This post translates an instructive example from “strmckr”, an expert forum contributor, and adds some observations about double common ALS_XZ.  Then Sudent comments on some Sudoku history that Strmckr included with his example.

In his comments on the April 10 post on a possible WXYZ vs. BARN settlement, Strmckr supplies a very helpful example of a double common ALS_XZ. Forum participants, through a long history of text communication, transmit the grids to each other in unformatted text.

Here is Strmckr’s example, more accessibly marked for Sysudoku readers.

Many ALS_XZ are BARNs, but is this double common one a “double barn”? No, this isn’t a “barn size 4” at all.

But it does work.  It’s a valid ALS_XZ, and also a BNS0 Robert Hanson’s Bent n-Set 0.  The 4-set remainders are disjoint, and every value is therefore locked. That itself guarantees the removals, including 8r3c8.

Strmckr was especially considerate to point out that his example is also a Sue de Coq, evaluated as we do in Sysudoku, with remainder ALS. The 3 or 6 not in SEc8 must come from 36r8c7, and the 7 or 8 not in SEc8 must come from 78r1c8.

But getting back to the double common ALS_XZ above, Strmckr states for us the fact that all values in both ALS become locked. That’s true of all ALS_XZ. He could add that with double commons, the uncommon values in each ALS are locked as well. Each ALS loses a common value and both are therefore individually locked in each uncommon value.

But that’s not all. The locked sets includes the double common values, but unlike the uncommon values, common values are locked across both ALS; not  within each ALS. That’s because we don’t know which ALS gets which common value, so a common value victim must see all common values in both ALS.

Thinking about this, I realize that I was overly restrictive in ALS Partnering, my lens prescription for spotting most ALS_XZ. Every member of a restricted common must see each other. That guarantees that when one of the ALS gets a common value, the other doesn’t. But the wink between common values in the two ALS can be an AIC. That can open the butterfly box for some amazing ALS_XZ. On the downside, it makes solver coding and comprehensive human spotting even harder.

My reply to Strmckr’s first comment asks his opinion on inference chains as weak links in XYZ. We missed out on that, because he took it as question on ALS nodes in AIC, which are covered in Sysudoku.  His statement that “xyz-wings are always constructed as a barn” prompted the question.  Apparently, Strmckr and the forum remain unaware of Sysudoku irregular XYZ-wings, and irregular victims of toxic sets.

Or perhaps, it’s another instance of Sudoku coders avoiding the weak linking AIC, a.k.a. forcing chain, as a subject of discussion. It’s burdensome to program them for a solver. Line and column “seeing” does encompass a simpler and more easily coded world.

Strmckr’s comment claims agreement with the subject post, but actually he disagrees. There are two disagreements in his statement introducing the double common ALS_XS rule:

“ALS-xz double linked rule does apply to wxyz-wings, aka ‘barn size 4’ “.

Strmckr does not explain, or give any reference to explain, how the ALS-XZ double common rule applies to the WXYZ wing that is not also an ALS-XZ.  And the statement ends by describing BARN 4 as a commonly agreed synonym for WXYZ-wing. It certainly is not that .

Strmckr believed he was rightfully extending the WXYZ in his original BARN post.  But actually his “single bent value” BARN is equivalent to Bob Hanson’s Bent 4-set 1, leaving no doubt that Strmckr was inadvertently mislabeing Hanson’s BNS1 as a particular form of WXYZ.

The BARN and the BNS1 are two means of spotting the same thing. If a single value is bent, then the bent area remainders share a single value, and vice-versa.

In his comment, Strmckr takes credit for dismembering the 4-candidate hinge of the WXYZ.  That abandons the primary feature that characterized the wings. The forum should not have agreed so readily.

Strmckr comments that his WXYZ modifications were for “opening up its very narrow search window”. The very narrow search window is due to the increasing size of the toxic set with increasing wings, aggravated by an unnecessary limiting of weak links to house membership. “Seeing” the entire toxic set gets harder very rapidly.

But this is a logical consequence of the XY, XYZ, WXYZ, . . . definitions, and there is no advantage gained by compromising the WXYZ identity to relieve it.

Programmers have a better way to extend and maintain interest in higher wings. It’s AIC weak links.  Humans don’t find them all, but they do have fun with them.  Have you seen what they do for the XYZ?

Next post starts a new review, with a twist. The book is inaccurately titled “World’s Hardest Sudoku Book” from Sudokubooks, the unidentified publisher who is, by my guess, Moito, Rebecca Bean, and A.D. Ardson as well. The review puzzles are the hardest 10 by the rating assigned by Sudoku books, which is the same format as the others just listed, but not explained by any of them.

Your homework is the only 1.00 rated, #25. The twist is to use the review to decide where the ALS Partnering and APE Type 2 should be placed in the Sysudoku order of battle, right after unique rectangles, or just before AIC hinges. I’ll try for easy ones in both places, and after reviewing checkpoints, you’re invited to find some I missed.