A WXYZ vs. BARN Settlement

In this post, progress is reported on the Sysudoku Guide, and on  a possible territorial agreement between the WXYZ wing and the Bent family.

The Guide, that link up there on the menu bar, is taken shape. The Basic pages are complete, with all the elements of Sysudoku Basic explained and demonstrated. That includes the bypass, 3-fills, slink marking, box making, line marking, subsets, and box/lines.  On the Advanced Side, there are pages now on unique rectangles,  Sue de Coq, APE, Bent n-sets, ALS toxic sets and XYZ-wings.

That advanced sequence may not fit your order of battle, but it is meant to delay the construction of visual aids until they are needed.  With XYZ Wings, the sysudokie beginner encounters the first of these, the XYZ map. Later will come aids for XY chains, X-chains, fish, coloring and patterns. My grid and aids are ©PowerPoint and ©Word template files.

Writing up the XYZ page led me back to the friendly argument I enjoyed with Andrew Stuart and the EnjoySudoku forum writer Strmckr, about the overlap between the WXYZ-Wing and a discovery by Strmckr.

It is basically a labelling controversy. Labels help us sort out the interplay of ideas and we all know that misapplied labels get in the way. What happened in this case is that the WXYZ wing, starting life as a logical extension of the XYZ wing was already compromised and losing its label to overlapping methods, when Strmckr noticed this property of an overlapped type of WXYZ-wing.

One of the WXYZ  overlappers is the ALS_XZ.  A lesser known one is something labeled here as a bent n-set, a collection n cells containing only n values (hence naked), sharing one box and one crossing line. I learned about it from Robert Hanson’s Sudoku Assistant report.

Here is the sysudokie view of the WXYZ -wing and one of its victims. The dashed lines are weak links (winks) between like-valued candidates in the hinge and the wings. OK, the wings can also be in larger ALS packages, but the winks remain. Any outside Z that “sees” (winks at) all four Z’s is toast, because its looking at a true candidate. If Z is not placed in any wing, it goes into the hinge. That’s identical to the XYZ-wing rationale.

The thing is, if weak links are limited to candidates sharing a box or line, its hard to find a sysudokie WXYZ wing, and even harder to find a victim. Instead, you need split hinge and a crimped wing arrangement to make it work, and those arrangements are bent naked 4-sets within the box and line of the hinge. Also, they are often ALS-XZ, but let’s leave that out of it here.

Bob Hanson stated an easily checked condition for the bent n-set to have a locked, and therefore toxic value. That value has to be the only n-set value shared by the n-set cells in the bent region’s remainders. In sysudokie speak, the box/line intersection is a chute, and the units minus the chute,  remainders. An n-set meeting this Hanson condition a Bent n-Set 1, or BSN1. That’s almost Hanson’s term, but with “n-set” where Bob said “naked subset”, which it isn’t. You know, labels.

By the way, if the remainders have no n-set value in common, you have a BSN0, with n toxic sets. The candidates of every n-set value see each other. It’s a subset, not of a box or line, but the bent region.

Now we get to the heart of the matter.  By diligent observation, Strmckr discovered another, equally easy condition for a BNS1:I  If the n-set candidates of only one value are bent, i.e. not “restricted”, i.e in both remainders, then these candidates are locked, i.e. must include a true candidate, i.e. are a toxic set. The “almost restricted” condition guarantees the BNS1 condition, and therefore has a basis in proof as well as observation. Hanson explains why the BSN1 works. My latest attempt at a simpler explanation is on the Bent n-Set guide page.

I had thought that, at the time of Strmckr’s “almost restricted WXYZ” introduction on the EnjoySudoku forum, the WXYZ had already been compromised with a distributed hinge, and BNS1 with n = 4 were being mislabeled as WXYZ, and that Strmckr had little choice but to  label his discovery as a property of the WXYZ wing. In his comment below on this post, however, he points out that the dissolution of the WXYZ hinge was his proposal, made to add flexibility to the WXYZ. Now his forum thread on the alternative condition for BNS1 is properly titled Bent Almost Restricted n-Set, the sysudokie amalgamation of Strmckr description and Hanson terminology, and he freely describes it as “the barn” without caps.

All of this is actually relevant to your homework, to solve Andrew Stuart’s example 4, In his WXYZ page under ‘Bent’ Sets.  When Andrew challenged his Sudokuwiki readers to find four WXYZ -wings, I thought there might be one of those hinge cell, unit seeing, classic WXYZ wings among them. Did you find one?

Here is the basic trace, with an active and interesting bypass to make the remaining stages routine and easy.






Prior to the first WXYZ, here are two of Andrew’s Y-wings (XY wings), and a regular XYZ wing. No iXYZ or or XYZi were available.






An AIC almost nice loop is enabled, and leaves the 4-set shown in Stuart’s WXYZ example 4.



The 4-set 1289 is in the Wr6 bent region. It is a BNS1, with remainder values 18 and 289. The single common remainder value 8 is locked. The 4-set is also a BARN, with single unrestricted value 8.





The removals leave a 4-set in bent region Cr5, with 4-set remainder values 349 and 89, and common value 9. The toxic set 9 removals in r6 leave a Cr4 boxline triggering the collapse.

Oops. It takes only two “WXYZ wings”, instead of four.


Maybe somebody will show me the solver’s other two. I have to recuse myself from using any solver. Hopefully, Andrew Stuart can now agree that these two “WXYZ wings”  should be classified as BNS1’s or BARN’s, depending on how you identified them.

By the way, all of the sudokuwiki WXYZ examples are also ALS_XZ. My next post updates sysudokie readers on that.

Seriously, there is really no good reason why the classic form of WXYZ should not be given the exclusive use of the name. That would create space for iWXYZ, WXYZi, and iWXYZi pursued with inference chains as weak links. In concession to the bent side, n-sets are not limited to n = 4, and should not be tied to a label that is.

Also, the bent family rationale is logically worthy, and needs to be respected and passed on to Sudoku enthusiasts without asssociation with an entirely different rationale. The work of Robert Hanson and his mentors, and Strmckr’s BARN perceptions, need not be muted by prior inaccurate labeling.

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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4 Responses to A WXYZ vs. BARN Settlement

  1. Strmckr says:

    Thank you, this post I agree with.

  2. strmckr says:

    | 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 2368 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 | 1389 38 2 |
    | 36 1 9 | 8 7 2 | 36 4 5 |
    | 468 248 248 | 59 3 15 | 16789 678 69 |

    case study puzzle:
    interesting things from this puzzle:

    als-xz double linked rule does apply to wxyz-wings, aka “barn size 4”
    which is: if there is 2 restricted commons between set A&B then all cells become locked sets and all peer cells for each candidate with in A&B can be eliminated}

    Almost Locked Set XZ-Rule {double link rule}: A=r1c8 {78}, B=r79c8,r8c7 {3678}, X=7,8 => r3c88, r7c73, r9c796

    {you might recognize this as a Sue de cog}

    I may agree with the majority of your post however:

    – i never mislabeled WXYZ in my original post, –
    what i did was remove the 4 digit count on the hinge cell and all the bivalves “pincer” cells could have more then 2 digits, opening up its very narrow search window.

    what i did do was petition to have others follow my suggestion and treat wxyz-wings the same as xy,xyz wings and have all of them follow the same als-xz rules {which the other 2 types already did} instead of restricting its ability to be useful.

    by the way:
    xy,xyz, wxyz “w” is the next letter in reverse order for increasing size of als-xz functions.
    hence its name and not a mistake.

    additional stuff that makes things more confusing for people:

    xyz-wings are always constructed as a barn.{all 3 cells are in the same chute or band}

    xy-wings are only a barn when they use a box + Row/Col configuration. {all 3 cells are in the same chute or band}

    • Sudent says:

      Thank you for both comments. I will expand your example graphically to help regular readers absorb its lessons, in a post or the Guide page. If you have the givens for it, I could post it as reader homework. Not necessary, of course.

      There was never any question about the W in WXYZ, but I had the history wrong about breaking up the hinge into multiple cells. I said it was already done when you introduced the barn, so that was not your doing. I take the hinge breakup as mislabeling, because it breaks the succession of wing patterns associated with the labels, and the toxic set rationale of the wings. That left Andrew with the bag.

      The wings rationale probably holds when wings are combined into larger ALS. If so, that was a good move. Now what is your opinion of using inference chains as the weak links in the wing patterns? You know I’m doing that under the XYZ-wing label.

      • Strmckr says:

        The wing rational does hold for all n sizes 1-8, I coded and tested my barn search engine for that including the double linked rule.

        However the codes limited search to single stacks or bands, is the only real difference between als-xz and barn.

        Using established als functions as an inference link we have been doing that for about 10 plus years over at daily sudoku, and enjoysudoku
        known as fin transport.

        It’s not heavily documented as it often referred to as a kraken (pattern with out side help) on enjoy sudoku but it is on the rise featured in our usersubmitted daily puzzle on the forums.

        I did write on the subject:

        I helped another fellow user code the barn expansion version


        The expansions of moves with digit transport applies to all subset move types, I have successfully coded the idea onto many of them for my own solver.

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