the outlier cell on a different row/column than the pivot cell, though in the same block. Is it legit to have it in the same row (row 7 in this puzzle)? And, if so, could it be outside the block as long as it was on the same row (so that the pivot cell could still see it)? Or is it required to be in the same block?

(2). It seems to me that candidate 8 can also be eliminated from C8, because if it were true, then there would only be 3 candidates for the 4 cells of the WXYZ. So, is there a broader definition here that I'm missing? Can I take all the stuff about the non-restricted common candidate and what the outlier doesn't see, and replace it with: "Any cell that can see all of the WXYZ cells can have its candidates eliminated that match the WXYZ candidates"? So that, to the cell that sees all of the WXYZ cells, those cells act just like a Naked Quad. While other cells that can see all instances of a particular candidate in the WXYZ cells can have that candidate eliminated (e.g. candidate 9 in C4). Or am I leading myself astray with false assumptions?
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smcroberts

Posts: 6
Joined: 08 January 2020

### Re: WXYZ Wing questions

Hi!

I cannot directly answer your questions, but I'm also interested in what you wrote here. I have some questions myself seeing what you used. Usually I refer to this pattern as a Bent Quadruple. It visualizes the logic easier. I've also heard of the terminology of "Sue De Coq". It doesn't seem like they're exactly the same, but both go about Bent Quadruples and make eliminations. Anyone who can extend on that?

It doesn't seem impossible to me that the pivot cell sees a cell that it can eliminate from (https://imgur.com/a/IUstMjH ; from https://www.sudokuwiki.org/WXYZ_Wing). I think all of that depends on conjugate pairs and bi-/tri-value cells. You're practically looking for a candidate in a different cell that break the quadruple. Nor do I see how it would be impossible to have the outlier cell next to the pivot cell. In a different row sounds harder to me. Because you might not get eliminations - the box doesn't see the pincer/bent cell.

To give another example: (https://imgur.com/a/IMcdXt3 - String: 298.....65.126.9787.6198....29.85.34.5.....8938.9..56.8623794..475812693913...827). What you're looking for is other digits (in the box) that break the quadruple so a cell ends up being dead. Usually, if not always, there's a bivalue cell in the bent quadruple that can either be abused to make a different cell dead, or can be made dead itself.

Unfortunately I'm not able to offer you more as I do not recognize and know this pattern enough to immediately see it and its eliminations. Perhaps this info helps you. Either way I hope we will both be helped and learn from this. I do not see an immediate resolution to the "8" candidate from the Bent Quadruple pattern.

Edit: looking at some examples it seems the main difference between WXYZ-Wings and Sue de Coq is that the former is an actual locked Bent Quadruple, while the latter doesn't have to be. It can be an almost locked - and can still remove candidates. Verification for this would be nice.

Robin
Last edited by FairyTailed on Thu Jan 09, 2020 9:42 am, edited 4 times in total.
FairyTailed

Posts: 8
Joined: 02 January 2020

### Re: WXYZ Wing questions

Hi smcroberts,

Welcome to the forum.

smcroberts wrote:(1). All the examples I've seen of this technique have had the outlier cell on a different row/column than the pivot cell, though in the same block. Is it legit to have it in the same row (row 7 in this puzzle)? And, if so, could it be outside the block as long as it was on the same row (so that the pivot cell could still see it)? Or is it required to be in the same block?

I don't even try to answer this question, because I don't really know (or care) what you mean by outlier cells and base cells, and I find such pattern-specific concepts unnecessary and confusing. You'd be better off thinking in more general terms which allow you to see the logic without relying on memorization. For example, a WXYZ-Wing can be seen as an AIC, or an ALS-(X)Z pattern, or an Almost Distributed Disjoint Subset, or a Rank 1 set logic pattern. If you learn any of those concepts, you'll understand how WXYZ-Wings (and almost all other patterns) work automatically without having ever heard of WXYZ-Wings.

If you instead try to memorize pattern-specific elimination rules, you'll end up making mistakes and/or wasting your time memorizing narrowly applicable specifics when generics would give you much more. (That said, you can of course use a bottom-up approach to learn generic concepts by studying specific examples.)

I'm sorry if that answer didn't help you. There are others here who are probably more willing to discuss such pattern-specific details, if you insist on that (arguably weaker) approach.

(2). It seems to me that candidate 8 can also be eliminated from C8, because if it were true, then there would only be 3 candidates for the 4 cells of the WXYZ.

No. You'd have three digits for the four cells, but one of them (1) can exist twice, so all the cells can still be filled -- as long as you don't remove the 1 (or more than one of the other digits). It means that any one (but only one) of the other digits can be completely removed, including the 8. Only the 1 is mandatory in the pattern, which is why it's the only one that can eliminate something because we know that it must be in at least one of the pattern cells. In other words, any external candidate that sees all the 1s in the pattern can be eliminated (only 1r8c3 in this case). None of the other digits in the pattern have that property.

On the other hand, if all of the digits could only exist once in the pattern, then you couldn't remove any of them. It would be a Distributed Disjoint Subset, i.e. a Rank 0 pattern (like a Sue de Coq), which allows you to eliminate any external candidates that see all instances of any digit in the pattern. As a chain that would be a (continuous) loop. We don't have that here because the 1 can exist twice, making it a Rank 1 pattern.
-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."

SpAce

Posts: 2560
Joined: 22 May 2017

### Re: WXYZ Wing questions

SpAce wrote:I don't even try to answer this question, because I don't really know (or care) what you mean by outlier cells and base cells, and I find such pattern-specific concepts unnecessary and confusing. You'd be better off thinking in more general terms which allow you to see the logic without relying on memorization. For example, a WXYZ-Wing can be seen as an AIC, or an ALS-(X)Z pattern, or an Almost Distributed Disjoint Subset, or a Rank 1 set logic pattern. If you learn any of those concepts, you'll understand how WXYZ-Wings (and almost all other patterns) work automatically without having ever heard of WXYZ-Wings.

If you instead try to memorize pattern-specific elimination rules, you'll end up making mistakes and/or wasting your time memorizing narrowly applicable specifics when generics would give you much more. (That said, you can of course use a bottom-up approach to learn generic concepts by studying specific examples.)

I'm sorry if that answer didn't help you. There are others here who are probably more willing to discuss such pattern-specific details, if you insist on that (arguably weaker) approach.

I agree with you, but the way in which you write this seems mildly unnecessary passive agressive .

With outlier cell he seems to indidicate the part where the quadruple bents - which is connected by the pivot. This is usually called a pincer cell.
FairyTailed

Posts: 8
Joined: 02 January 2020

### Re: WXYZ Wing questions

FairyTailed wrote:I agree with you, but the way in which you write this seems mildly unnecessary passive agressive .

That wasn't my intention at all. I have zero reason to be any kind of aggressive towards new members who have valid questions. If anything, I'm annoyed with the majority of the teaching materials which tend to promote such pattern-specific thinking, because I find it a very inefficient way to learn and to solve. That's why I make it clear what I think about that My annoyance is not directed at the victims of such teachings. I regret if it sounded like that.

SpAce

Posts: 2560
Joined: 22 May 2017

### Re: WXYZ Wing questions

FairyTailed wrote:Hi!Usually I refer to this pattern as a Bent Quadruple. It visualizes the logic easier. I've also heard of the terminology of "Sue De Coq". It doesn't seem like they're exactly the same, but both go about Bent Quadruples and make eliminations. Anyone who can extend on that?

I wouldn't necessarily call WXYZ-Wing a Bent Quadruple, at least without qualifications, even though it looks like one. A normal quad is a Rank 0 pattern while a WXYZ-Wing is a Rank 1 pattern, which makes them fundamentally different (Rank 0 being much more powerful).

On the other hand, a four-digit Sue de Coq is more like a Bent Quad because it's also a Rank 0 pattern. The only difference between that and a normal quad is that a Sue de Coq resides in two sectors (a box and a line) while a normal quad is completely contained in just one. In both cases all digits can eliminate in their corresponding sectors.

Sue de Coq is a Two-Sector Disjoint Subset which is a special case of generic Distributed Disjoint Subsets (all Rank 0 patterns). Thus all Sue de Coqs, and possibly some of the larger DDSs as well, can be seen as Bent [Quads|Quins|Sextuplets...]. [UVW]XY[Z]-Wings look similar but are much weaker, so I don't think they earn that name. Perhaps Almost Bent [Quad|etc]?

Edit: looking at some examples it seems the main difference between WXYZ-Wings and Sue de Coq is that the former is an actual locked Bent Triple, while the latter doesn't have to be. It can be an almost locked - and can still remove candidates. Verification for this would be nice.

I think you've got it exactly backwards. Sue de Coq is fully locked while WXYZ-Wing is not. That's the difference between Rank 0 and Rank 1.

Btw, Multi-Sector Locked Set (MSLS) is a further generalization of DDS where the same digit can be locked into multiple sectors, which makes much larger patterns with tons of eliminations possible. The max size of a DDS is nine cells because there are only nine digits, but MSLS has no such restrictions (20-cell MSLS patterns are normal).

SpAce

Posts: 2560
Joined: 22 May 2017

### Re: WXYZ Wing questions

Since my attitude was questioned, I'll take another stab at the original question and the part I didn't answer

(1). All the examples I've seen of this technique have had the outlier cell on a different row/column than the pivot cell, though in the same block. Is it legit to have it in the same row (row 7 in this puzzle)? And, if so, could it be outside the block as long as it was on the same row (so that the pivot cell could still see it)? Or is it required to be in the same block?

Yes, yes, and no. The pattern can be spread in many different ways, but different arrangements cause different constraints on what can be eliminated. For example, imagine that the "outlier" cell were in r7c4 instead of r7c2. If everything else stayed the same, nothing could be eliminated because there's no external candidate 1 that sees all 1s in the pattern. However, if the 1s in r27c3 were eliminated first, the pattern would work because then 1r1c4 would see both 1s left in the pattern (r1c3 and r7c4) and could be eliminated. (Never mind that it would in that case get eliminated more simply with a pointing triple in r1c123 -- imagine that there are other 1s in box 1.)

SpAce

Posts: 2560
Joined: 22 May 2017

### Re: WXYZ Wing questions

Thanks for the replies, Robin and spAce!

My aha! moment from this was the quote:

No. You'd have three digits for the four cells, but one of them (1) can exist twice, so all the cells can still be filled -- as long as you don't remove the 1 (or more than one of the other digits).

That makes it clear to me why the 8 and the 9 cannot be eliminated.

You'd be better off thinking in more general terms which allow you to see the logic without relying on memorization. For example, a WXYZ-Wing can be seen as an AIC, or an ALS-(X)Z pattern, or an Almost Distributed Disjoint Subset, or a Rank 1 set logic pattern. If you learn any of those concepts, you'll understand how WXYZ-Wings (and almost all other patterns) work automatically without having ever heard of WXYZ-Wings.

You're right: I am currently doing exactly what you caution against: trying to cram my head with all the specific techniques. But I have been wondering if there are broader principles at work that could be learned and applied with less likelihood of going astray. So, I have begun looking into the TDP system of Robert Mauriès...

You indicate that the broader principles reside in some patterns I have not yet come across. So, do you have any recommendations as to where one could go to learn such things as Rank 1 Set Logic and Multi-Sector Locked Sets, etc.? I do know of AIC's and ALS's, but have considered them last-resort techniques due to their complexity. The number of possible paths of an AIC, for instance, seems so overwhelming that learning to spot specific patterns seems more comfortable. But I'm willing to alter my approach and expand my mind; learning this stuff is what makes Sudoku fun for me.
smcroberts

Posts: 6
Joined: 08 January 2020

### Re: WXYZ Wing questions

smcroberts wrote:My aha! moment from this was the quote

You're right: I am currently doing exactly what you caution against: trying to cram my head with all the specific techniques. But I have been wondering if there are broader principles at work that could be learned and applied with less likelihood of going astray.

Good! There's nothing wrong with starting with specific techniques and building up from there. It's the bottom-up approach I mentioned, and it can work just fine. Almost everyone starts like that and it gets you going quickly. However, at some point you should start seeing similarities in those techniques and wondering if there are some underlying principles that make the patterns work -- and that's exactly where you seem to be right now!

Personally I aimed for that point very quickly when I started studying advanced sudoku techniques because I hate memorization. It's not because I have bad memory (I don't) but because I think it's stupid when there's probably a simple overarching logic to be discovered (and there is).

You indicate that the broader principles reside in some patterns I have not yet come across.

Glad you asked. It might come as a surprise that there are only two major paradigms that can explain almost all existing sudoku techniques: chaining and fishing. Chaining can exist as if-then-else type of implication chains or as static boolean logic in the form of AICs, and it includes not only linear chains (simple) but also branching nets (complicated). By fishing I mean not only the common single-digit fishes but the omnipotent multi-digit set logic developed by Allan Barker.

Every chain/net can be represented as a fish, and every fish can be represented as a chain/net, so theoretically they have equal powers and one could choose to learn just one. In practice, chaining is much simpler and it's the bread and butter of advanced solving, at least for most people. However, some heavily branching patterns are very poorly suited for chaining, and they're much easier to express and think in terms of set logic. Even basic fishes larger than X-Wing fall into that category, as something like a full Jellyfish would be very hard to express with chains. Some much bigger and more complicated patterns like MSLS would be almost impossible. In those cases, set logic is invaluable, but it can simplify other situations too. So, both is best, but I'd definitely start with chaining and concentrate on that for a long time before dwelling deeply into set logic.

So, do you have any recommendations as to where one could go to learn such things as Rank 1 Set Logic and Multi-Sector Locked Sets, etc.?

Above I gave the link to Allan Barker's set logic (and his excellent XSudo solver which, unfortunately, I haven't tried personally). Here's also a brief intro to set logic and Rank 0 patterns I wrote some time ago. But, like I said, it's not where I would start. Most definitely I wouldn't start with something like MSLS In fact, it was the last well-known technique that I learned myself, and a pretty good understanding of set logic was a prerequisite for that (imho). (J)Exocets are even more complicated but they have better documentation and don't require set logic (though originally found through that). Both of those techniques are needed only in very difficult puzzles, so learning them can be safely delayed (possibly indefinitely, depending on one's ambitions).

I do know of AIC's and ALS's, but have considered them last-resort techniques due to their complexity.

Well, someone's basic techniques are another's last resort techniques. For me, and most other solvers here, AICs and ALSs are the bread and butter of almost any non-basic solving. They're not complex at all once you understand them and know how to look for them. Of course it might take a while to get to that point. For me it took a few months to get from a mediocre basic solver into a decent AIC/ALS solver, but it was because I skipped studying all those zillions of named patterns and concentrated on what I recognized as fundamental. It worked. After that it was trivial to learn all the named patterns as well because they were so easy to understand.

(Why did I bother to learn learn those patterns if I didn't really need them for solving? Well, it's fun, and knowing them is pretty much a necessity for effective communication. Spotting common patterns also makes solving faster and provides new opportunities for advanced solving when you learn to spot their almost-forms that can be used as nodes in chains. So, it's not like patterns are useless. Generic methods are just more important.)

The number of possible paths of an AIC, for instance, seems so overwhelming that learning to spot specific patterns seems more comfortable.

Seeing useful chains directly is not easy but there's an almost unfair tool to find them: coloring. It's the easiest way to find chains and nets, and it was another thing I recognized as a fundamental skill very early on. These days I can find chains without it (a recent example) but I rarely bother if I'm in a hurry. My coloring technique is mostly explained here. It's not the simplest, but I pretty much guarantee that it's the most powerful humanly-applicable coloring technique there is.

But I'm willing to alter my approach and expand my mind; learning this stuff is what makes Sudoku fun for me.

That's a good attitude. Either way, there's nothing wrong if you want to stick to the pattern-based methods, or go for something like TDP. I can't guarantee that my approach works for everyone, but I know it's worked for me.

So, I have begun looking into the TDP system of Robert Mauriès...

It's one way to do it if you really don't want to learn any patterns. Robert's method is an example of a very general approach to solving, and in fact it can probably solve pretty much everything if taken to the extremes. Many of the same concepts most of us here use can be found in TDP as well, with just different names and notations. For example, his "anti-track" is basically a replacement for an AIC. Similarly his "conjugated tracks" is similar to what we call Krakens. One major philosophical difference is that his approach automatically accepts branching nets (which we try to avoid until really needed) and contradictions (which we really try to avoid). Both of them make solving easier and faster, but the question is: are they elegant and do they make it more fun? Another downside is that such approaches are really difficult to notate in an understandable way.

The biggest selling point of TDP is simplicity, although it's not reflected in the über-complicated documentation (I've given feedback on that, and Robert has listened). However, not everyone is interested in such extreme simplicity because it takes away a lot of the fun of learning and solving with more interesting techniques. For someone like me there's no point in switching to TDP because I'd consider it downgrading. On the other hand, it could be very helpful for someone who doesn't yet know any advanced techniques. On my part I can also freely admit that I've learned some new tricks and perspectives from Robert, so I'm glad he joined the forum and introduced his approach.

I really hope I'm giving a fair assessment of TDP. Even though it's not for me, it might be a perfect fit for you or someone else. I like Robert and I have no reason to badmouth his technique. I think variety is good, and TDP is a welcome addition to the zoo of techniques to choose from.

--
Last edited by SpAce on Thu Jan 09, 2020 8:33 am, edited 1 time in total.

SpAce

Posts: 2560
Joined: 22 May 2017

### Re: WXYZ Wing questions

http://forum.enjoysudoku.com/wxyz-wings-t30012.html

since i am the creator of the advanced version of the wxyz- technique ill chime in

wxyz is an als-xz technique aka bent quad also found under my header B.A.R.N.s {bent, almost restricted N set: wxyz wings are size 4}

with some simplifications to the als-xz cravats: its 2 als sets found on a row/col/box where both sets intersecting in a box which total 4 digits and 4 cells

this is why space mentioned understanding als-xz or more complicated techniques can make this technique irrelevant as this technique set restricts the main one its derived from. moreover, when programing hierarchy for complexity these can appear early in the tree.

note:
wxyz - wing is the first als-xz technique that can have 2 restricted commons which can add to the confusion :
but this where they mimic sue de coq's and become vastly more powerful in the number of eliminations they can induce.

to start: these are the basic operations of the als-xz technique.

set A has N celles with n+1 digit
set B has N Cells with n+ 1 digit
set A & B has 1+ digit that is common to both A & B but can only exist in one set at a time these are the RC refereed to as x

when the RC is placed in A Or B
then set a has N digits in n cells, or set b has N digits in n cells. {placement of 1 digit locks one of sets}
thus any common digit found in A & B is locked to both sets any other cell that is visible to all these digits can be excluded
this digit is refereed to as z

in the case of 2 restricted commons: { as placement of 1 RC forces the placement of the other RC: then set a has N digits in n cells, & set b has N digits in n cells.}
set a is reduced to a locked set all peers of the A digits cells not equal to RC may be eliminated
set b is reduced to a locked set all peers of the B digits cells not equal to RC may be eliminated
all peers of the RC digits cells may be excluded
all peers of the shared digits cells of A & B may be excluded

Codeing notes; sets can overlap cells hower no rc can exsit in the overlap cells.

for als-xz reference see:
http://forum.enjoysudoku.com/post285035.html#p285035
Code: Select all
`+--------------------+-----------------+---------------+| 1689  169    (189) | 1589  2    4    | 7    3    158 || 5     4      (189) | 3     7    89   | 2    6    18  || 2     3      7     | 1568  15   568  | 159  189  4   |+--------------------+-----------------+---------------+| 7     12569  1259  | 59    3    259  | 8    4    156 || 69    2569   3     | 4     8    1    | 59   279  567 || 19    8      4     | 579   6    2579 | 159  129  3   |+--------------------+-----------------+---------------+| 3     (12)   (128) | 1678  14   678  | 46   5    9   || 148   7      58-1  | 1568  9    3    | 46   18   2   || 1489  159    6     | 2     145  58   | 3    178  178 |+--------------------+-----------------+---------------+`
yes this is a valid wxyz : A=r127c3 {1289}, B=r7c2 {12}, X=2, z=1 => r8c3<>1

Removed; to complicated example so i removed the example amd it lacked the digit cell count for a proper wing size i listed in error.

Edit: looking at some examples it seems the main difference between WXYZ-Wings and Sue de Coq is that the former is an actual locked Bent Triple, while the latter doesn't have to be. It can be an almost locked - and can still remove candidates. Verification for this would be nice.

the difference is the 2 restricted commons that are required for the sue de coq to function which is where this technique and all als-xz techniques over lap it
same goes for als-xy the triple linked Rule = death blossoms

easier more practical example for a double linked wxyz-wing
wxyz-wing: A=r1c8 {78}, B=r79c8,r8c7 {3678}, X=7,8 => r3c8<>8, r7c7<>3, r9c79<>6
Code: Select all
`+---------------+------------+--------------------+| 5    478  1   | 2   6   3  | 789     (78)   49  || 34   6    247 | 1   8   9  | 237     5      34  || 9    238  28  | 7   5   4  | 2368    236-8  1   |+---------------+------------+--------------------+| 2    9    3   | 6   4   8  | 5       1      7   || 48   478  478 | 59  19  15 | 236     236    36  || 1    5    6   | 3   2   7  | 4       9      8   |+---------------+------------+--------------------+| 7    38   5   | 4   19  6  | 189-3   (38)   2   || 36   1    9   | 8   7   2  | (36)    4      5   || 468  248  248 | 59  3   15 | 1789-6  (678)  9-6 |+---------------+------------+--------------------+`

or :
Sue de Coq: r79c8 - {3678} (r1c8 - {78}, r8c7 - {36}) => r7c7<>3, r9c79<>6, r3c8<>8

edit:
updated add a simpler als-xz dl rule show casing wxyz-wing and that its equivalent to a sue de coq.
Last edited by StrmCkr on Sun Jan 12, 2020 9:12 am, edited 2 times in total.
Some do, some teach, the rest look it up.

StrmCkr

Posts: 1170
Joined: 05 September 2006

### Re: WXYZ Wing questions

Thanks for all the replies. It's more info than I ever expected to get, and I really appreciate the time and effort. It will take me some time to digest all this and check out the resources...
This forum is great!
smcroberts

Posts: 6
Joined: 08 January 2020

### Re: WXYZ Wing questions

Hi StrmCkr,

StrmCkr wrote:wxyz is an als-xz technique aka bent quad

Did you see my point about the "bent quad" above? I don't think it's a good synonym for WXYZ-Wing because unlike a normal quad it's not a Rank 0 pattern. A normal quad is a One-Sector Disjoint Subset, which means that the logical "Bent Quad" is a (four digit) Two-Sector Disjoint Subset, i.e. Sue de Coq. Just like a normal quad, it's Rank 0 and it guarantees that the four cells of the pattern will contain exactly four digits in the solution.

On the other hand, WXYZ-Wing is an Almost Two-Sector Disjoint Subset, which behaves very differently from a normal quad. For one, it's elimination logic is Rank 1 (weaker) and it doesn't guarantee that the four cells will have four different digits in the solution. So, it's nothing like a normal quad. The same goes for any of the larger wings.

with some simplifications to the als-xz cravats: its 2 als sets found on a row/col/box where both sets intersecting in a box which total 4 digits and 4 cells

I've come to the conclusion that (UVW)XY(Z)-Wings can be defined even more simply. They're special cases of ALS-XZ which can be described as what I call ALS-Z. It means that the pattern can be seen as two strongly linked naked sets without thinking of restricted commons (X) at all. That's not possible with all ALS-XZ patterns, but it's possible with all patterns that I accept as wings. For example:

Code: Select all
`.---------------------.-----------------.---------------.| 1689   169    *189  | 1589  2    4    | 7    3    158 || 5      4      *189  | 3     7    89   | 2    6    18  || 2      3       7    | 1568  15   568  | 159  189  4   |:---------------------+-----------------+---------------:| 7      12569   1259 | 59    3    259  | 8    4    156 || 69     2569    3    | 4     8    1    | 59   279  567 || 19     8       4    | 579   6    2579 | 159  129  3   |:---------------------+-----------------+---------------:| 3     *12     *128  | 1678  14   678  | 46   5    9   || 148    7       58-1 | 1568  9    3    | 46   18   2   || 1489   159     6    | 2     145  58   | 3    178  178 |'---------------------'-----------------'---------------'`

As a normal ALS-XZ pattern it can be seen two ways. Written as AICs:

(1=2)r7c2 - (2=891)r712c3 => -1 r8c3 // X=2, Z=1
(12=8)r7c23 - (8=91)r12c3 => -1 r8c3 // X=8, Z=1

As an ALS-Z it's simpler:

(12)r7c23 = (891)r712c3 => -1 r8c3 // Z=1

In other words, either we have Naked Pair (12) or we have a Naked Triple (189). Since the 1r8c3 sees all 1s in both, it can be eliminated.

Every pattern that I agree to call (UVW)XY(Z)-Wing can be seen that way, and vice versa, any N-digits-in-N-cells pattern that can be seen that way is one of those wings. I think that's the simplest possible definition. It also automatically excludes any edge cases where no one is quite sure whether it's one of those wings or not, including all Rank 0 patterns (see below). I think that's beneficial because named patterns should be unambiguous and simple to recognize and verify as such.

wxyz - wing is the first als-xz technique that can have 2 restricted commons which can add to the confusion

Exactly, so let's not!

but this where they mimic sue de coq's and become vastly more powerful in the number of eliminations they can induce.

If it does have two restricted commons it becomes a Rank 0 pattern and shouldn't be called WXYZ-Wing. No Rank 0 pattern should be called "wing" because that term implies Rank 1. X-Wing is already an unfortunate misnomer in that regard, so let's not add to the confusion.

when you move the barn size up to size 6{cell count}: uvwxyz- wing

Yes, but shouldn't the number of digits match the number of cells in a wing? Note that you have 5 digits in 6 cells in your example. As far as I know, that's against even the normal definition of a UVWXYZ-Wing. Furthermore:

we'll find this example of the double restricted common on the same puzzle alsofound as a sue de coq /als-xz

Like I said, I wouldn't accept any Rank 0 pattern as a wing, because it's a confusing misnomer. In this case the mismatching number of cells and digits is an even more obvious red flag.

Code: Select all
`+--------------------------+------------------+----------------+| 168-9  169       189     | 1589  2    4     | 7    3    158  || 5      4         189     | 3     7    89    | 2    6    18   || 2      3         7       | 1568  15   568   | 159  189  4    |+--------------------------+------------------+----------------+| 7      (1256-9)  (125-9) | (59)  3    (259) | 8    4    16-5 || (69)   25-69     3       | 4     8    1     | 59   279  567  || (19)   8         4       | 57-9  6    257-9 | 159  129  3    |+--------------------------+------------------+----------------+| 3      12        128     | 1678  14   678   | 46   5    9    || 148    7         158     | 1568  9    3     | 46   18   2    || 148-9  159       6       | 2     145  58    | 3    178  178  |+--------------------------+------------------+----------------+`

Almost Locked Set XZ-Rule: A=r4c2346 {12569}, B=r56c1 {169}, X=16, z=/ => r4c9<>5 r5c2<>6 r4c2<>9 r4c3<>9 r1c1<>9 r5c2<>9 r6c4<>9 r6c6<>9 r9c1<>9

Your Doubly-Linked ALS-XZ is valid, but it's one of the most complicated examples I've ever seen (is it really the best sample for ALS beginners?). First of all, I'm not sure if I've ever seen an ALS-XZ with cannibal eliminations (9r4c23). I can see how they're justified, but it really makes this a very complicated example. Secondly, as written I can't see how your ALS-XZ eliminates the 9s in box 5 directly (9r6c46), although they get eliminated with claiming right after. As far as I see, only these are direct eliminations:

(1=9'6)r56c1 - (6=259'1)r4c4623 - loop => -69 r5c2, -5 r4c9, -9 r19c1, -9 r4c23 // X=1,6

To get the 9s in box 5 directly you need a different ALS-XZ (with the same cells but different ALSs):

(2=9'5)r4c46 - (5=169'2)b4p4723 - loop => -69 r5c2, -5 r4c9, -9 r6c46, -9 r4c23 // X=2,5

In other words, I can't see that a single ALS-XZ can logically get all of your eliminations at the same time. It's even worse with set logic. I need three different combos of cover sets to get all the eliminations:

{56N1 4N2346 \ 16b4 25r4 9r4 9c1} => -5 r4c9, -6 r5c2, -9 r19c1
{56N1 4N2346 \ 16b4 25r4 9b4 9b5} => -5 r4c9, -6 r5c2, -9 r5c2,r6c46
{56N1 4N2346 \ 16b4 25r4 9r4 9b4} => -5 r4c9, -6 r5c2, -9 r4c23

Note that the cannibal eliminations are Rank 1, while the others are Rank 0. Like I said, a very complicated example.

Sue de Coq: r4c23 - {12569} (r56c1 - {169}, r4c46 - {259}) => r4c9<>5 r5c2<>6 r1c1<>9 r4c2<>9 r4c3<>9 r5c2<>9 r6c4<>9 r6c6<>9 r9c1<>9

Nope. There's no Sue de Coq here. Can't have the same digit locked in two different sectors in any DDS. It's an MSLS, though.

SpAce

Posts: 2560
Joined: 22 May 2017

### Re: WXYZ Wing questions

Ill remove the example as it to complicated for learners.
Retracted further comments as it detracts from this threads topic hindering more then helping.
.
Last edited by StrmCkr on Sat Jan 11, 2020 7:40 am, edited 2 times in total.
Some do, some teach, the rest look it up.

StrmCkr

Posts: 1170
Joined: 05 September 2006

### Re: WXYZ Wing questions

StrmCkr wrote:Ill remove the example as it to complicated for learners.

Ok. I keep it in my comments, though. I think it's an interesting pattern and actually a perfect bad example. Bad examples are often better educators than good ones, so there's no reason to remove it. This one is great exactly because it's very close to the patterns it was meant to depict, but not quite. Such pitfalls are good to know. What's really great is that it still has valid eliminations, which would be missed if one relied on those specific patterns only. It also has value as a simple example of MSLS. The pattern also contains an actual WXYZ-Wing (and a degenerate VWXYZ-Wing) which are fully on-topic:

Code: Select all
`.----------------------.-------------------.---------------.|  1689  169      189  |  1589  2     4    | 7    3    158 ||  5     4        189  |  3     7     89   | 2    6    18  ||  2     3        7    |  1568  15    568  | 159  189  4   |:----------------------+-------------------+---------------:|  7     1256-9  *1259 | *59    3    *259  | 8    4    156 || "69    2569     3    |  4     8     1    | 59   279  567 || *19    8        4    |  579   6     2579 | 159  129  3   |:----------------------+-------------------+---------------:|  3     12       128  |  1678  14    678  | 46   5    9   ||  148   7        158  |  1568  9     3    | 46   18   2   ||  1489  159      6    |  2     145   58   | 3    178  178 |'----------------------'-------------------'---------------'`

WXYZ-Wing (1259): (91)b4p73 = (259)r4c346 => -9 r4c2
VWXYZ-Wing (12569): (961)b4p743 = (259)r4c346 => -9 r4c2

And it lacks the digit count to be the wing i listed which is a fail on my part. Thanks for that catch.

Ok, good. So we agree that a multi-letter wing should have an equal number of cells, digits, and letters in the name (except XY-Wing which has two letters but three cells and digits).

Ps mutiple sue de coq programs identified that move as one and all the eliminations as well the sue de coq posted is a copy paste.

That's why I trust my brain more than any sudoku software. That said, I don't know what programs you're using, but neither Hodoku nor SudokuWiki sees any Sue de Coqs there. And they shouldn't, as the definitions of both the original Sue de Coq and its DDS generalization make it very clear that each digit is confined to a single sector:

Sue De Coq wrote:iii. The most general form of the pattern is as follows.

Consider the set of unfilled cells C that lies at the intersection of Box B and Row (or Column) R. Suppose |C|>=2. Let V be the set of candidate values to occur in C. Suppose |V|>= |C|+2. The pattern requires that we find |V|-|C| cells in B and R, with at least one cell in each, with candidates drawn entirely from V. Label the sets of cells CB and CR and their candidates VB and VR. Crucially, no candidate is allowed to appear in VB and VR. Then C must contain V\(VB U VR) [possibly empty], |VB|-|CB| elements of VB and |VR|-|CR| elements of VR. The construction allows us to eliminate the candidates V\VR from B\(C U CB) and the candidates V\VB from R\(C U CR).

Obi-Wahn wrote:this thread is about the idea of generalizing theTwo-Sector Disjoint Subsets (Sue De Coq) technique to more than 2 sectors.

The rule can be put very simple:
A subset of N cells sharing only N different digits is disjoint, if all occurrences of the same digit share one common sector.
This means that none of the N digits can be true more than once in the subset, because all instances of this digit within the subset can "see" each other.

Every candidate outside the subset that shares a sector with all occurrences of the same digit within the subset can be eliminated.
This follows because if such a candidate was true, it would eliminate this digit completely from the subset, leaving us with only N-1 digits for N cells.

So, I don't think there's any question that whatever software you're using is wrong if it marks this as a Sue de Coq. Clearly the pattern is not that or any DDS as per the definitions above. Like I said, it requires a further level of generalization which is MSLS (more specifically MSNS). As such, it's an interesting specimen, because it allows to see the principles of MSLS with a small and relatively understandable pattern instead of a 4x5 cell monster. Sue de Coq (and DDS in general) is a subtype of MSLS with the specific constraint that each digit can only exist in one sector. When that's not the case, but it's otherwise a cell-based Rank 0 pattern, it's a generic MSLS. That's what we have here.

Same with the als xz i listed.

The Doubly-Linked ALS-XZ was correct as such. Only the elimination list was partly wrong, because those particular ALSs didn't justify all the eliminations directly. That's another risk when you trust software blindly. The eliminations may be valid for the cell pattern as a whole but not necessarily justified by the presented logic. Btw, Hodoku shows the eliminations correctly, i.e. it needs two separate DL-ALS-XZs (X=1,6 and X=2,5) to get them all like I said it should be. However, it misses the valid cannibal eliminations with both, but that's much less of a problem than listing too many eliminations (the former is just inefficient, while the latter is incorrect).

canabolistic eliminations are possible, and not as rare as you might think.

Ok. It's quite possible that I've just missed such opportunities, because I don't think I would have seen them here either without prompting (in fact, at first they looked incorrect). That's why I found your example very interesting, even if it failed at what it was supposed to depict. Do you have other examples of cannibalistic ALS moves? (Perhaps this is not the right thread to post them, though.)

SpAce

Posts: 2560
Joined: 22 May 2017

### Re: WXYZ Wing questions

Retracted to not ditract from this posts intent to help as it was going oftopic.
Last edited by StrmCkr on Sat Jan 11, 2020 7:41 am, edited 1 time in total.
Some do, some teach, the rest look it up.

StrmCkr

Posts: 1170
Joined: 05 September 2006

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