Please Give WXYZ Some Space

This post takes issue with recent redefinitions of the WXYZ-wing It suggests a name for Andrew Stuart’s update, apparently proposed by StrmCkr, and recalls the Bob Hanson’s Bent Naked N-set method in Student Assistant. The derivation of the toxic set in these two methods are contrasted with the WXYZ.

Getting a timely alert from my friend Gordon Fick, I revisited Andrew Stuart’s page on the WXYZ-wing. The landscape has shifted underneath the WXYZ. My early post on the regular XYZ-wing just passed “W” off as the rare cooperative enterprise of four bv, but during my lapse of attention, it became something else. Andrew’s second example fit my earlier conception, but his current first one, which he calls a “classic” WXYZ, shook my hammock:

Stuart Bent WXYZ 1Tw0 of the wing bv are replaced by a single ALS! I didn’t panic, though, because the ALS supplies the not(29) & not(59) logic of the WXYZ.

In the old WXYZ, a victim has to actually see the Z in the hinge and three bv wings . These four Z’s are a toxic set because at least one of them is true. Otherwise the hinge cell, r4c3 here, gets stripped of all candidates.

Andrew doesn’t make the ALS argument, but attributes the elimination to a rule by a forum correspondent StrmCkr, which says, if effect, that in four cells in the union of a box and line(a bent region), containing only the candidates of four numbers(the naked set), if candidates of three numbers are restricted commons (i.e. all see each other), and the candidates of the fourth number are not, the fourth number candidates are a toxic set. Any candidate seeing them all is removed.

How’s that again? How about a picture?   barn schematicIn this one, the locations of all candidates of numbers a, b, c, and z are represented by closed curves. The arrows point to the four 4-set cells. The a candidates are row restricted, and the b and c cells are box restricted. The 4-set represented is

{a,z}, {a.c}, {a, d}, and {bz}. The victim sees the two z’s, and is eliminated.

Great rule, but why is that? StrmCkr doesn’t say why, except that by definition, this is a form of WXYZ-wing, or at least it would be if Z could be omitted from the hinge.  I’ll go into how the “almost restricted naked set” rule was discovered in a later post, but here is why it works.

Let’s call the naked set an n-set. The contents of the n-set is a sample drawn from the candidates of n numbers in the bent. In the solution, the n – 1 restricted numbers can only fill n – 1 of the n cells of the n-set. The filling candidate has to be the true candidate of that number in the line or box. Each must solve to a different number, because you can’t have two true candidates of the same number in a restricted set. The unrestricted n-set number Z must fill the last cell, and the fill candidate must be the true Z of the line or the box, or both. These n-sets cannot occur just anywhere!

WXYZ figureAs I mentioned, Andrew does verify that his revised WXYZ is still about the old classic version with hinge and three bv wings, by including one as his second example. I’m going with the Sysudoku WXYZ schematic to make clear that all such WXYZ do not necessarily fit within a bent region, and are therefore not necessarily almost restricted 4-sets. There’s now an example in my post of February 2, 2016.

And I do irregular XYZ wings all the time, in which a wink attaching a wing  is a forcing chain. When such a wing in the WXYZ falls outside bent region, the WXYZ is not an almost restricted 4-set. It’s still a wing.

Above we are introduced to the fact that an ALS can serve up multiple wings of a WXYZ.

Stuart Bent WXYZ 3Andrew’s follow up “WXYZ” examples drift  away from the classic WXYZ. His third example seems to have an ALS hinge and a three candidate wing.


At first I feared maybe the pressures of the sudowiki site had gotten the best of Andrew, but then I read through the EnjoySudoku forum thread initiated by StrmCkr with his four cell, three restricted common idea, under the title WXYZ-wings, way back in May of 2010. Andrew provides a link.

StrmCkr claimed that all bent almost restricted 4-set cases are all WXYZ-wings, and uses w, x, y, and z in a display of all possible cases.  Andrew Stuart, for the moment, has gone with that in his Strategies page, presenting them as WXYZ-wings differing in structure from the definition still published there.

The toxic set rationale given above for the bent almost restricted n-set bears little resemblance to that of the WXYZ wing. It depends on the bent region, and WXYZ doesn’t. The number of cells and numbers can range from 3 up. To cover its many shapes, StrmCkr had to go much farther than his original proposal, which was only to allow Z to be removed from the hinge.

I’m saying that here is a practical new method for human solving, not too hard to spot and easy to verify. It needs an apt name, and an identity of its own in advanced solving literature. Let’s get off WXYZ’s back and let it be. I propose giving StrmCkr’s corollary  a new name: Bent Almost Restricted N-set, or BARN for short.

But as another reason for getting off WXYZ’s back, let’s recall another worthy but initially misnamed technique that also overlaps the Stuart examples, and many of StmCkr’s wxyz cases. It is just as entitled to a room in WXYZ’s condo as the BARN, which isn’t much.

In Bob Hanson’s explainer site for Student Assistant, Bob called it the bent naked subset and I got after Bob for his misuse of the good word “subset” for something that is not. There I suggested it be known as Hanson’s bent naked N-set, or BNS. I also got on Bob’s case about using the BNS to explain the XYZ-wing, which it is not. Bob had his reason for doing that. It was to make as much advanced solving as possible fit under one grand principle. But I saw this theoretically commendable objective to be at cross purposes with clear exposition of human solving methods.

The characterizing term “bent” is already in use on Stuart’s “Strategies” page. The N-Set in the suggested name refers to the fact that the method applies to n cells containing n numbers which are not a subset as normally defined. Bob normal perhaps, but a little wild.

In the BNS, Bob calls the N-set cells of the box and line intersection the hinge, and the cells of the box and line remainders, the wings. The wings then define toxic sets by how they divide the candidates. This differs distinctively from BARN, but the effect can be the same.

BNS1 and BNS0Bob defines the BNS in two flavors, which I call BNS1 and BNS0.

If the N-set cells in the remainders have more than one number k in common, say k1 and k2, then all the k1’s of the N-set still not guaranteed to include a true value. There can be true k2’s in both remainders, and consequently, no true k1’s among the N-set cells.

In a BNS1 with a single common number (k), this is not possible. There must be a single true value in the k’s of the N-set.

In a BNS0, with no common number, candidates of every value see each other. The N-set is locked. Candidates of any of its numbers are toxic. I would still not them a subset, because they are a locked set of the N-set, not a box or line.

In application, BNS is easier to spot and apply than the BARN. But understanding and explaining the BNS1 rationale is heavy lifting, I admit.

Hansons BNS0I’ll close with Bob Hanson’s BNS0 example from Student Assistant explainer, done up in Stuart style but with Sysudoku pencil marking. It’s a five number case.

Notice that cell r4c8 might have been substituted for r4c6 to remove 7r4c6 and promote 1r3c1, but this fails because 8 and 9 would then be common in the wings.

The fact is, the BARN isn’t any more worthy of moving in with WXYZ than Hanson’s BNS. Both of them are flexible on the number of cells and values. The WXYZ-wing, XYZ-wing and VWXYZ-wing form toxic sets the same way. That’s a live-in family. Throw everybody else out.

Next, we begin a very instructive (for me) review of the extensive Hodoku Techniques page, which takes us back to the days when a WXYZ-wing was a WXYZ-wing. I hope to have you come along.

About Sudent

I'm John Welch, a retired engineering professor, father of 3 wonderful daughters and granddad to 7 fabulous grandchildren. Sudoku analysis and illustration is a great hobby and a healthy mental challenge.
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13 Responses to Please Give WXYZ Some Space

  1. Good to see this review of WXYZ-wing – makes me want to re-visit the strategy and improve the documentation. I got a lot out of talking with StrmCkr but I disagreed with his approach which defines ten or twelve different patterns. I knew there must be a generalization I could make and code up and was pretty happy when I did so. The thought of looking for all those patterns individually was a bit daunting. Although its hard to prove I have all bases covered, I got a big gain on positive hits – and to check which pattern they belonged too would be to re-visit the individual pattern identification problem, so I can’t be statistical about my matches with his.
    I’m aware those I should be nudging towards a bent-sets family so what I’m reading here is interesting and I’m prepared to straighten up my definitions. Do get in touch directly. I’ve not visited your blogs before and I can see I should have.

    • Sudent says:

      Andrew, I’m pleased and honored. Well before starting this blog I’ve been relying on your site, and then was lucky enough to get a copy of The Logic of Sudoku.

      I appreciate your positive response to the WXYZ post. My focus is on the human solver armed with laptop and office software. My basic notations and visual tools evolved from that. In most cases where we differ on advanced methods, I’ve been carrying on where you may have left off, perhaps temporarily. I certainly will get in touch directly.

    • Strmckr says:

      Andrew it’s one pattern for solving,
      Four cells four digits with one restricted common, all cells that see all restricted Commons can be eliminated.
      Exactly like an Als-xz rule.

      I documented all formation patterns they aren’t searched individually.

    • strmckr says:

      “This post takes issue with recent redefinition of the WXYZ-wing It suggests a name for Andrew Stuart’s update, apparently proposed by StrmCkr,”

      first off I would like to dissect your understanding of my rehash expansion of a wxyz- wing.

      “if candidates of three numbers are restricted commons (i.e. all see each other), and the candidates of the fourth number are not, the fourth number candidates are a toxic set.”

      the above is completely wrong to the commonly shared view of all wxyz-wings, xyz-wings, xy-wings etc

      there is only 1 restricted common and 3 common digits to a wxyz wing

      in essence if the restricted common digit is placed outside the 4 cells, then one of the cells will be empty.

      wxyz wing:
      is a combination of 4 cells containing 4 candidates

      there is 2 sectors used by a wxyz wing:

      sector A : Which can be a Box, Row, Col
      sector B: Which can be a Box, Row, Col

      the cells that are in both sectors A and B are referred to as a Hinge

      Sector :A uses N cells containing N+1 candidates
      Sector: B uses N cells containing N+1 candidates

      where N = 1 to 3
      where the total number of N = 4 between sectors A and B

      sector A and B are linked by one candidate Z which is refereed to as the restricted common

      Candidates wxy are common to both sectors A and B
      however not all candidates wxy may be in both sectors A and B

      Elimination of Z candidate occurs in every cell that sees All cells Containing candidate Z but are not part of sectors A and B

      based on the above setup my definition of a wxyz-wing
      removes all of the needless restrictions placed on the original definition and broadens its solving scope. {ie number of digits per cell, where and what can contain Z}

      my original post seen here

      contains the exact same definition as outlined above which I included herein with more detail.

      the examples of types 1-4 seen in the above linked page are enumeration of the potential patterns and where their respective eliminations are derived.

      My solving function dose not search explicitly for each pattern and label them {as suggested by andrew} it is a single algorithm the pattern enumerations are merely representations of all combination forms.

      depending on construction of a wxyz wing there can be 1-3 hinge cells

      for example:

      {this was included in my first post on wxyz-wing before the original crashed}

      { you may see this as a hidden or a naked quad
      by function it is also 4 different wxyz-wings }

      | wxyz wxyz wxyz | . . wxyz| . . . |

      has three hinge cells: cells {123}
      sector a is a box, sector b is row
      w,x,y,z are restrictive common and w,x,y,z are common

      all the “.” cells wxyz

      Now if you prefer the extremely restricted case scenarios that the original wxyz-wing definition finds then feel free to objectify this technique as a

      ALS -XZ rule {almost locked set}
      {pretty much the oldest post I can find on the topic}

      for further information exploration of the technique

      which 100% covers pretty much all base wings from size 2-8
      xy-wing, xyz-wing and so on

      from this post forward

      the thread topic is regarding the possibility of having 2 restricted commons in a wxyz-wing which is possible in theory

      however its easier to view it as an als-xy wing and should not be considered a wxyz-wing

      after that topic I did I attempt to quantify a more rigorous almost -almost-xz rule to solve it regularly, from there I left the topic as is.


      • Sudent says:

        Strmckr, Thanks for your extensive comment, particularly the links to early definitions of the WXYZ-wing. If your comment is about my post, then it’s not a reply to Andrew. You didn’t say my characterization of your proposal was wrong. I’m saying it’s not a WXYZ-wing by derivation of the toxic set, and deserves a name of its own. I offer the argument that it works by virtue of the almost completely restricted commons, and suggested the name of Bent Almost Restricted N-set, N-set being a generic label for related methods. In the comment, are you calling your almost restricted proposal a rehash of the WXYZ-wing? I’m not. Whether or not a BARN can exist, and not be a WXYZ or an ALS-XZ, that’s another matter. I don’t know.


  2. Howard says:

    Dear friends, I have been playing Sudoku for some years after retirement. My standard is not high. I recently feel that a strategy, very useful to me, is the WXYZ Wing and the one extended to VWXYZ. It helps me to solve some of the difficult , diabolical and nightmare levels. Comments on these Wing Strategies are welcome!

    • Sudent says:

      Howard, you and me both. It’s just me on the blog.

      I’d say your Sudoku talent is very high, to be finding the high numbered wings regularly. Perhaps you can fill us in on your spotting tricks. Do you use a map of bi-value cells? Want some templates? See the tools page. In the post I’m trying to shoo off techniques derived by different means, but I haven’t really demonstrated a practical technique for finding all high numbered wings.

      • Howard says:

        Glad to communicate with you.
        Using the WXYZ-Strategy, I endeavour to identify 4 connected cells with 4 candidates WXYZ that meet the following criteria:-
        (a) There is a hinge or pivot
        (b) Each candidate must appear at least twice in the wing
        (c) Not all the candidates are required in the hinge
        (d) Not all the candidates may be present in any of the cells
        (e) There are non-restricted candidates, ie those that cannot see one or more of the same candidate in the wing
        Once when the above criteria are met, I would exclude the non-restricted candidates from the cells that can see all the cells of the wing. That is my approach. Please advise me if the rules above are 100% correct for WXYZ Wing.
        Regarding VWXYZ Wing, I am still exploring it.


      • Sudent says:

        Howard, your rules don’t describe the WXYZ-wing I am defending in the post. Maybe one of the variants Andrew Stuart proposes. Looking at my diagram in the post, candidate Z appears in all three wings. No other candidate appears in more than one. The wings are all bi-value cells. All four candidates are required in the hinge, which is a single cell. The dashed curves represent weak links. The victim must see Z in all three wings and the hinge.
        Maybe yours is a bent variant described in the post, a bent naked N-set(BNS) or a Bent Almost Restricted N-set (BARN). If it works, Andrew and I would really like to see some examples. Can you send me some, at ? If I can make them out, I’ll pass them on to Andrew.

  3. Strmckr says:

    You missed the point of my thread covering wxyz wings.

    Wxyz wings are four cell with four digits where there is not enough digits to satisfy n in n locations if one or more digits are contained out side the cells.

    My thread covers all combinations of the idea displayed as exemplars verified and confirmed from multiple users and Coded.

    The old definition of a wxyz wings is the four cell with four digits having one common restricted in the hinge, removing and exploring what if the hinge didn’t have the restricted candidate gives a more accurate display of all potential eliminations from this patter.
    If you don’t like my exploration on the subject then call it an
    almost locked set xz rule size four which is exactly what all bent sets are Als xz rule.

    As for Andrews comments below I originally mapped out all formation patterns as examples roughly 10 of them, then I compressed it down to four which circumscribed all of them, next I wrote one solving algorithm which solves all of them.

    So yes they are still wxyz wings by definition four cells and four digits with a single restricted common, the only difference from this to mine is how and where the restricted common occurs (in the hinge or not)

    The rest of my threat which tailed off was regarding an oddity when I noticed a Wxyz wing could contain a double restricted Commons more accurately viewed as an Als xy wing

  4. strmckr says:

    | 2 6 4 | 37 37 8 | 1 5 9 |
    | 1 59 59 | 4 2 6 | 78 378 378 |
    | 38 78 378 | 5 1 9 | 6 4 2 |
    | 4 789 1789 | 2 5 17 | 3 789 6 |
    | 59 3 12579 | 6 8 17 | 4 279 57 |
    | 58 2578 6 | 9 4 3 | 2578 1278 1578 |
    | 3589 2589 23589 | 1 379 25 | 25789 6 4 |
    | 7 1 23589 | 38 6 4 | 2589 238 358 |
    | 6 4 23589 | 378 379 25 | 25789 12378 13578 |

    Almost Locked Set XZ-Rule: A=r4c6 {17}, B=r4c23,r56c1 {15789}, X=1,7 => r4c87, r5c3,r6c2,r7c15, r6c28, r5c39

    it is a vwxyz -wing {used seldom and mentioned even less on most forums}
    most of the time the 5 cell+ size are not used nor are they documented extensively

    why this is not a wxyz wing is that a wxyz wing exclusively uses 4 cells

    99% of the examples are more commonly named after
    almost locked set xz rule
    which i did link to in my massive comment above,

    • Sudent says:

      Thanks for the example and additional comment. Can you identify the source of this example? I’m scheduled through January, but I’d like to follow up in a post in February. Any disagreement we may have comes from having very different objectives.

  5. Howard says:

    Sudent, my use of WXYZ Wing is extended to non bi-value cells. I am still exploring this strategy and will consult Andrew Stuart and Strmckr. I will send you some of my examples in due course.

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