Trial or Error?


This post compares Sysudoku trial strategy with the chain or disproof hypothesis strategy described in Bob Hanson’s Sudoku Assistant report. My opinion is that Bob’s strategy contributions and his Sudoku Assistant solving records are both compromised by a willingness to resort to trial-and-error. His acceptance of Gail Nelson’s blatant T&E method confirms my position on the T&E issue.

Trials have become an important part of the Sysudoku solving repertoire, as the targeted puzzles have reached extreme and monster class. These trials involve the assembly of set of candidates that would be true or false together, and marking the assumption that they are all true. A contradiction then has a good likelihood of confirming a corresponding set of candidates as true. To support trials I introduced a method of breadth first marking and tracing that permitted the trace writer to document graphically the shortest inference path to a contradiction.

HMEM trial gridA recent theoretical triumph of this strategy is the discovery of the double nasty loop that confirms the Hanson and Marans four slink loops and duplicates the “opening volley” SK loop’s removals in the HM Easter Monster.

Sysudokie trials are undertaken when direct methods seem to be exhausted, with one major exception being the verification of alternatives in the Sue de Coq. Other trials include colors, or alternative directions in chains, or alternative patterns, with one polarity being necessarily false when the other is proved true, and vice versa.

Bob’s SA report section, Hypothesis and (Dis)Proof, is about chains of two possible states, but he misses completely the idea of constructing as large a bipolar chain as possible, for trial of one or both sides. Instead Bob starts with an arbitrary candidate, assigns it an arbitrary true (Hypothesis) or false (Disproof) value, then marks chain nodes according to their confirmation or disproof of this guess.

Bob doesn’t seem to understand that neither the right or wrong state of his hypothesis or disproof guess tells us anything about the logic of the puzzle. It only tells us a piece of the solution. He compounds the error by suggesting his trial method as a way to solve a regular XYZ-wing. The point of this suggestion escapes me.

In Sysudoku bipolar chain trials, when we cannot identify the solution with simpler logic, we invest more of the puzzle constraints into larger sets of candidates. These investments are observed facts, not arbitrary supposes. If we have to go to trial, at least the conclusion has logical weight.

Even though it was not programmed into Sudoku Assistant When the report was written, Bob passes on another suggestion from a friend. I’ll be calling it bv bifurcation nets. Bob readily accepts this blatant trial-and-error method:

Pick a bv and simultaneously try both values as true, distinguishing by marking candidates confirmed by the two cell candidates. Bob says

“You often don’t have to go far in either direction to come to a satisfactory conclusion:

  1. The two logical chains converge to a particular value in a specific cell. Then that cell is that value.
  2. The two logical chains converge to two different values in the same cell, row, column, or block. Then every other possibility in that cell, row, column, or block can be eliminated.”

As he has been throughout the SA report, Bob is being unclear and incomplete. Let’s do it properly:

  1. If no contradiction is reached, candidates which both truth nets deem true are clues: candidates which the nets agree are false are removed.

This was the original rationale for forcing chain method that Sue de Coq thought would obviate her method. As Bob’s friend Gail Nelson undoubtedly did, and as Bob does here, the promoters of forcing chains from randomly chosen bv (Rule 1) carelessly or deliberately omitted the “if” clause..

  1. If no contradiction is reached, where the two truth nets confirm two different candidates in the same cell, all other candidates of the cell are removed.
  2. If no contradiction is reached, where the two truth nets confirm two different locations of a number in a unit, all remaining candidates of that number are removed from the unit.

Rules 2 and 3 (Bob’s 2) extend the power of the bv bifurcation nets, and may originate with Hanson . The idea goes even further. As they stand, rules 2 and 3 offer a systematic way to find eliminating and confirming ANL (almost nice loops). My post Digit Forcing Chains? revealed this Andrew Stuart fantasy in November 2012.

duel cell fcAbout the arbitrary choice of bv originating cell, I presented this Stuart example and made this comment:

“This simple AIC proves that either 3 in r1c2 or 9 in r5c2 is true, eliminating 3 in r5c2 that sees them both. Stuart picks the bv in r1c2 for his dual cell forcing chains, but the bv r1c9 and r7c9 could have served just as well. “

Stuart’s folly was to masquerade a  legitimate elimination method as a new method, marked as T&E by the arbitrary choice of starting points.

But note how the almost nice loop differs from bv bifurcated nets. The  ANL was constructed as a complete entity. The loop invokes only one of the winks from each bv candidate. The elimination in no way depends on the actual value of the any bv candidate.

Not clear on the difference? For a checkpoint next post, pick r1c9 and color 2 green, 8 blue. Then do the  two way truth nets that Bob describes. You can use colors, but this is nothing like Medusa coloring. Winks, not slinks, are in control. For contrast, we’ll also solve this Stuart example puzzle by a Medusa coloring trial in the next post.

But the bifurcated nets idea goes much further. If you had the time, you could initiate forcing chains in every candidate of a number in a line or box, as a way to find Stuart’s Unit Forcing Chains . I dismissed this idea as a human solving tool, but that is no barrier to computer solver programmers.

Getting back to the proper statement of the bv bifurgated nets list:

  1. When a contradiction is reached, then the bv value and all of its 1-3 conclusions remain, and the contradicting value and its confirmations are discarded. Be careful how you mark the removals of 2 and 3, so they can be reversed. Do not extend the truth net based on its removals.

Rule 4, the most likely outcome of a bv bifurcated truth net, reveals it as blatant trial-and-error method, requiring nothing but basic marking skill to get the solution of any puzzle that offers enough bv.

This leaves us with a profound question about the bifurcated truth net or multifurcated truth net:

If we expand the truth nets concurrently, and stop before a contradiction is reached, are we entitled to the confirmations and removals of rules 1 – 3?

Think about that for a minute. It is the kernel of the trial vs trial-and-error issue. I encountered this issue very early in the Sysudoku blog with Denis Berthier’s xyt-chains, arriving at rule that limited toxic sets generated by the rule that avoids T&E. This was possible because the origination of these chains is far from arbitrary. With this discipline, the xyt-chain becomes a legitimate extreme method, although I’ve not yet been desperate enough to use it.

In the trial vs T&E issue, there is Sysudoku ‘construction first and trial as late as possible’ on one side, and the ‘pick a cell or unit at random and generate truth nets’ on the other. I’m not judging, but for my own satisfaction of discovery, I consider arbitrarily located truth net results as forbidden fruit, with the arbitrary choice of origination as the poison.

In the last section of the SA report, Bob describes two levels of depth in SA’s backtracking code. +Depth is trying single entities per step, while ++depth is assembling entities into groups of identical polarity (true or false). For human solving of difficult puzzles, ++depth is the only practical answer. It adds spice to the human solving game, where it is a triumph to assemble trial sets decisive enough to cut the depth of trials to a humanly manageable size.

Bob ends his SA report with:

“Obviously this can’t solve all Sudoku puzzles. But so far no one has shown me a puzzle that can’t be solved using ++depth. Using the above techiniques together Sudoku Assistant has solved all of the puzzles I have thrown at it”.

These are encouraging words that support Sysudoku trials. Bob attributes the SA success to ++depth backtracking. The report offers no guarantees, but we hope that the SA codes order of battle promotes logically advanced and extreme measures ahead of ++depth trials, and that the implementation of +depth trials has been and will be resisted.

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Hanson’s Strong and Weak Hinges


Continuing our intense review of Bob Hanson’s web report on Sudoku Assistant, I found no useful additions to the Sysudoku repertoire in the next few sections, so they just get a mention in passing. Then we examine Bob’s simple but worthwhile human solving idea, strong and weak hinges. Also I unload on Bob’s Medusa variations.

If you’re following along in the report itself, I’m skipping the result tables on SA solving options at the end of Almost-Locked Ranges and the options on the level of ALS analysis in The Sudoku Assistant and Almost-Locked Sets. I’m not planning to run Sudoku Assistant myself.

This brings use to a truly befuddling section entitled 3D Medusa Analysis. After what looks like a reference to a network of forcing chains, we get to:

“What is significant is that in a “binary grid” every other cell along the grid either has one value or another — only two possible values. (The grid is made by specifically selecting only cells that have exactly two possibilities.) If any two such cells are in a single row, column, or 3×3 block, it is as if they were the ONLY possibilities for their cells.”

Is Bob talking about the bv map, or an alternating inference chain, or an XY-chain of bv cells? Are the “values” numbers, or TRUE and FALSE?   And is the last sentence describing a naked pair? This was covered earlier in the SA report.

Hanson binary 5 gridA translation of the example diagram that follows is:

From this, Bob must be talking about a strong link network with one candidate seeing both polarities. To my readers, it’s a coloring cluster and a simple trap.

OK, after admiring a 3-D picture depicting all slinks on all numbers, including bv cells in the vertical direction, let’s move on.

I thought I would finally see how Bob would explain alternative inference chains when I began his section, Weakly Linked Chains, but it was not to be. He is talking about AIC, but his treatment of winks and slinks is both restrictive and vague. He goes on about the “strongness” or “weakness” of chains and nodes, without explaining what he means. I gave up trying to make sense of it.

In the next section, 3-D Medusa Hinge I did find a humanly useful idea. It is based on this rather obvious observation: If two outside candidates combine to see all candidates of a number in a unit, then they cannot both be true. Profound, isn’t it?

Hanson strong hingeBob shows us what can be done with this idea. In his “strong hinge” example, the hinge unit is a box. The North box insists on

not(Xa and Xc)

but the forcing chains between them say that

Xa => Xc and Xc => Xa.

The only way out is not Xa and not Xb.

In Bob’s diagram, the forcing chain nodes are bv cells XA, XB, AND XC, suggesting that the technique requires bv cells. It doesn’t. Any forcing chain will negate its originating candidate. Like Paul Stevens, Bob is having cells linking, instead of candidates linking. It limits and obscures the technique.

Hanson weak hingeBob calls this idea a Weak Hinge when applied to a line. Once we agree that r2 contains no more candidates X, the row insists on

not(Xa and Xc).

Because of the wink, we can proceed only from Xc, but it follows that

Xc => Xa .

This time Xc is removed, but the news does not reach Xa or Xb.

It’s something to watch for, but just one more thing. The title of this section, 3D Medusa Hinge, suggests that a Hanson theme, the use of number values in a cell as a third dimension, applies to the hinge idea as well. In fact, Hanson says:

“Medusa variants also target cells in the same row or column, or target values in the same cell.”

To follow up on that, go back to the profound observation above, and imagine what the two outside candidates are, in these variations. Instead of fixing X on a single value, pick a row position. Then columns and number values are the other two dimensions. A box is the numbers in a range of 3 in the same column. OK, what is a crossing line? Now what is an outside candidate? Mind boggling, isn’t it?

Now consider this. The translation of the row/column version of a puzzle to row/number or to column/number versions is a simple computer algorithm, as is the translation back. All computer solving algorithms are good in that other space.

To me, Medusa variations of human solving techniques are interesting, but useless. But to a computer solving enthusiast, its hard core. I’ve been quite hard on Bob, but remember, he’s a leader in the computer solving camp. It’s not the same game.

Next time, I conclude the review of Bob’s Sudoku Assistant report. We’ll all be relieved. But fair warning, I’ll be on my T&E soap box.

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Hanson’s Polarity ANL


This post reviews the section of Bob Hanson’s Sudoku Assistant report titled Almost- Locked Ranges. It presents an effective solving idea, but with hazy definitions and a confusing name. Here I identify this idea as a familiar one in the Sysudoku repertoire, with spotting methods much more consistent with human solving.

Hanson ALR 1As usual, Bob leaves the definition of his “almost-locked range” to examples. His first one is built on the r6c9 removal from this distribution of 6 candidates:

The “almost-locked range” is identified as the intersection of r6 and c9. There is an ER change of direction from c9, enabling 6r6c9 to see both colors of the cluster. Unfortunately for this demonstration, there is already an obvious trap (light blue diamonds added) that confirms r5c7, making the ANL through r6c9 and the shortcut slink superfluous.

The idea of Bob’s removal is worth considering whenever any extensive AIC is constructed. Outside candidates seeing a member candidate directly may also have a forcing chain to a member of opposite polarity. That is the relevant idea!

But why does Bob describe the r6c9 intersection, in this case, a coloring trap on steroids, as an “almost-locked range”? “Range” was earlier used in the label “row/column range checking”, which could as well be called line checking. The section of that title was actually about boxline restrictions. In this example, it seems to be about line/line restrictions, and the ER periscope type of vision. My use of ER has been supplanted by the more general grouped forcing chain vision, which I found to be easily represented in a more explicit and consistent way.

So I’m going forward with the theory that Bob’s “almost-locked range” is a combination of direct and ER or forcing chain vision of toxic polarities.

Sudo FrankenBob’s next “almost-locked range” example is the Franken Fish, which I analyzed in a post of 8/05/12, based on Andrew Stuart’s reporting. It’s certainly a good example of a 90 degree change of inference direction, but it’s not an ALR. In the Franken Fish, or SudoPod?, a slink forces one of two aligned groups(x’s) to be have a true candidate x. This forces one of the aligned groups A or B to have a true candidate x. A candidate in any red cell sees a true candidate from every possible combination true candidates.

But the winks don’t work the other way. The red cell candidates only force contradictions, not a polarity. Another mislabeling.

Next in the report is Bob’s enumeration of 15 possible configurations of the ALR. They are the empty rectangle(ER) box configurations I have reported on, distinguishing incoming and outgoing directions by red and blue colors. This enumeration confirms that the “almost-locked ranges” is all about seeing both polarities of AIC’s, by direct and ER vision. So instead of “almost-locked range”, let’s call the general version of this toxic set elimination method the “polarity ANL” method. “Polarity” is a concept in common with Bob’s strong link chains and my coloring clusters. In the Sudoku inference notation the ALR becomes an almost nice loop (ANL), with the removed candidate between the adjacent winks.

Hanson tlink 1Now let’s examine Bob’s four examples, as transcribed for sysudokie readers. In Sysudoku diagrams, I’ll use the burnt orange coloring for the removal and the targeted link.

From the puzzle top1465 #250, 1r1c3 sees both partners of a naked pair. Along with the naked pair r4np15, it is probably the ER path in the N box, along with the direct wink, that signals a possible polarity ANL.

 

Hanson PANL 3In Bob’s second example, from top95 #20, the grouping of the ER box trades in two winks for a slink.

Hanson PANL 4

 

 

 

 

 

In the third polarity ANL example, did I miss something, or did Bob use a forcing chain instead of an ER box? I got the image, but I couldn’t find my magnifying glass. If so, I can give Bob credit for forcing chain vision in the ALR, although he never mentions that, or any other form of AIC.

 

Hanson PANL 5Bob’s last example, from impossible520, #517, is not self evident. Five 9-candidates have been mystically removed from the completion field by Sudoku Assistant, the ones in the red font. These removals create bv and allow a blue green cluster to spread into the 8-candidates. The ER periscope allows the green 8-candidates to see each other. But this is not a polarity ANL. It’s a color wrap!   Blue is true, and green candidates must go.

It makes a big dent in impossible 517, but it’s a different animal . No targeted polarity difference. The only element in common with examples 1 – 3 is the ER box, and the previous abandons that. I’m sure I’ve done this kind of color wrap, but if not, here it is.

And so, what Bob’s “almost-locked range” really is, remains a mystery. But I’m entirely satisfied with Hanson’s idea, which he illustrates, but does not adequately define, in his SA report.

Three of Bob’s examples of targeted polarities were single slinks. This suggests looking around each new slink and naked pairs for an AIC closing the loop. This distraction in the basic solving process isn’t practical.  Instead of scanning outside candidates for those creating polarity clashes, I have a better alternative. That is, in the advanced solving stage, to expand the polarities of AIC loops and clusters by AIC extensions with X-panel analysis and coloring. Then the polarity ANL manifest themselves as X-panel ANL and as color traps. An example is the nice loop cluster of KrazyDad Insane v.4, b.6, n.5 in the Insane review.

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Suset Sashimi


This review of Hanson’s Sudoku Assistant shows that his grid ALS discovery algorithm, which I have long used as a scratchpad algorithm for human solvers, is very good at finding effective finned fish and kraken fish, particularly sashimi. Readers were invited to analyze six remaining grid ALS examples from his SA report , using the suset scratchpad algorithm, for checkpoint in this post.

Hanson c39 krakenThe previous post ended with Bob’s second grid ALS example, a column finned swordfish sporting three box fin victims. Bob’s ALS combines susets 3/79, 6/278, and 9/2789 to form the suset 369/2789. Five potential kraken victims escape by not seeing the fin.

You may have constructed the ALS bottom up, like I did, reaching suset 39/279 for a kraken 8-wing. The victims are all kraken, and the escapees are different.

 

Hanson r14 krakOr you may have gone to rows with suset 14/458, and let them all get away. Sorry, every suset is not guaranteed to work.

 

 

Bob’s third example below is coded as “A 1s”, a link on his site bringing it up on Sudoku Assistant.   The column suset list includes 1/16, 4/17, 9/367. I started with 14/167, and duplicated his removal, 6r6c8, with no kraken victims.

Hanson A1s fin wingThe kraken forcing chain has 6r6c8 seeing the fin, and that is normal enough, but then this victim joins the three escaping candidates on a path verifying the fin. So the real reason for removing 6r6c8 is that it forces another candidate true and false, hence the diamond.

 

 

 

Hanson A1s cookedFor some reason, SA chooses the larger column ALS, 149/1367. Now the example is cooked more thoroughly, as four candidates in r6 force both 6-candidates out of r9.

 

 

 

Now for the four “outside” examples that Bob collected to show grid ALS at work.

 

Hanson 273From the collection impossible520, #237, with an ALS list of

16/258, 18/358, 58/357, 69/248

Bob picks 69/248.

The kraken victim also forces both 4 candidates from c1.

 

 

Hanson 202In his next impossible520 selection #202, Bob identifies 4r5c5 as the guilty party for removing all 4 candidates from c4, via column ALS 79/358. The sashimi kraken analysis finds all three Cr5 potential victims guilty, confirming 4r5c7.

 

 

 

 

Hanson 121aBob’s third impossible520, #121, provides a host of grid ALS and several versions of the truth. From a list of 17/159, 34/257, 39/247, 47/259, SA picks 27/259, and then banishes 4c2c3 because it forces c3 to 5 and 7 to 59 and therefore to 9, thereby forcing both 4-candidates from c1, a unit(c1) forcing chain removal. My question is, why r2c3? We can’t do that analysis for every candidate. 4r7c9 forces c7 to 5 and c4 to 25 and therefore to 2. So now we can look for the damage that 4r5c7 and 4r2c4 can do? It’s not humanly practical.

 

Hanson 121bSo now make the same suset selection and consider it to describe the sashimi 4-wing with fin at r9c7. Two candidates are now singled out for examination, as potential victims. 4r5c5 escapes because it confirms the fin. Bob’s victim, 4r2c3 is removed because it sees the fin, or if you notice that it also has a forcing chain that confirms the fin, it is removed because the fin has to be true or false. It is an Andrew Stuart digit forcing chain, or more simply, the removal of an almost nice loop.

 

But wait, the suset 47/259 describes two distinct sashimi 4-wings, depending on which row holds the fin.

Hanson 121cPlace the fin in r2 and 4r9c1 replaces r3c3 as a potential victim. It does see the fin, and is dismissed, but also because it verifies the fin by a different path, completing an ANL that removes it. The other potential victim, 4r5c5 escapes again, by confirming the new fin.

 

 

 

 

Hanson 89Bob’s last grid ALS example uses a row based suset 29/134, but he mentions only row 1 to point out the counter clockwise inference path that completes the removal of both 6-candidates from r2. Remember my posting that sashimi wings are skyscrapers? More reliable signposts to this removal are the 6-chain skyscraper or the simple 6-chain ANL.

But failing these, suset 29/134’s sashimi wing would uncover it (below), bringing along with it two kraken suspects to be questioned and released.

Hanson 89 sashimiIn closing this exceptionally long post, I commend you on your patience, and say that Bob’s examples, though incompletely represented in his report, do confirm the power of his grid ALS concept and confirm our admiration for his independent Sudoku thinking. I’ll be getting out my scratchpad in x-panel analysis to seek the sashimi among the thin lines.

But that isn’t all. Stay tuned.

 

 

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Hanson Finned Fishing with ALS


Here Bob Hanson’s technique of identifying finned fish through almost locked grids, as reported in Sudoku Assistant – Solving Techniques, is explained and recommended as part of the Sysudoku x-panel analysis. Kraken analysis, neglected by Hanson, is demonstrated to produce additional results with Bob’s examples.

In the previous post, I noted that Almost Locked Sets, or ALS, can be enumerated with the algorithm that I call the Suset scratchpad algorithm, that also lives inside Bob Hanson’s Sudoku Assistant. Here we are going to review a section of Bob’s SA report on the use of ALS to find effective finned fish. Finned fish violate the “lock” of n candidate locations along n lines, by having exta location, while nevertheless causing removals.

Hanson fin wing 0Bob’s first example is a finned 8-wing on two columns of this panel. Perhaps you’d like to find one of your own before peeking below. In the suset scratchpad starting list I selected the columns by increasing size, resolving ties left to right:

5   8     1     2       4       6       7

16, 25, 146, 126, 237, 1467, 1235

Susets are combined by taking the set union of column numbers and location(row) numbers. Our generated list includes ALS:

15   25     156     258

146, 126, 1467, 1256

That’s far enough. We’ve got two possible X-wings and two swordfish. Bob’s example is generated by the 25(columns)/126(rows) suset. Marking columns c2 and c5, a finned X-wing jumps into the boat. Now we search out the victims.

Hanson fin pvThe process is described in Casting for Regular Fish, posted 4/3/12. We mark the columns in a spare row(|) and mark the victim rows in a spare column(+), then mark potential victims(v) in the victim row and not in the wing columns. Bob’s victim 8r1c1 is the one in the fin box. My post Krakening a Finned Fish of 4/10/12 tells what to do with the other potential victims, and why. All victims go free except those that “see” the fin. That accounts for Bob’s victim, but for the others, the potential kraken victims, we use the full definition of “seeing”, including ER and forcing chains.

Hanson fin wing 1Every finned fish has potential kraken victims. I endorse Bob’s technique as a way to find finned fish, but why is it that he never mentions the possibility of kraken fish? Here is the finned 8-wing, along with the results of the kraken analysis. I’m doing it myself these days. Virginia is away at camp.

All of the potential kraken victims escape, by seeing all other 8-candidates in one of the fin’s (f) units. Thus when each potential victim is present, it makes the fin a legitimate competitor for an 8 location.

Let’s note that Bob doesn’t mention the fin box, and gives a different rationale for the removal. It’s that 8r1c1 would remove both 1 and 2 from the sets of the ALS, namely {126, 16}. From this, it appears that Sudoku Assistant analysis of finned fish is limited to the special case of fin box removals.

Hanson fin wing 2But tell us, Bob, what would happen if we picked one of the other grid ALS that we found? Oh, never mind. I want to do it right here, to demonstrate what a productive idea your ALS finned fish finder really is.

The 15/146 suset gives us a surprise. It’s another finned fish with a victim in the fin box. But it’s a different box and a different victim!

Maybe the c25 8-wing victim was enough to solve the puzzle, but this is getting fascinating. What about the two finned swordfish that our ALS suset list beckons us with?

Hanson k swordAgain, we learn something by asking. The 156/1467 induced finned swordfish finds the same ALS uncovered victim, this time as a kraken swordfish. This demonstrates that finned fish go beyond   fin box removals.

 

 

 

 

 

Hanson c5 fcBut the 258/1256 finned swordfish goes a little crazy, as my kraken analysis of the c25 8-wing victim 8r1c1 reports that it forces both 8-candidates from c5. Bob’s grid ALS example is valid but the removal is justified more directly as an Andrew Stuart unit forcing chain, with either candidate in c5 forcing it out.

 

 

 

 

Hanson fin swordBob’s second example, a finned swordfish on columns, is derived by

3/79, 6/278, 9/279 => 369/2789 .

I’ll bet you are curious about the alternative,

3/79, 6/278, 9/279 => 39/279 .

Go for it. I’ll checkpoint in my next post.

 

 

I’m curious about the further examples of Sudoku Assistant analysis that Bob presents in his SA report, so I’m devoting the next post to a checkpoint of your complete suset finned/kraken fish analysis of these examples. Get busy, there’s a lot to discover.

 

 

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Sudoku Assistant’s ALS Toxic Sets


This post reviews Bob Hanson’s illustrations on ALS toxic sets, in Sudoku Assistant – Solving Techniques. For these illustrations, Hanson uses the bent naked n-set examples of the previous post. Strangely enough, this extreme economy brings insights.

Bob Hanson designed his Sudoku Assistant solver to bypass by default the enumeration and analysis of Almost Locked Sets. His reasons are similar to my reasons for placing this task well back in the Sysudoku Order of Battle. For one thing, as Bob points out, there are so many ALS in the typical puzzle. They slow SA down so much, that unless the user requests it, SA bypasses them. Also, they produce toxic sets of various sizes, and the larger they are, the fewer removals they produce.

Hanson wing ALSBob re-uses the 145-wing of the previous post to illustrate the standard restricted common form of the ALS toxic set. You’ll find examples that are not regular XYZ-wings in my ALS Toxic Sets post of two years ago. If two ALS share a number and all candidates of that number see each other, the number is called a restricted common. Shared candidates of any other number in the two sets form a toxic set.

Bob points out that an XYZ-wing is also a pair of simple, related ALS with a toxic set of three candidates. The 1-pair is the restricted common, making the other 4-candidates of the two ALS a toxic set. Of course, we don’t look for this in a cloud of other ALS. It’s much easier to find as an XYZ-wing.

The example shows clearly why the other common candidate must be shared by the two ALS. One of the ALS ultimately gets the restricted common number, but we don’t know which one. The ALS that doesn’t get it cannot give up another number. The victim would take a number from both ALS.

Hanson dbl rc ALSThe second BNS example adds another ALS toxic set wrinkle. Bob uses it to illustrate something new: that a pair of ALS can share two restricted common links and thus become a single locked set, in which every number defines a toxic set. Bob doesn’t say why, but it’s because, in the solution, each ALS loses one restricted common number and gains the other. We don’t know which is which, but we know that neither ALS can give up a number.

In this case, the 5 eliminations are relevant, being directly attributed to the double link of the ALS. The 7-candidate that Bob left out also gets clobbered. As a glider pilot, Bob is a lot more careful.

In the next posts, I will be reviewing more ALS applications from Bob’s report. His interest in them may come from the fact that enumeration of ALS comes as a byproduct of the computer algorithm finding locked sets. That scratchpad list leading up to the locked set, or failing to find one, contains all of the possible ALS of the unit. It’s all there in Subsets by Susets. Pretty neat, eh?

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Hanson’s Bent Naked n-Set


This post examines Bob Hanson’s Bent Naked Subset rules in Sudoku Assistant – Solving Techniques, and provides a more relevant rationale – and  a more appropriate name – for his technique. Also, XYZ-wings not covered by the BNS are identified, as well as types of BNS not covered by XYZ-wings.

Last post, I objected to the use of the term “subset” in Bob’s name for this formation. I’ll use the term n-set instead, the n standing for the number of values in the set of cells. In the name Bent Naked N-Set ( or BNS) “Bent” refers to the two intersecting units, and “Naked” further restricts the BNS to n cells containing only candidates of n numbers. The cells of the intersection of the units is the “hinge”and the naked set cells in each of the two subunits outside of the intersection are the “wings”. Bob describes two types of BNS. I’ll label them BNS1, having exactly one of the n numbers common to the two wings, and BNS0 having no numbers common to the wings. One or none numbers in common is the requirement for toxic set eliminations

Hanson 145-wingFor his only example of a BNS1, Bob uses this 154-wing.

Then, where you’d expect a proof of this assertion to be, Bob explains why this simple XYZ-wing removes 4r6c3. His explanation lacks the generality of his BNS1 rule, and is a little off base even for the wing, stating that either value of r2c3 forces 4r6c3 out. Using a forcing chain argument to explain an XYZ-wing, to illustrate a BNS? All backwards.

Bob’s unproved BNS1 assertion reads:

“If a bent naked subset contains one and only one candidate k that is present in both of its nonintersection subdomains, k can be eliminated as a candidate in any cell that sees all the possibilities for k in the subset.”

We’re left to decide if he means one and only one candidate, or one and only one value(number) , in each wing. Bob often says “candidate” when he means “number”. I’m guessing the less restrictive “number”.

Hanson BNW1Is there a proof of Bob’s BNS1 assertion? After showing that an XYZ-wing works, Bob represents a bent naked wing by one of his generalized method diagrams. It represents the naked set candidates in each unit and both units. Bob explains that if all of the candidates of k, the number common to both wings, are removed from A and B, no remaining candidates can be duplicated to make up for the missing number in the naked set.  Do you understand what “duplicate” means here? Let me explain what Bob didn’t. I went around in a puzzled state for a long time.

The “one and only” condition of the BNS1 rule is necessary because Bob’s “subset”, the naked n-set, is not a subset. It does contain n and only n numbers in n cells, but some of those cells are in different units, allowing solved cells to contain the same clue number. This generally allows the true k-candidate to be absent from the naked n-set.

Hanson BNS2With two (or more) common numbers, k1 and k2, the k1 candidates of the n-set are not a toxic set, because in the final solution, a duplicate of k2 true candidate could fill a k1 cell in the other wing. But with only one common number in the wings, no duplicate is available.

This kind of bent naked n-set is a generalization of the regular XYZ-wing whose wings are attached by unit induced winks, including those whose victims see toxic candidates by means of forcing chains.

IN 415 136 boomerangIt does not cover XYZ-wings whose wing-to-hinge winks are constructed from forcing chains, such as this one, found in KrazyDad Insane v.4, b.1, n.5.  Here, 3r2c9 sees the hinge 3r9c4 by a grouped forcing chain, creating the 367 wing that leaves three removals when the smoke clears.

 

 

 

Hanson BNS0Now consider Bob’s other rule, for BNS0. If there is no common wing numbers, the naked n-set behaves like a naked subset. That is, candidates of every number form their own toxic set. That is clearly the case, because all candidates of any n-set number are in the same unit.

The BNS0 is nothing like the XYZ wing, because its wings have no common number.

Hanson BNS2 exBob provides a real BNS0 example. In his example grid, three of the completion candidates are missing, namely theshaded ones 9r4c3, 7r4c8 and 2r9c8. The omissions don’t affect the example. There is a self verifying Sue de Coq Wr4 = 4(1+7)(8+9) +489 that removes 7r4c8.

The BNS numbers are 15789. The wings and intersection are marked in blue, green and orange.Notice that 89r4c8 is left out of the naked set. This in itself shows the naked n-set is really is quite far from a subset.  By the BNS0 rule, 9 has to go because it sees all 9-candidates in the naked set. It was very convenient to be able to leave out the r4c8 cell.

Bob points out that this 9 would also reduce Wr4 to 471, reaching a contradiction. That’s interesting, but not actually relevant, because we are looking for logical methods of making removals, not for candidates that cause contradictions.

To summarize:

A Bent Naked n-Set, or BNS, is a set of n cells with candidates of n numbers, contained in two intersecting units. Two types of BNS produce toxic sets:

In type BNS1, the two wings contain one number in common. The BNS candidates of the common number are a toxic set.

In type BNS0, the two wings have no number in common. The BNS candidates of each BNS number are a toxic set.

I have no generalized BNS examples yet. Before analyzing Hanson’s report, I was not aware of that possibility, and thought that Hanson’s bent naked whatever was simply an XYZ-wing. That’s what out-of-the-box thinkers do for us.

But now that you’re on to it, I’d be happy to publish your examples, with full acknowledgement, of course. You can attach to sysudoku@gmail.com .

Next we explore Bob Hanson’s views on almost locked sets. Without looking, you could review the Sysudoku posts of July 2012 and anticipate what he’s going to say about the ALS in the example above.

By the way, there’s a new page, titled Find It. There, you will be able to scan titles and key words on a complete list of posts, to find the one back there somewhere, that you want to review again.

 

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