An ALS Toxic Set Update

This post reviews Sysudoku history on the problem of spotting ALS toxic sets, and presents the updated  recommendation in the ALS Toxic Set page just completed for the Guide.  This spotting process, to be referred to here as ALS partnering, will be applied in future reviews, and updates

In sysudokie speak, a restricted set of like valued candidates guaranteed to contain a true candidate, a solution placement, is a toxic set.  “Toxic”, because any outside candidate seeing all members of the set is removed.

In a pair of ALS sharing a restricted common value, the like valued candidates of a different value in both ALS are a toxic set. The expert community uses the term ALS_XZ for this toxic set, understandably because almost all of the ALS toxic sets actually spotted by human solvers are ALS partnered with bi-value cells.  The XZ is a reference to the bv wing of XYZ or WXYZ wings.

The logic of the ALS toxic set, that one of the ALS loses the common value, and becomes locked, is best explained in those general terms.  On the other hand, Sysudoku has struggled with the obvious fact that the complexity of detecting a general ALS toxic set is too much for human solvers to undertake, especially those with the knowledge that methods easily spotted with template aids, like XY chains, fish and coloring, come next.

On my first encounter in July 2012, the Sysudoku remedy was to graphically display all ALS, then comparing all promising pairs. By October 2015,  I was ready to admit defeat, with a compromise, updating the 2012 post to generate one ALS at a time, looking forward in the sequence for promising partners. As an algorithm that saves half the ALS vs. ALS comparisons. Also it means that only the most promising, as a partner of a given ALS, are marked on the grid for testing.

It didn’t work. Over the intervening 2 ½ years and 150 posts, I have found very few ALS toxic set eliminations. My talented friend Gordan Fick, on the other hand, has spotted many for me.  My problem is, being neither a computer or Gordon Fick, I need an explicit signpost procedure, something I can recognize, that leads me to the crucial test, and will lead other ordinaries to find enough ALS_XZ to inspire them.

So now Sysudoku takes a third swing, an ALS_XZ spotting technique, is available in the Guide. I’m just hoping to get on base with ALS Partnering.

This is applying the construction vs. search principle to ALS toxic set.  Rather than looking for two ALS that fit together in the right way, look for possible first ALS components. With each one, you are now engaged in one or more guided constructions that will arrive at an ALS_XZ if it exists.

This prescription was tested with the blog cases, all of which it covers. Then to test evaluate the scanning effort and spotting improvement, the selected review puzzles from Castillo’s Only Extreme Sudoku were scanned immediately after line marking. This confirmed that ALS partnering is a large effort.

As to results, In Only Extreme 218, a regular 671-wing,


is duplicated by an ALS_XZ.



A more significant result occurs in Only Extreme 347. The 6’s are locked in the 3 value BARN, but there are no victims.  Also, this is not a BNS1, the remainder values being 569 and 569. But it’s undeniable that the victim 5 is looking at all 5’s in the two ALS.

Missing this removal in the review is significant as well. It enables a Sue de Coq and avoids a coloring trial.

Although ALS partnering does provide a clearly marked trail to follow, it still takes considerable time per productive ALS_XZ. In the Sysudoku order of battle, it is placed with other bv scan methods, to be  completed  prior to template constructing methods. That avoids the appearance of unnecessary scaffold construction. Aside from that consideration, ALS partnering might be better positioned after coloring.  The time it takes might be better spent on a puzzle actually requiring it.

Next week, a sysudokie version of the case study puzzle in Strmckr’s comment on the April 10 post, and some insights it brings. To bring up comments on a past post, click Comments at the bottom.  Also in the next post, a little more on the WXYZ, and some accounting for our honest differences.

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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.

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Departing Highlights on Bean’s Extremely Hard Sudoku

It’s time to wind up the volume 10 review, with a few highlights behind the review table.

The first one is the very unusual basic trace of IV-12, the homework puzzle of the previous post.

You have the characteristic Extremely Hard stinginess in the bypass, then a near collapse in the 1: list in box marking, with normal box marking completing the collapse. The 3-fills are plentiful and decisive.

Here’s the grid as box marking resumes.





Along with bypass solutions, here is something else an Extremely Hard collection should avoid: a routine basic, and a solution by means of:

a regular 475-wing, rare, but not extremely hard. That’s in Extremely Hard 10 I-3.

The Guide explains when this happens and when it doesn’t.

Then capper in I-3 is a 5-chain skyscraper.

The collapse begins with C5.







Extremely Hard 10 IV-47 is another puzzle embarrassed to be found in an Extremely Hard collection.  Advanced fun starts with a remote pair that’s hard to miss.





Next, another regular 251-wing. The grouped 1-chain victim is advanced level hard, but hardly extreme. The knockout blow for IV-47 is a short XY-chain ANL.




And finally, there is Extremely Hard v.10 VI-9, starting with a beautiful spiral, . . .

. . .and moving, Hodoku batting cage style, to a remarkable nice loop, which triggers the collapse inducing 1-wing.



In closing this review of volume 10 of Rebecca Bean’s Extremely Hard Sudoku, I recommend its highest ranking puzzles as an advanced collection, with the possibility of an extreme challenge.



Next week, as part of a progress report on the Sysudoku Guide, checkpoint your solution of Andrew Stuart’s fourth and most challenging example in the WXYZ-Wing page of the ‘Bent’ Sets section of his Strategy Families page. In case you don’t want a peek at his example first, here is the starting grid.

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Bean Gets No Second Extreme

This post describes how I-41, in Extremely Hard v. 10 by Rebecca Bean comes close, but fails to force a deferred Single Alternate Sue de Coq trial. It does force a review of AIC hinges and grouped forcing chains.

Here is the basic trace. These are always displayed with analyzed puzzles, to vet the basic solving  behind the candidates.

The first advanced method to be applied is an AIC almost nice loop eliminating 1r7c9. The AIC hinges are left on the grid, even though they weren’t involved in the ANL.

The SASdC chute SEr8 is left in case it is needed later. On my first pass on I-41 it was. The test of a 1 and 9 missing failed, and the Sue de Coq removed 1r2c8 and 1r7c7 as well as 1r7c9. But these removals were indecisive.

It was only when I returned to write up this post did I realize that having one member of the ANL toxic set in a cluster is lead worth investigating further. What would you do with it?

Well, first I noted that any 1 seeing 1r6c9 and any blue candidate is eliminated.  Then I started looking at forcing chains starting on any other 1’s in c9. That led to the grouped 1- chain to the blue terminal that makes the NE 1-group a victim.



The collapse confirms blue, and since the SASdC was not required, it holds volume 10 of Rebecca Bean Extremely Hard to one Sysudoku “extreme”.

Next week, the review is concluded with a few unreported highlights, starting with IV-12 here. Come prepared with your basic trace.

 After that, the plan is to post several reports on the development activity on the  Sysudokie Guide, a set of pages on Sysudoku methods and related issues. The Guide is accessed via the menu bar above

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Colorful Trials for Bean’s Extremely Hard v.10

This post calls on pattern directed coloring trials to solve IV-24, a puzzle in Rebecca Bean’s Extremely Hard, vol. 10. This one that seems to defy almost all advanced methods, and therefore, to deserve the rank of “extreme”. This also makes IV-24 a good candidate for refining or expanding the Sysudoku Advanced repertoire as the Sysudoku Guide is put together in the pages. In Bean’s rating system, however, it stands 7th.

More accommodating to the bypass than most v. 10‘s, VI-24 clams up beyond that in basic.

 Basic arrives in the grid below, which is destined to be a test plot for refining methods prior to coloring.

The reason for that is, I found only this indecisive X-chain ANL before attempting to  classify the bv into coloring networks.



There seems to be four clusters with some bridging implications:

not(red and tan) => orange or aqua

not(green and lavender) => blue or yellow

not(lavender and tan) => yellow or aqua

No bridge eliminations come up. That would be a candidate seeing both orange and aqua, blue and yellow, or yellow and aqua.

We do notice two colors appearing twice in the color conflicts above, tan and lavender. If we put tan on trial, orange and yellow take sides with tan.  If we try lavender, blue and aqua will jump in.

We go with tan, because it looks less scattered.

Yellow and orange rat on tan, confining S 2’s to r8, and placing SE2 in r8.

I needed a trial trace here.





Unanticipated but welcome, aqua brings in blue on the cleanup crew. We call upon a freeform diagram drawn earlier for a possible toe hold. Blue has three possible patterns for 4, one through yellow and two through lavender.




We choose the more decisive, yellow. In a trial, we want a quick contradiction, leading to a immediate confession. Call it the Perry Mason strategy.

It’s not so quick, but yellow fails, bringing in red, but placing two 9’s in c4. This leaves two 4-patterns for lavender to resolve.

 Lavender brings in orange, and it takes both to place NE4 to reject the dotted pattern.

Whew! Extremely Hard.

Next week, the Rebecca Bean Extremely Hard v.10 review is completed with highlights from the last three review puzzles.

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Almost Extreme Bean ExHdv10 II-29

Here we have a regular XYZ wing with two irregular victims, multiple coloring clusters and wraps, one, by an AIC almost nice loop. Rebecca Beam’s Extremely Hard v.10 II-29 is borderline extreme.

Basic is tough, but routine.


Except for the 4-wing as line marking begins.


Line r7 was filled in box marking with a match occurring on the r2, the first line marked. The wing fins stay in place as the remaining lines are marked, prohibiting the entry of 4’s in columns c2 and c6. To report what happened, I put them in and mark them as removed.



The 4-wing removal enables two naked triples, one in the North box and one in c6. The latter triggers a 9-boxline, and a 67 subset in c4.







The growing bv field produces a rectangular XYZ-wing, which yields nothing to unit winks. Here two victims look around the ER corners to see two of the three members of the wing’s toxic set.


When a target is suggested, you can often construct a chain to see it.

With coloring, II-29 begins to  look more extreme.  Coloring breaks the bv into three disconnected clusters. When AIC hinges are added, an AIC ANL wraps orange with a trap in r2c7.




One wrap leads to another, and another, in the collapse of Extremely Hard v.10 II-29.


At this point I’ve completed the remaining review puzzles, and can share the review table for Rebecca Bean Extremely Hard v.10 with you.

Next is one of the two Sysudoku extreme rated puzzles of the ExHd v. 10 review, way down at a Moito rating of 0.93.  It is VI-24. From me it demanded coloring trials. Maybe you can do it straight up. Notice how I’m assuming the Moito and Bean ratings are the same. Where’s the explanation of either or both?

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The Volume 10 Rebecca Bean Extremely Hard Sudoku

This post begins the second review of a Rebecca Bean collection, this one prompted by a clever cover, and similarities in format to recently reviewed collections attributed to Moito Publishing, A.D. Ardson and James D. Glover. Here we resolve Extremely Hard v.10 I-34, assigned last week for your homework.

In my first Bean review, of 600 Extremely Hard Sudoku Puzzles with Answers, there were 12 sections of 50 puzzles each.  I selected the #6 puzzle from each section. There were only two advanced puzzles in that selection, and nothing extremely hard. Later I reviewed Moito’s Sudoku Road to Mastery, which proved to be much more challenging. The grids of the Moito, Ardson, and Bean books are identical. None of them provide any background accounting for the similarity of production or anything else.

But significantly, Moito and Bean ExHd v.10 share something else, a rating system covering an identical number range. The system also appears in Glover’s unacknowledged Sudoku Puzzles Evil Killer.  In the Moito review, I selected 10 highest rated puzzles, and therefore have reason to believe that doing the same with Bean v.10 would produce a fairer evaluation.

One more bit of context before we plunge into Bean’s ExHd v.10. A look at Bean’s Amazon author page reveals over 50 collection books carrying “Extremely Hard Sudoku” in the title! And I thought the “volume 10” was a promotion device!

Many have clever covers, but there is no statement inside about duplication. I found one match between my two ExHd’s by picking one and scanning the other book for a duplicate. Also, in view of the deluge of similar books,  I’m not persuaded that  Rebecca Bean and A.D. Ardson are people. Could they be computers in a Moito back room?

Representative or not, the two Bean reviews and two Ardson reviews will be my last, unless readers have reason to suggest otherwise.

Now for Extremely Hard I-34, Bean rated 0.94. The basic trace is typical of the volume 10 collection: resistance to the bypass, normal box marking, and a tough line marking.

The Sue de Coq chute  is described by   SEr9 =8(6+9)(1+7) with a removal in each remainder.








On the 3-panel, a grouped 3-chain ANL reduces the fins of a swordfish to one, and the fish removes a 3 in the fin box.

The removal leaves a decisive boxline. See it?





Yess! 3r6c1 goes as well, setting up a naked pair that takes five 9-candidates with it.








Then more X-chain ANL, one on the 6-panel, and one on the 8-panel, bring the collapse.

Advanced, but not extremely hard. Did you do it that way?





Next week, it’s another 0.94 but tougher, with a special WYZ-wing and a heavy dose of coloring. Its Extremely Hard v.10 II-29.

There are new pages off Sysudoku Advanced in the Guide. One’s on unique rectangles and other uniqueness methods. Off the BV Scan, there’s a new one on Sue de Coq. The Sysudoku approach differs from expert consensus in a fundamental way.

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