Decoding Hodoku Chain Notes

This post comments on the first part of the Hodoku “Chains and Loops” page. It reveals the premise dependent view of chains to be based on computer coding, and unsuited for human solving advice. It explains Hodoku’s chain notations as a paraphrase of the code for chain following.

Bernhard Hobiger begins the Hodoku Chains page with a general introductory section applying to all forms of alternating inference chains. Too bad this important page begins with, “What is a Chain?”, a series of mistaken pronouncements that confuse Sudoku computer solving with human solving. It begins with

“A chain is simply a stream of implications that lead from a premise [e.g. candidate x in cell y is not set] to a result.”

If you are a computer, this is true.

Hodoku XY wingHobiger’s first chain example is the XY-wing.

Sysudokies represent the 57-wing eliminating 2r2c6 in black as an XY-chain, which it is. As if to emphasize that, this chain can be extended to an AIC ANL knocking out another candidate.

Hodoku explains the chain first as an XY-wing, saying, with my annotations:

XY wing“Cell r1c3 (the pivot) contains candidates 5 [X] and 7 [Y]. Cell r1c6 shares row 1 with (sees) the pivot and contains candidates 5 [X] and 2 [Z], cell r2c1 shares box NW with (sees) the pivot and contains candidates 7 [Y] and 2 [Z]. Cell r2c6 (victim) sees both pincers (r1c6 and r2c1). It cannot contain 2 (Z).”

Once you get past cells seeing each other – which they don’t – this matches the rule for an XY wing. It would help to include a schematic in  Hodoku here, and to prove the rule somewhere.

But to Hobiger, this wasn’t enough. Hodoku adds this: 5r1c3 =>2r1c6; 7r1c3 => 2r2c1. Guided by my previous post, we can immediately recognize this to be XY-wing detection code, a simplification of XYZ-wing detection code.

By preceding this with “The logic is simple:”, Hobiger is claiming that playing computer this way is easier than just recognizing the wing and its toxic set and noticing that 2r2c6 sees both wing cells. Considering how often this logic fails to uncover a wing, this claim is extremely misleading.

But somehow, this wasn’t enough. Hodoku adds his paraphrase of chain following code:


Well, not exactly. I think the programmer has to actually tell the computer about the bv slinks, so the more accurate paraphrase might read:

(r2c1<>2) => (r2c1=7) => (r1c3<>7) => (r1c3=5) => (r1c6<>5) => (r1c6=2)

This is known to students of Sudoku as the way you prove that one of the matching slink terminals of an AIC is true. Once you are convinced of this fundamental fact, you construct chains, re-use them at will, and never follow the alternating inference node by node again.

But Hobiger seems unconvinced by the XY-wing rule (theorem) or the “AIC end” rule (theorem). He leaves that impression when, while applications of chains as links abound in Sudoku, Hodoku can include the statement quoted above, and can explain complex chains by writing out chain following notations.  It’s the common failing of being too close to your work.

In computer code, you write a routine to follow an inference chain, given a premise derived from the technique at hand. This premise is not arbitrarily chosen, like the one’s Hodoku seems to be recommending when he omits the purpose of the code he is paraphrasing.

The routine may explore many branches of the chain before returning a result. The chain itself is not saved, only the result. Later, to search for something else, the same chain may be followed again and again.

This is the only way a computer code can deal with the logical reality of a chain, by always repeating it exactly from the same starting point. To a computer, a chain is the corresponding chain-following routine, not an object that we picture on the grid.

Human Sudoku solvers are not so limited. They regard chains as objects. They don’t require a simple premise to trigger their “follow chain” routine. They can assemble a chain from dissimilar parts and from any promising starting point. They can construct a chain to satisfy a higher purpose, without having to check out every possible branch. And to humans, following the same chain again and again is insanely inefficient.

Hodoku grp AIC 4 471wingHere is an example from Hodoku, illustrating the higher purpose of the premise leading humans to find chains. It was presented on Hodoku to illustrate a grouped AIC chain, but it would never have gotten that far in sysudokie territory. A 471-wing is found on the sysudokie XYZ-map, but in a configuration with no possible unit-wink victims. But 1r2c5 and 1r9c8 see two wing 1’s. A chain from 1r9c8 to 1r4c2 fails, but 1r2c5 has a grouped 1-chain view of 1r9c9, and has to vacate the premises. The Hodoku premise would have been “1r2c5 is true”.

Scan back over the last few posts, to see chains used extensively in URs, XYZ-wings, ALS-XZs and color wraps. In Hodoku Techniques, however, chains are limited to direct eliminations and confirmations. They do not function as winks and slinks in other methods.

Except for one other purpose. Bernhard manages to make another error of fact in the innocent looking statement above, and immediately amplifies the damage. He says that the purpose of a chain is to prove or disprove a simple logical premise that a candidate is true (“set” in Hodoku speak) or false. He does not refer to any purpose of the premise in the current task , misleading many to conclude that you just make it up. Then he deliberately throws logic under the bus, with “If a chain does not produce a contradiction, it proves nothing.”

I’d like to excuse Hobiger by saying that he only forgot to explain what he meant by “contradiction”. But Bernhard makes that impossible, advising that if the chain does not produce a contradiction, try the opposite premise. Now it’s just “pick a candidate and try a value”. We try to convince beginners not to forego the true pleasure of Sudoku by doing that, and here is a computer solving expert carelessly promoting it.

The general lesson is, be especially wary of what computer programmers advise you to do. They may be coding and you’re not a computer. That explains the Hodoku notes on XYZ-wings, and it explains many of the other chain following notations in Hodoku. I’ll not be parsing any more of them . Instead, I will try to identify the logic by which a human solver is motivated to find the chain in the excellent Hodoku examples.

Not all Sudoku programmer/writers suffer from this occupational hazard. Andrew Stuart, on his very informative site, and in The Logic of Sudoku, is a notable exception. Bob Hanson, in his explanation of Student Assistant, sought to refine human solving methods.

OK, we got past “What is a Chain?” The next post examines the Hodoku chain mechanics and examples on chain links and nodes

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Hodoku’s Limited XYZ Wings

In this post reviewing the Hodoku Wings page linked under Techniques, we see how an expert programmer can go wrong in explaining how to recognize where a technique applies and why it works . The omission of irregular wings is another update challenge for the Hodoku project. For realistic solver training, it’s now required to include the possible employment of grouped AIC winks in any technique.

Hodoku SdC 5Did you step into the cage and take a swing at the “Extended Variation” Sue de Coq? It turns out to have a multi-cell ALS in each remainder, but still manages to do some damage. A little more than Hodoku marked up, in fact. The contents of SEc7 can be described as


The row remainder is responsible for 4 or 9, but it so happens that r5c7 is left holding the bag for 2 and 8, and the ALS in c7 must supply 1 as well as 4 or 9. That wasn’t very nice.

To cap off a bad day, the Hodoku Wing section steps out with “An XY-wing is really a short XY-chain”. That is true. So why not treat the XY-wing as a side comment in the discussion of XY-chains? It’s time to do it that way, and I will, in the next post.

After some nice XY-wing examples, the wing section continues with the XYZ-wing, calling it “an enhanced version of the XY-wing”, which it isn’t. The extra Z in the hinge is more handicap than enhancement.

That is followed by “Now the pivot contains not only X and Y, but Z as well”. Well, why is that important? Nowhere does Hodoku mention the fundamental fact that one of the three Z’s must be true. The reason he doesn’t is explained shortly, but therein lies the advantage of XY-chains over XYZ-wings, two candidates, instead of three, that a victim has to see, no matter how long or how short the chain.

Hodoku XYZ wingHodoku’s first XYZ-wing example offers a side demonstration of this advantage. In a generous field of bv, XY-chains, the generators of 2-candidate toxic sets, run wild with opportunity. Here, only a few 3-candidate hinges were available, but a regular 457-wing did appear. I use thick burnt orange arrows to mark the hinge and wings. But the XY railroad gives two XY-chains making the same elimination, one with a bonus elimination and clue.

More on the Hodoku chains to follow, but my job here is to evaluate what Hodoku’s example note is telling readers about the XYZ-wing . After identifying pivot(hinge) and pincers(wings), it’s this: .

“If r7c2=4, r2c2=7 => r9c2<>7; if r7c2=5, r7c1=7 => r9c2<>7; if r7c2=7 => r9c2<>7”.

There is a full explanation of the notation in another section, but to use actual words:

Try each value in r7c2. No matter what r7c2 is, r9c2 is not 7.

What is this? Is it the way an XYZ works? No, it’s not about seeing 7’s in both “pincers” and the hinge.

Is it how you spot an XYZ? No, it’s not about pivot and pincers sharing numbers and seeing each other.

Is it something you do with every three-candidate cell, to see if it is an XYZ pivot and what it’s victims are? Astoundingly, yes.

The Hodoku diagrams sorely need explanatory notes, but this one has nothing to do with the XYZ- wing as a human solving tool. It actually describes a computer search for XYZ-wings! The algorithm actually goes something like this:

Hodoku XYZ wing codeNow if you want a practical way to describe this to nonprogrammers, just identify the pivot and pincer cells, and write the Hodoku note above.

But think about how impractical this is for human solving. Such notes are not telling you how a person could find the wing or how it works in this case.

XYZ wingMy post on regular XYZ does provide these essentials. In the following one, on irregular XYZ, there is another major update project for Bernhard. It’s in inclusion of grouped AIC winks in the XYZ link structure. This introduces many more ways for the victim to strip the hinge.

And what about the other winks of the schematic, the ones connecting the “pincers” to the hinge? Yes, these can be forcing chains and ER’s.

A remarkable case from Sue de Coq’s original post is reported in my XY-chains post of December 2011.   Of course this insight applies to all applications of seeing, that is to say, all winks in technique structures. It led me to promote the concept of the toxic set, a set of candidates other than a unit, guaranteed to contain a true candidate of a number. Toxic sets are generated in many ways, but all are exploited via the same “seeing” methods. It is important to recognize the forcing chain as a wink, a way of seeing.

Addicts 134 irregular XYZHere is an example from Paul Stephens Sudoku Addicts, puzzle 134. These wings don’t exist for most experts reviewed in Sysudoku, including Stephens and Hobiger.

Hodoku does grouped AIC winks. For example of grouped AIC winks in the XYZ, and in the fatal vision of the victim, have a look at the review solution of KrazyDad’s Insane v.4, b.3, n.5. Such cases are not nearly as hard as they look after the fact. The search is awesomely pre-filtered by having a given start and ending point, and is guided by bv and slink markers along the way. The initial spotting process, aided by a map of the bv cells, is simple for the human visual system.

Hodoku passes up on the WXYZ-wing, but really should include it, and a relative, the Death Blossom, with the BARN and BNS in any update.

W-wingThe last wing of the Wings page is the W-wing, which Hodoku describes as “two bi-value cells with the same candidates, that are connected by a strong link on one of the candidates. Actually his two examples reveal an AIC wink between two of the bv candidates, and simultaneously, an AIC slink between the other two. The W-wing is an AIC almost nice loop. It is an eliminating ANL, but if one of the “Discontinuous” winks is a slink, the W-wing is an AIC confirming nice loop for one of the X-candidates.

One of the Hodoku W-wing examples has been added in Sysudoku notation to in my December 2011 post on XY-Chains. Two more found by Gordon Fick appeared in a recent revisit of Weekly Extreme 435 from the Competition review.

The next posts will be about Hodoku coverage of chains. Some topics are handled well and the examples are very worthwhile. Unfortunately, Hobiger’s notes continue to describe only an inner loop in the coding of Sudoku techniques.

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Hodoku Sue de Coq in 2015

This post reveals the Hodoku view of Sue de Coq, reflecting an earlier expert consensus, to be dated. His classic examples are good, but the examples of Hodoku’s “Extended Variation” of Sue de Coq are better interpreted as simply the use of multi-cell ALS along with bv. Except for the last one, which steps outside of the traditional definition to expand further the role of ALS in Sue de Coq. Any update of Sue de Coq must now  include the SASdC, the single alternate form that is a basis for trials. The concept of trials is a Hodoku update issue in itself.

Hodoku begins the Sue de Coq page with the statement that SdC is a variant of subset counting, a mechanical search process totally unsuited to human solving. Actually the question is, what isn’t?

Hodoku SdC 2The examples begin with classical SdC with chute matching bv in box and row remainders. Hodoku covers the two classic types, single clue with four numbers and the clueless with five. In his definition, Hobiger makes the unnecessary restriction that only one of the bv can occupy a remainder. And he omits the two clue, single free cell variation with a single alternate.

Here is the second example, a clueless classic with chute SEr9 contents described by 4(2+7)(5+6). This type of logical description is missing from Hodoku, as it was from the original SdC post.

Extra 2’s are removed from the box remainder and the extra 5 and 6, from the line remainder. The chute must contain true 4, therefore the remainders don’t.

“Sue” thought that SdC was soon to be overshadowed by forcing chains. Experts of the day thought it rare and overcomplicated. Hodoku goes along, filing Sue de Coq as the only topic under Miscellaneous – an undeserved snub. SdC is an easily spotted technique exploiting bv. It is among the first techniques I look for on the line marked grid.

A mistake common in Hodoku’s time was to overlook the role of multi-cell ALS in Sue de Coq. The Hobiger description, and presumably his program, continues to confine “basic” Sue de Coq to bv. In contrast, Andrew Stuart in his The Logic of Sudoku (2006) features a two cell ALS in his primary classic SdC example.

Hodoku followed the wrong experts, and missed multi-cell ALS, even though he was demonstrating elsewhere a command of ALS.  Citing an elaborate formal definition from programmer forums , Hobiger proposed an “Extended Variation” SdC, which is actually classic SdC with multi-cell ALS. I’m revising my Sysudoku Sue de Coq post, to include one of his examples. Here is the other one.

Hodoku SdC 4In this triumph of puzzle composition, two ALS define the classic Sue de Coq alternate pairs, eliminating 1 and 9 from the West box, and 2 and 3 from the c1 column. Of course ALS have the disadvantage in the SdC of using up cells that could have removals. But since an ALS can give up one number only, they have the advantage of containing true candidates of any extra number. Here that advantage is illustrated in both remainders. The green ALS supplies (1+9) and must contain 5, removing three in W. The blue ALS supplies (2+3) and must contain 4, removing five candidates, and confirming SE4, SE2, and W4. Tell ‘em where you saw it.

It wasn’t on Hodoku. That version currently misses the remarkable reach of the blue ALS and 4r8c1. His code for the “Extended Type” of Sue de Coq removes only the other 4 in remainder, 4r7c1.

A similar, but less spectacular failing occurs in the next Extended Type example, which you might like to analyze yourself. Again, it’s a classic Sue de Coq with two-cell ALS in place of bv. I’ll checkpoint you on that one next post.

Hodoku’s current last example in this section is something that Sue never imagined. It fits under the “formal “ definition of SdC that the forum experts devised to cover all cases, and which is quoted for your convenience and wonder on Hodoku . The definition allows a cell of the Sue de Coq chute to be incorporated into one or both of the remainder ALS. Here is the Hodoku example:

Hodoku SdC 6It is the contents of r34c4 that are described by a logical expression. The column remainder bv 39 is joined by the N box ALS 1678 in restricting those contents to the logical expression r23c4=(3+9)(1+6+8). We don’t need to wade through the formal SdC definition. What we know about ALS is enough:

Since the box ALS must supply one of 1, 6, or 8 to r34c4, the other cells of the box remainder cannot have the 1, 6, or 8. Or even the 7, because an ALS can give up only one number. The wipe out in the box is too cluttered to even draw. You can verify that the collapse follows immediately.

If you tried to transcribe this from the Hodoku site, you might notice that I treated the blue 7r8c8 as a given. Otherwise I don’t know where this clue comes from.

I take this chute raiding variation to be fully in line with Sue de Coq logic. I have yet to look for one of these in my bv scan, so my blog is dated on that.

Bernhard’s examples illuminate classic, ALS aided Sue de Coq well. He needs to put away his Extended Variation labeling with the ALS driven classic approach illustrated here, bringing ALS detecting routines to bear on it. And he needs to give Sue de Coq a better treatment on the Hodoku Techniques list.

Also, at the meta solving level, Bernhard needs to consider the place of Single Alternate Sue de Coq, and other logically constructed trials, in Hodoku. His starting premise for every technique, which actually conceals a large number of arbitrary premises, will be a major topic in this review.

By now I’m sure you are less than thrilled with Hodoku color coding as you go back and forth between web pages. A challenge for Bernhard to introduce curves and pencil mark coloring to Hodoku might appeal to him. Bernhard, you’re welcome to the Sysudoku drawing conventions. I do them by hand in ©PowerPoint.

Next time, I’m reviewing the current Hodoku Wings page. Hodoku dismisses the WXYZ as too rare for serious attention. You and I can agree, based on personal experience, but let’s put our order in for a Hodoku repertoire expansion into BARN and BNS under separate headings.

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Hodoku Unique Rectangles

This post lists the Hodoku UR techniques in a convenient reference sheet, and illustrates many of the examples in a more explicit Sysudoku format. I review Sysudoku principles for UR exploitation, and call attention to the Hodoku omission of forcing chain winks.

My review of Hodoku advanced methods begins with unique rectangles.  Hodoku compares well with the sources of my post of January 2013, The Unique Rectangle. Extended rectangles are omitted on Hodoku, but Bernhard goes further than either Andrew Stuart in The Logic of Sudoku, or the now defunct Sudocue site on UR types. I found this table useful in making comparisons.

Hodoku UR table

Hodoku UR type 2Type 1 is too obvious to highlight, but here is Hodoku’s second example of Type 2 and your homework. Did you find it?


My UR post notes how  Hodoku Type 3 logic resembles Sue de Coq.

Hodoku t3 n1Hodoku illustrates the more traditional technique of completing a naked subset. The roof extras are combined into a “cell” with two other cells in c8 to form a naked triple.

A Type 3 UR from puzzle 110 is among the highlight snapshots from Sheldon’s Master Class. The post offers interesting interpretations.


Hodoku UR t3 nastyHere is a simple Type 3 from the review of Frank Longo’s Absolute Nasty IV collection. In puzzle 71, the roof extras form a naked pair 35 in r6 to remove the raider 5r6c3.

Alternatively, 5r6c3 is found to remove both 3 and 5 from the UR roof, forming a deadly rectangle. Naughty, naughty.

Another often reported UR formation is the Hodoku Type 4, in which a UR candidate is slinked on the single side with extra candidates. Hodoku’s explanation becomes almost unintelligible, as Hobiger builds in the definition of a strong link without referring to it by name. This is happens because, for some reason, Bernhard only recognizes strong and weak links as components of chain techniques. It’s one of the now peculiar deficiencies Bernhard shares with mentor Paul Stephens.

Hodoku UR t4 #1Here is the first Type 4 example. Since one of the slink partners is true, it cannot be joined by the other UR partner, so both of the UR partners in the slinked cells must go. Arnold Snyder was in command of this one  in his “slipknots” treatise.


In identifying Type 5, Hodoku simply shows that it is possible for candidates to see extra guard candidates – those preventing the deadly rectangle – in diagonal corners or three corners. It’s hardly news, but if you doubt it, look at the Hodoku examples.

Hodoku UR t6 #1Hodoku’s last special UR , Type 6, occurs when there are extra candidates on the diagonal, and one of the UR partners is confined to the rectangle, as in the Hodoku first example here. Either UR candidate on the extras diagonal will force a deadly rectangle in the solution.

Hodoku t6 from Sheldon 120An unrecognized Type 6 UR was exploited in a different way in puzzle 120 of the Sheldon Master Class review. Following the Type 6 rules, diagonal 7’s are removed, giving two clues.

Missing this, I was led to see if any 4 seeing the two 4’s in r78c8 could also force out 6r7c8. It worked. Going back, the combined effect of the UR removal of 4r5c4 was the collapse of 120.

Hodoku t6 fcThe forcing chain removals illustrate a serious omission remaining in the Hodoku and Stuart UR discussions. With the “types” for guidance, sysudokies look for guard candidates that cannot be removed, and for raiders that can remove all guards. That way, they are open to forcing chains of all types from raiders to guards.

Hodoku t3 NastyForcing chains are a possibility where winks are needed in any technique.

Yes, I do have a case in which a Hodoku Type fails, but a forcing chain is found to exploit the UR. It’s a Type 3 wanna be from Longo’s Absolute Nasty IV #91, where no naked subset incorporates the extra candidates 2, 3 and 5. Seeing 5r4c3 remove extra 5’s then 3’s, I went on to discover it removes the extra 2 as well. Such behavior cannot be tolerated.

To close out the list, below is one Hodoku version of Stuart’s Hidden Rectangle. with a corresponding schematic on the grid above. One corner is free of extra candidates, but one UR candidate is slinked in both directions from the opposite corner. The UR partner in the opposite corner, by removing one end of both slinks, would force the deadly rectangle.


Hodoku Hidden URIt was a long post, but covered one of Hodoku’s best Technique pages. You might grab a copy of the Types table for reference.

The Avoidable Rectangle is the idea that guards against completing a deadly rectangle with existing clues cannot be removed, and raiders removing such guards must be removed. Spotting such opportunities without wasting too much time is the problem. Follow up after spotting isn’t.

Next I have some worthwhile observations to make on Hodoku Sue de Coq. You’ll find it on the Techniques page linked under Miscellaneous. In the order of battle, SdC is contemporaneous with UR. It deserves a better entrance on Hodoku .

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An Unusual Review

This begins a review of the advanced techniques of Hodoku, Bernhard Hobiger’s puzzle generator and solver, as described on the Hodoku site’s Techniques page. The post describes the Hodoku program and the special nature of this review.

Hodoku is primarily a learning tool for advanced solving techniques. It’s a drill facility, something like a batting cage. You can dial up curves, fast balls, or sinkers. The program will compose a puzzle to challenge a user on any technique selected from its extensive advanced repertoire, and solve it up to the point where the selected technique is needed. After finding the technique and identifying eliminations, you get a color coded grid displaying the principle candidates of the technique and its removals, with a brief note of explanation. A remarkable feature of Hodoku puzzles is that they usually come out of basic solving with an example of the ordered technique ready to be found on the grid.

From the example displays on the Techniques page, you can link to the pre-solved grid on which the technique is to be discovered, and try to find it yourself. You have to spot the technique on Hodoku’s keypad pencil marked grid. You can even choose to do the pre-solving yourself. The given clues are in black on the example grids. Generally speaking, line marking is a bit tough. In this review, I transcribe all Hodoku grid displays to Sysudoku notation.

Hodoku UR t2 n2 LMAs an example of all of that, here is the line marked grid for a Hodoku unique rectangle, Type 2. It’s the same as Andrew Stuart’s Type 2, in which two candidates of the same number on one side stand guard against the deadly rectangle of a multiple solution.



You could do a personal grid searching experiment by loading the example puzzle into Hodoku. Click on the grid for directions.

The conception and execution of Hodoku is admirable. But as you see here, the batting cage solving experience is not the same as you have on a newspaper or collection puzzle. One reason is the “composed to order” characteristic of the puzzle itself, but it is more the fact that you know what to expect. A batting cage is no replacement for a talented pitcher at game time.

The descriptions and examples the Hodoku Techniques page are there to identify and explain the repertoire of the system, which I believe, reflects the general consensus of human solving experts at the time of the Hodoku launch. Hobiger’s major influence seems to be Paul Stephens, whose recent books are reviewed here. Bernhard’s aim was to create accurate examples, and solve them by technique rules and to trace solutions, for the current techniques of the day. He did an excellent job of that.

Hodoku is not a puzzle collection, and there will be no rating table of ten preselected puzzles. However, a primary reason for the review is Hobiger’s excellent selection of examples to illustrate his training system. The review is an opportunity for me to recall, and link back to additions to human solving repertoire made since the Hodoku launch. It is an opportunity also to correct misconceptions often spread by computer solver programmers, Hobiger included, about solving Sudoku puzzles with neurons rather than CPU’s.

The Hodoku active training system provides teachable moments, but I don’t recommend any solver for your general use in your Sudoku adventures, as opposed to training for them. To my mind, it’s pointless to have your computer solve a Sudoku with mysteries to be discovered. If the solution were all that important, we’d just look in the back. We get to do it for fun. With the emergence of Sysudoku basic solving, especially the bypass, I wouldn’t even consider letting my computer clutter the grid with too many starting candidates.

The Hodoku batting cage doesn’t help much with basic solving. Examples show pencil marked grids in keypad style, with the advanced technique marked. In Techniques pages, Naked and hidden subsets are defined and illustrated. No basic procedure, with stages, is suggested. Not even number scanning. The candidates appear, courtesy of Hodoku.

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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 1Three of the wing bv are replaced by a single ALS! I didn’t panic, though, because the ALS supplies the not(25) & not(29) & not(59) logic of the WXYZ.

But wait! In the old WXYZ, a victim has to actually see the Z in all four wing bv. 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.

The victim 9 does see all 9’s in the new WXYZ wing, but why are these 9’s toxic?

Andrew doesn’t say, only attributing the elimination to a rule by a forum correspondent StrmCkr, which says, if effect, that in four cells 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.

Great rule, but why is that? StrmCkr doesn’t say either, but sometimes you only need the right question. The answer is a fundamental argument in set link theory, and is arithmetical. It is that each of three restricted common numbers can supply only one true candidate of the four required, leaving the “unrestricted” number Z obligated to supply one, which the toxic set victim sees.

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 WXYZ schematic to make clear that all such WXYZ obviously meet the StmCkr restricted common requirements. Four numbers, all but Z in restricted common.

But does that make the above example a WXYZ-wing? I don’t agree that it does.

Stuart Bent WXYZ 3Andrew’s follow up “WXYZ” examples drift even further from the old 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.

It is StrmCkr who claims that his technique is a WXYZ-wing. He uses w, x, y, and z in a display of all possible cases. In this thread the only detailed challenge was that his example could be covered by overlapped ALS, another non-wing alternative. There was some mild grumbling about calling it a WXYZ, as the thread trailed off.

The toxic set mini tutorial above suggests that anyone that familiar with base and cover sets should not have bought into StmCkr’s labeling his idea as a WXYZ-wing. And I hope Andrew backs away from it as well. Look at the derivation of the bottom line of the two methods, the toxic set. The candidate stripping logic of the WXYZ is reviewed above. It has little in common with StmCkr’s 4-to-3 toxic set logic. And besides, the StrmCkr proposal applies to the XYZ-wing as 3 numbers, 2 restricted , and can expand to 5 numbers, 4 restricted, and beyond.

I’m saying that here is a new method, not too hard to spot and easy to verify. It needs an apt name, and an identity of its own in advanced solving lit. Let’s get off WXYZ’s back and let it be. I propose giving Stuart/StrmCkr’s new WXYZ 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 for most solvers.

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, 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, then if an outside k1 sees all of them, the N-set could still be filled, because a k2 in each remainder could be true,. No toxic set.

In a BNS1 with a single common number (k), this is not possible. If all k’s are removed from the N-sets, no other number can fill two of the naked cells, so one N-set cell goes begging in a solution.

In a BNS0, with no common number, the N-set, including cells in the hinge, is locked. Candidates of any of its numbers are toxic. I would still not call it a subset, because its set is not a unit (house), but the N-set.

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

Clearly, Andrew’s examples above, are BNS1, as well as BARN. But every WXYZ is neither. Grouped AIC winks allow restricted commons outside of the box/line. When I get an example, I’ll revise this post to include it.

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 question is, why is the BARN any more worthy of moving in with WXYZ than Hanson’s BNS? Maybe ALS chains should move out as well, but that’s an issue for another day. If you start a thread about it, could you please send a comment to let me know? I don’t track the fori (high school Latin).

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.

Posted in Advanced Solving, Expert Reviews, Hanson, Stuart | Tagged , , , , , , | 2 Comments

Arnold Abandons Reason

This post calls out Arnold Snyder’s embrace of arbitrary guessing as a way around the inadequate advanced Snyder methods in Sudoku Formula 3. I also deplore his misguided endorsement of Peter Gordon’s Sudoku Guide.

If the puzzle survives the “Snyder Method”, Arnold advises that you “forget all the difficult stuff”. If it has plenty of bi-value cells, use Arnold’s ultimate weapon, the grandly titled Impossible Force. Arnold doesn’t say what to do, if it has only a few bv. Maybe that’s what Sudoku Formula 4 is all about. I’m not going to find out.

So what is the Impossible Force? Following in the footsteps of his mentor, Peter Gordon, Arnold doesn’t tell you what anything is, but only in great detail what you do to verify that his example works. Or in this case, he does describe the process, because it’s not difficult. Here is his overall description of how you “find” the impossible force: “You find it by taking any cell that has only two possible candidates. Assume that one of the candidates is the actual number, then follow the trail to see where it leads.” That means solving until you reach a contradiction or a solution? No, as you read on in Sudoku Formula 3, you discover that “where it leads” means solving until you reach a contradiction, a solution, or a “dead end”. Arnold doesn’t believe in learning “the difficult stuff”, so he reaches many dead ends. That’s why he goes for the impossible force only when he has many bv cells to try.

That excuse for arbitrary guessing is embarrassing enough, but Arnold also argues that the impossible force is actually a logic based method. He gives us several examples where the choice of the wrong bv partner leads to a conveniently quick contradiction. One of them is the preview puzzle #19, with the Sysudoku basic trace:

Form3 19 basic tr

Form3 19 swordfishArnold’s order of battle yields a swordfish which is quite evident on the Sysudoku 8-panel. His next comment, though, is telling. He notes the lack of more fish, slipknots, or classic cycles, then asks,

“Do we have to start looking for the really weird patterns like jellyfish or squirmbags? Do we need to find some erratic cycle pattern that’s not rectangular? Do we need to look for a different type of pattern that we’ve seen discussed online but that’s not even mentioned in this book, like an X-Y-wing, or and X-Y-Z-wing?” (Arnold’s names.)

Arnolds Impossible FarceArnold avoids this “difficult stuff” by “assuming” that 4r4c1 is the “actual number”, and following the trail to see where it leads. Conveniently it leads to a contradiction in a nice rectangular pattern, ending as 3r9c4 implies 6r9c1 in an “impossible force”.  Arnold then announces that the collapse from 4r4c1 can be verified by the answer in the back of Sudoku Formula 3.  Case closed, and Arnold moves on to more examples, even using Peter Gordon’s Repetitive Bilocation Cycle as a “too difficult” strawman he can bypass by arbitrary guessing. I won’t waste your time with that, but of what “difficult stuff” in Formula 3 #19 does Arnold’s “impossible force” leave him unaware?

Form3 19 railroadWell, the XY-chain railroad, would have shown him a large number of his “classic cycles”, and he might have realized why they are often ineffective in a cloud of related bv.

Since Arnold does “classic cycles”, he could do certainly find XY-chain ANL, if only he undersood alternative chain logic. He would know all about “hard stuff” XY-wings as a bonus.

Form3 19 three ANLAnd by being no more systematic than he is in pursuit of his “impossible force”, Arnold could find many ANL (almost nice loop) eliminations. Here are three of them, with toxic ends marked. Extensions to the inner chain remove 6 and 8-candidates, but the removal of the inner chain, the 4-candidate, removes the other two and finishes the puzzle.

And of course, with such an extensive network of bv, why not color? Here, Snyder has allowed himself to be mislead by the one writer he endorses, Peter Gordon. Gordon has him believing that Medusa coloring is a way of seeing where his guess leads.


Form3 19 coloringHe has totally missed an easy way to exploit the puzzle’s network of strong links, made extensive by the bv. That network is a fact on the ground, like the bv themselves. It is there regardless of which candidate Arnold guesses is true.

In this case, the easily applied cluster covers the bv field. It traps two candidates for one clue, and forces two green candidates in r8c1, and two green 6-candidates in r8, a color wrap that declares all blue candidates true. So easy. So decisive. Such a testament to the willful Sudoku ignorance of “experts” Arnold Snyder and Peter Gordon.  Instead, Arnold finishes his puzzle with another “Impossible Force”

Form3 19 iXY-wingOn a brighter note, I can report that Arnold’s #19 also produced the only example I have so far of an irregular XY-wing, and this with an interesting kicker. The wing, in red, is an XY-chain of length 3, with toxic end 6-candidates. The 6r6c6 victim sees one of them directly, and the other, by ER, illustrated by a grouped forcing chain. This is the kind of fun in store for Arnold when he invests more deeply in Sudoku.

Then look at what the iXY-wing removal leaves behind, a BUG +1 resolved by Stuart’s three candidates per unit rule.

Next, we defend the honor and living space of a seldom seen but still remembered friend, the WXYZ-wing. We hope to give two rivals their own apartments, so they can move out and leave our friend some breathing room.  After that it’s on to a Hodoku review. Now get out there. Summer’s waning.

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