Touring the Bypass and a Human Engineered Inscription

This post continues the visit of Sudoku Satisfaction readers to Sysudoku with a demonstration of the slink marking bypass, line marking inscription, and basic subset spotting.

How was lunch?  . . .  Good. OK, we’re ready to continue. Here is the grid of our puzzle, Saturday 2, after Sysudoku box marking.

This box marking, as is customary in Sysudoku now, was performed in two phases.  The first phase, slink marking bypass,  is essentially box marking without writing the slotting pencil marks for slinks and aligned triples.  We do mark naked pairs (twin pairs) or other subsets (partnerships).

New York Post Sudoku puzzle maker Wayne Gould is the inspiration for the Sysudoku bypass. He advised “shaking free of pencil marks” in order to see the true beauty of Sudoku.  After the bypass, we do put the pencil marks in, but it is a joy to go as far as possible without them. 

As it turned out here, the bypass produced all of the clues (uno’s). The pencil marks above, except for the naked pairs Enp14 and c2np59, were added  afterwards. Later, on your copy of Saturday 2, read the bypass trace, accounting for every effect. In the bypass, the numbers are still taken in increasing order, but combined into one list.

Many of the bypass effects depend on the slinks shown above, but pictured  mentally while doing it.

In Sysudoku box marking, we bypass more than the slink marking. We do not attempt crossing or c-u-c on lines of more than three unresolved cells. It’s not that we’re lazy. It’s because we have a very efficient lazy man’s inscription process, that incorporates partnerships.

Before taking that up, let’s talk about the kind of partnership we can handle in box marking, i.e.  before inscription. Partnerships are more generally known as subsets. A subset is a set of n cells of the unit containing candidates of exactly n numbers. In the solution, each of the n cells will contain one of the n numbers.

The naked pairs (twin pairs) we’ve been discovering are naked subsets with n = 2. They are marked when two slinks of different numbers converge on the same two cells. Naked (plain)subsets are so named because there are no other numbered candidates present. In box marking that can happen with higher values of n, as long as we know that other numbers cannot be added later to the subset cells. So we can have naked triples, naked quads, etc.

Of course, when you spot a naked subset, you can remove any of its numbers from all other cells of the unit. If there are any other cells. If not, it’s not exactly a subset, is it?

After inscription, when all remaining candidates are known, hidden subsets are also possible. This occurs when candidates of numbers other than the n numbers required are present.  How do you spot them?  Look for n numbers are confined within n cells. To confirm difficult cases, Sysudoku provides a scratchpad algorithm to find them. When you find one, all the other numbered candidates are removed from the subset cells. These cells are spoken for, and the extras are locked out.

Let’s move on to line marking the Saturday 2 grid above. As we go through it, I’ll point out the best time to look for subsets of both kinds. After that, new ones can appear when removals are made.

At first reading of Satisfaction, I thought that inscribing was done to end basic solving with all candidates, and partnerships came after that. Then I came to realize that inscribing and line marking have the same input and output except for the c-u-c and crossing done before that. Then it finally dawned, why do the harder c-u-c beforehand? You’ll see in line marking a better way to do it.  I crossed off crossing ahead of line marking very early on, when first announcing line marking in 2011. I showed that no naked single escapes the normal line marking. And the sweet part is that you can spot your other inscribed subsets in the line marking.

So let’s look at the grid above. Computers have a simple algorithm to do it, but most people number scan every unresolved cell. That means crossing each one, and the box slink pencil marks are of little help in doing that. Only marked unos and marked subsets are exempt.

OK, let’s at least do the units with fewer unresolved cells first. We may add unos and make eliminations on the harder ones. But we can do better. Let’s do lines only, in which the cells have largely the same needs. Every cell gets covered, right? 

Heck, let’s write down the numbers needed to fill the line, and apply the list to every cell.  If we start with a copy for every cell, we just take away those appearing in the crossing line and box of that cell, and we have the candidates to inscribe the cell.

Good grief! In ©PowerPoint I can place the fill needs in a string on the side or bottom of the line, make a copy with a Cntrl-C, paste a copy in the cell with a Cntrl-V and edit out the unwanted digits. In this systematic process, the slink marks help a lot. These are numbers already placed. Slinks along the line eliminate digits from the line’s fill string.  They’re already marked. Numbers in three boxes, as clues or box slinks, are omitted from the fill string.

Now on top of all that, go back to that original thought and line mark first the lines with fewest unresolved cells. You could do it from here without any more coaching. Here is the trace:

You start with the 3f: list.  The label means three free (blank) cells. I go down rows, then left to right on lines of the same number of free cells. The order can change when unos and subsets occur. You may have to refer to the finished product below the first time through, but it comes naturally very soon. You may want to write for a ©PowerPoint template (free). See the Tools link on the menu line above for the email.

The event on the row 9 marking is a box/line interaction where the lone pair of 5’s in r9 in the S box means no other 5’s can be in the box.  The lone 5’s are slinked because one of them must be true, for r9’s sake. 

The last list names the lines in the cross direction that are left when lines of one direction is completed. Even though all cells are marked, extra tasks performed in line marking remains to be done for these lines. 

One of these is marking the cells containing two candidates. They play an important role in early Sysudoku advanced methods.

Also, there is the repositioning of marks to denote line slinks and triples. An example is the pair of 8’s in c1. In this, box marks retain their positions. A low corner mark is especially significant, and the partnering box mark is easily found.

And then there is the review of the marked line for subsets. If the line is the third to cover the bank or tier boxes, it is the right time to examine these boxes for subsets. This closing action on the SW box would have shown the alternate form of box/line when the 5 slot takes out 5r7c6.

The line marking ritual also includes another action for every new line slink. That is to spot a matching line slink in a parallel row. That would uncover a X-wing, the first fish you will find in many puzzles. That’s the case, for example, in Satisfaction’s Friday 2.

Break time. You get 30 minutes. Bathrooms are down the hall to the right.

We’ll be back (in next week’s post) for a quick look at some of the advanced methods and tools that follow line marking in the Sysudoku Order of Battle, and contribute to the solution of Saturday 2. We’ll also show how some of them are covered in the Satisfaction Solution Triangle methods.

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The Sudoku Satisfaction Tour

In the next three posts, I introduce Sudoku Satisfaction  by Richard Nicholas Seemel, by inviting sysudokies to attend a tour for Satisfaction readers. The tour will highlight differences between  two systems that agree on basic solving approach, but not on its fundamental procedures.

Last November I was delighted to receive and examine an autographed copy of Rick’s Sudoku Satisfaction. Rick is a retired engineer(civil), and clearly values the human engineering of Sudoku solving.  This is the first opportunity I’ve had, after Hidden Logic, to review this book.  Rick understands what I do, and has the generous spirit to encourage me to do it with his book. My review will be in the form of a tour for Satisfaction readers, explaining what they can do with the basic solving knowledge and experience gained from Sudoku Satisfaction.

Before the Satisfaction people arrive, a word about their book.

Sudoku Satisfaction is thin on the shelf, but packed with definitions and examples, mostly on basic solving.  Like another one-of-a-kind, Carol Vorderman’s  Master Sudoku, Satisfaction does not acknowledge advanced  techniques based on linking relationships between the remaining candidates(canos), the techniques I have spent so much of your time on.

A mark of Rick’s independent approach to Sudoku is a barrage of coined terms for many basic  concepts. I cope here by placing Rick’s term after my customary one where it first occurs, and outside of that, adding quotes around the Satisfaction coins, as in “canos” above.

Satisfaction has an Order of Battle (the guideline), somewhat like Sysudoku basic, with phases that are followed systematically.” The “guideline” is designed to find easy clues (unos) before number scanning(inscribing) the remaining candidates into cells(squares). Scans are done repeatedly, consisting of an “inspection” to identify the most promising box, followed by the chosen action. Actions consist of modified slink marking (slotting), box marking (lacing), line filling(completing the count), or a number scan on a cell (crossing). When no further actions seem possible, a full number scan (inscribing) is performed. Slotting leaves aligned slinks and triples in place, for use in inscribing.

The one major difference is that, in Satisfaction, subsets (partnerships) are “advanced”, i.e. defined following “inscription”.  Satisfaction does a form of pencil marking, placing “cano” digits in keypad style. They mark aligned box slinks and triples, calling it “slotting”, and . . . Uh Oh, they’re coming in. Please move back and give them room in front.


We’re so happy you could come for a tour of our Sysudoku resolution factory, where we manufacture solution traces and diagrams for the toughest Sudoku puzzles in the world. We’re standing now at our box marking line .  Down there is our puzzle loading bay.

We’re starting the tour by walking through a Sysudoku box marking of your 4-1 example puzzle, looking at both grids and traces.  Box marking “flow” is by increasing number, so here we’re missing Satisfaction’s 9 run and resulting 1 run, but on completing the 4’s, we are making progress, with “uno”s and “slots” on the grid.

Actually the pencil marks are not exactly slots. Most are aligned strong links defined by boxes.

Candidates of a number in a unit form a strong link when there are exactly two of them in the unit. If one is found to be false, the other must be true. That’s the logical definition of a strong link.  Like Rick, Sysudoku uses selfie names for important concepts. To us, strong links are slinks.

We also mark triple candidates in a slot as in Box 6.  I should tell you, to us, Box 6 is the East box, or E. Boxes are named for compass points, with C in the center. So E appears in traces, not MB RT (Middle Band, Right Tier), or B6.

Now let’s look at the corresponding box marking trace. This is how it starts:

The trace has a list of marking effects for each number. I know you’re used to scanning combined effects of numbers, one box at a time. We look for “lacing” effects of clues and slinks on all boxes, for one number at a time. When they sweep into boxes along a bank or tier, we call it a double line exclusion, or dublex. When they come from two directions, it’s a crosshatch. But we seldom use these terms. Here’s why:

In Sysudoku traces, every effect is shown, so we must be brief.  Compared to Satisfaction, we save tons of space by reporting the “what”, but not the “why”.  In Sysudoku basic, the “why”  is supplied by the reader. They know, but don’t particularly care, whether it’s a dublex or a crosshatch.

Each effect depends on the state of the grid at that moment, so looking at the grid above, you need to know the state it was in as you read every effect. For that reason, you read a trace by starting with the grid of givens, and filling it out as you read. Also, the trace leaves the exact cell (square) of the box unspecified.  Of course, you have to know some simple abbreviations as well. The “m” stands for slink “marks”, “t” is “triple”, “np” is “naked pair” (twin pair). A number alone marks a “uno”. The “m” with no number means it’s the list number.

Sysudoku traces show cause and effect, and the order of solving, what you know as “the flow”. Causes have their effects listed beneath them, indented to the right.  When there are multiple effects from a cause, the list of effects is in parentheses. Causes are slid to the right to create space for their effects beneath them.

In the trace, Sysudoku flow is therefore left to right, and depth first going down. Every effect from a cause is explored before the next effect on the list becomes a cause. The 5: trace illustrates all of this well. Now it’s time to get out the handout on trace reading.  The grid is already posted for lists 1 through 4, check this out for a minute, and we’ll follow the roller coaster on list 5. . . .

Ready?  After the box slink in NE, a triple in C makes a small run. Next, we post SW5, and its four effects. Only S3 continues the run. Do you have a good eraser? OK then, in the list of S3 effects, post the S9 and S8 as pencil marks in the center of the cell (square). You can replace them with full size digits when they become causes. It’s to keep track of where you are. Meanwhile you have the SE3m effects to post, with small font SE3, unless you anticipate that NE8m has no effects.

C3 ends the run, and we go up to promote S9 to full size, then S8. These font size changes are less messy for me, because I do it all in ©PowerPoint.

That’s another point I wanted to make. Sysudoku is for tough puzzles. Pencil and paper is fine for easy puzzles or casual solving. But the art of tough solving is detailed and error prone. ©PowerPoint is a great pallet, tracking system and record book. Many resolutions go on,  sheet after sheet.  The sysudokie focus on tough puzzles explains the special machinery on the floor here. We display selected information and construct candidate (cano) networks to enhance the vision of the human solver, and minimize ineffective searching.

I see that everybody gets the idea, and most of you finished posting the handout grid, while I was rambling on. Here’s what it should look like now, less a few smudges.

Rick’s run of the 9’s in Satisfaction 4-1 shows up in two places. At SE7 we had an hidden single c9 for a NE9, but when I looked on the trace  for the last move that enabled it, I discover that it was there in the givens! You can do that kind of thing with a trace. In the previous grid, we had another earlier cause for the single NW9 from a line marking (c-u-c) on c3, but I just left it in reserve for the 9: list. I do that for numbers higher than the current list, except when the unit filled had three unresolved (blank) cells (squares) or less.

Question from the back? . . .

Yes, you see the 8r8c6, and what happened to the crossing that found it in Satisfaction? Well, we don’t do number scans(crossings) in box marking, simply because it’s harder than box marking, and we can do it easier later. Number scans are done a line at the time, in line marking, the major stage after box marking.  There, we take advantage of box marking result, and common fill needs of remaining free cells on the same line.

That brings me to the other major difference in basic solving between Sysudoku and Satisfaction. I mentioned the strong link being important enough for a coined name in Sysudoku. Another type of link between candidates has one, the weak link, or to us, the wink.

The logical connection between wink partner candidates is, well, weaker. If one is proved true, the other is false. They can both be true, or both false, but they cannot both be true. All candidates of the same number in a unit are weakly linked.  Another term used for winks is “seeing”. Wink partners “see” each other.  The slink is stronger because, if another candidate “sees” either partner, it is doomed.

Seeing is the essence of the melting of partnerships illustrated in Satisfaction.  It is also the essence of commonality and crossing square patterns. A square “sees” like numbered candidates in its CSP. Sysudoku concentrates more on the logical properties of candidate links, because many advanced methods depend on winks and on chains of candidates that alternate slinks and winks to extend seeing across the grid.

I’m hearing that lunch is ready, so I’ll just say that 4-1 collapses in marking the “6:” list. The trace gets you well into the collapse. Here is the solution, in case you need it. On the way  to lunch, please wash your hands




After lunch (and to be reported next week), we’ll demonstrate the slink marking bypass, and line  marking, a more efficient inscribing procedure, and how “partnerships” are handled in Sysudoku basic, not after it.

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A Closing ALS Postscript on THLS

This post demonstrates ALS-XZ as an alternative to hidden XY chains, and concludes the review of The Hidden Logic of Sudoku by Denis Berthier.

In THLS, Denis goes outside the Royles 17 collection only a few times for illustrative examples. Sudogen 17-3403 is from a large collection generated by random and analyzed by SudoRules. It was used to compare two “resolution paths”, one with regular xy chains of 7, 8, ,13, 11, and 4 cells, and a hidden XY chain alternative path with chains of four cells, including two XY chains, a hidden XY chain, and an xyt chain. It was selected to thus demonstrate the superior logical simplicity of hidden chains.

sudogen-3403-quod-bxlWith advanced fireworks to come later, let’s start with a checkpoint of sysudokie basic on 3403. Here is the almost completed line marking grid, with its aligned triple and naked quad induced box/lines.

On reviewing Chapter XIV on XY chains, I passed on the 3403 example, the point of it being the reduced logical complexity of the shorter hidden chain path.

The Sysudoku railway extends XY chains with so little effort that this measure of logical complexity is itself obsolete. There is not rationale here for maintaining and reviewing the extra grids required for hidden chains.

Then I received a New Year email from sysudokie contributor Gordon Fick with the outstanding ALS-XZ of the January 17 post. It was accompanied by Gordon’s findings for 3403, including an ALS toxic set barrage made possible by the bv rich environment of the THLS examples. I decided to include it in the review, to counter the impression that highly complex example might have conveyed about the difficulty of the these examples.

sudogen-3403-urFirst, there’s an instructive Type 2 UR (see Tools page) that must wait on the THLS last resort tarmac by Berthier decree.



Next is Berthier’s 7-cell XY chain:








sudogen-3403-xy13and a shortened  version of the  xy13, courtesy of the UR.

In the thinned out grid, ALS master Fick begins to work his magic.







Two column ALS share single 8’s for the restricted common, and 1r3c9 sees all three 1’s they contain.  The 8-victim sees both ends of the restricted common. It looks easy, after you find it.





Two more ALS-XZ trigger a precipitous collapse, bypassing the 11 and 4-cell XY chains. The 7 and 8 victims see the toxic set; the 2 victim, the red/orange restricted common.

Or so I thought. Actually we have to deny the red/orange ALZ-XZ that victim. My diligent proofreader, Guenter Todt,  pointed out that the collapse from (Ehs8,Chs8) leaves 2r7c9 as a clue! So what went wrong?


The problem is my mistaken notion (not Gordon’s) that the restricted common candidates of one of the ALS must include the true RC candidate.   Reader Mittleman set me straight on that.  I’ll have to go back and figure out how I argued myself into that one, and what earlier damage that might have done. Guenter also caught my less serious error in leaving out  the 8r8c5 removal.

Just restore 2r7c9 above, and the collapse follows. The size of it is untypical of THLS examples.




Anyway, the point is that  ALS-XZ is a much better investment than hidden logic for the human solver. Now to close this very long review with a summary of its conclusions.

First, I do thank Denis Berthier for The Hidden Logic of Sudoku, a monumental effort bringing AI expertise and scholarly experience to bear in a unique way.  What Denis disclosed of his background and intentions were reported here to establish that this work, while succeeding as an AI project, was not intended as instruction in human Sudoku solving, and doesn’t serve this purpose well. In fact, it was undertaken little regard for practical human solving techniques available at the time.

Also, Berthier was not following the expert system practice of capturing the skills of an expert in a set of rules. Instead, he was expressing his own ill informed expertise in human Sudoku  solving.

THLS directs little attention to basic solving, the derivation of candidates. Its only detailed disclosure was reverse engineered in this review to reveal a rigid algorithmic technique well suited to computer solving, but outstripping human time and patience. Unreasonably, diligent readers of THLS must utilize their own basic technique to follow the “resolution path” traces of THLS examples.

It is ironic also is that most of the solving in in THLS examples  is basic solving. The candidate field generated by these puzzles is generally a simple one, with may binary valued cells.

The primary thesis of THLS, that symmetric versions of the puzzle state offer unique solving techniques unavailable to human solvers on the standard Sudoku number/row/column grid, was thoroughly refuted in this review, using THLS examples.  Mapping of hidden XY chains to AIC in nrc space was demonstrated.  Mappings of regular fish to subsets in Suset space, without symmetry grid maintenance, was also demonstrated, refuting another claim for the necessity of symmetry spaces. Alternative solution paths were displayed in detail for all puzzles selected to support the claim of necessity for hidden logic.

This review recognized THLS xyt and xyzt chains as possible human solving tools. The speculative nature of these chains was noted, but they remain useful as extreme resources. It’s just that a comprehensive exploration of every linked bv pair for a possible “t” chain, while reasonable in a computer solver, is not practical for a human solver.

The xyt and xyzt chains are extensions of XY chains, but the innovative involvement of the starting value and the victim in these techniques actually applies to all AIC types. That is a broad, if seldom needed, avenue available to the human solver, compared to the embedded XY railway offered by THLS.

In my view, Berthier’s effort was very hampered by the commonly held but mistaken notion linking cells by their candidate content, as opposed to linking candidates directly. He missed the general implications of alternating inference chains and the slink network of Medusa coloring. He missed finned and kraken fish, and almost locked sets and their application to toxic set elimination and fishing.  He mixed X-chain grouping.  He missed pattern analysis.

Berthier’s solver, SudoRules, is built on a unique and innovative technology, but it is one of many such systems that depend on the computer’s capacity for searching.  SudoRules does not emulate an expert human solver. It’s actions do not follow resolution paths applying well known humanly practical techniques, as was demonstrated repeatedly in this review. I am thankful to be able to report that THLS does not show that its “hidden logic” is necessary, or even suitable, for human solving of Sudoku puzzles.

Next is a brief review  of a basic solving guide by a fellow engineering retiree, Rick Seemel, titled Sudoku Satisfaction. Rick approaches everything a little differently, and with his own distinctive nomenclature.

Satisfaction comes with a two week collection of puzzles from Rick’s local paper. Does that sound like anybody you know? Think Dave Green. The review solves two puzzles, comparing solving concepts along the way.

example-4-1Here is the first, a puzzle that Rick uses to introduce Satisfaction techniques. You could have your own box marking ready to compare as we host a tour for Satisfaction readers.  The second post will solve the toughest puzzle in the Satisfaction collection, which goes beyond Sysudoku basic. It will serve to illustrate some techniques that could add advanced Sudoku satisfaction for many of Rick’s readers.

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Berthier’s xyzt Chain

This post concludes my reading of Denis Berthier’s The Hidden Logic of Sudoku, with two example puzzles to illustrate his xyzt-chain. While it is good to be aware of Berthier’s “t” versions of classic XY chains, I don’t recommend a comprehensive search for them.  I’ll explain why.

17-9373-basic-trHere is the basic trace of the homework puzzle, Royle 17-9373. It was selected to introduce the xyzt-chain in THLS.





To the requirements for the xyt, Denis allows one exception for the xyzt.  In only one internal chain cell, there can be an additional candidate of the starting cell number. This makes the construction of the embedded XY-chain dependent on the victim, along with the directional assumption of the chain starting with a false candidate.  A new thing in itself. Before using the cell, you need to make sure there is a potential victim seeing the starting candidate. Then the victim must see the ending candidate as well.

Does this work?  Yes, if the added candidate is false and the starting candidate is false, the chain continues and the end candidate is true. A true candidate cannot see all three. Having a 3-candidate toxic set is a disadvantage shared with the xyz wing.

Is this a good deal for the human solver? No. The xyzt allowance adds very little to the chain construction budget, in return for the victim’s  having to see a third candidate. Berthier has reported many of these xyzt-chains being built, in this a previous chapters of THLS, but I have yet to see one not cooked by a more easily spotted technique.

17-9373-xyztIn Royle 17-9373, Berthier’s “type 2” xyzt-chain is found on the line marked grid. Starting on r8c5 with assumption 5r8c5 is false.  Selection of r8c6 as the third cell anticipates a victim on r8 or c6, and gets the chain to r8c3 for the final slink.  So 5r8c7 if false, whether 5r8c5 is true or false, as it allows the chain to create the embedded XY-chain that removes it.



The only reasonable way to look for this is to start an xyt chain on every possible pair, go as far as you can with the xyt, then look for a linking cell that would be next, except for a candidate of the starting number, and a corresponding victim. 

17-9373-finned-5-wingThe xyzt removal  enables a  5-chain, leading to a collapse.  But here is a simple kraken 5-wing bringing the same collapse. One of the wing’s victims sees the fin, 5r5c7.  So the victim, like xyt and xyzt victims, is false if the fin is false (clean fish), but also false if the fin is true. The kraken logic is very similar to the logic of  the xyt-chain.

Pardon the side lecture, but if the fish’s victim is in the same box as the fin, it’s just the  finned fish we saw earlier in the THLS review. If its not, and forcing chain “seeing” is required, it’s a kraken fish.

17-4601-xyzt-1For another of Berthier’s “simplest” xyzt-chain examples, here is the first one in Royle-4601. The victim see’s the candidate that is preventing its removal by the embedded XY chain. Strangely enough, it is the terminal 9r9c2 that provides the rational for the removal. It removes the victim directly if true, and by XY chain with the victim’s cooperation, if false. You could call it a suicide xyt chain.

17-4601-xyzt-2The removal is indecisive, but another victim is strapping on the vest. The starting candidate partner 9r2c4 removes one barrier 9r2c9, but spares 9r1c1 for the assumption chain, with the victim stepping in to make its removal possible.

17-4601-xyzt-3Next in the THLS trace, but after the removal of 8r1c8 by a hidden hxy-chain, a third xyzt-chain, starting in r3c1 like the first chain.

This time the solver somehow sees the enabled terminal cell waiting and that a victim will allow the chain through r2c9.  In the defiance this flabbergasting spotting, 17-4601 demands  another xyt-chain and an xy-chain for its collapse.

But wait. There are much more reasonable alternatives.

17-4601-xy5The first one illustrates how badly Berthier’s logical complexity fails as an order of battle for human solvers. There is a simple XY-chain of five cells that wipes out 17-4601, all by itself. Sorry for displaying it, after having you laboriously trace out all these extreme four cell chains, but here it is, fresh out of line marking.


Or, with 7 of 9 boxes containing bv only, you might take an even easier path, with your crayons. Stay within the lines, please.

17-4601-coloringColoring the line marked grid, I have two small clusters connected by the 5’s. The bridging logic is easy:

Row r3 says,

not red and green.

Column 1 says,

Not orange and green.

But one of these is true. We must have red or orange, so it can’t be green.

With blue candidates, 17-4601 folds up like an accordian.

We’ve gone far enough with xyt and xyzt chains. For human solvers, it makes no sense to work through the uncertainties of the “t” chains when there is certain progress to be made with techniques Berthier ignores or rejects. This review has shown why and how these chains work. I advise trying to find them yourself to appreciate just how speculative they are. Then forget’em, until you are really desperate.

sudogen-17-3403The review of The Hidden Logic of Sudoku closes with the next post, with a return to XY chains for a reminder on ALS toxic sets, a.k.a. ALZ-XZ. This is another widely known advanced method ignored in  the design of SudoRules, a human emulating solver. The puzzle for this demonstration is one of those generated and selected by Denis Berthier to compare regular and hidden XY resolution paths, Sudogen 17-3403.


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Berthier’s xyt-Chains Rated Extreme

This post suggests a limited role for Denis Berthier’s xyt chain in human solving. I found no THLS examples of longer chains not requiring prior xyzt or hidden chains.The last prospect, Royle 17-33442 proves to be both unjustified and unnecessary. In the Sysudoku Order of Battle, the xyt chain is therefore will be rated as extreme, and place among last resorts. Being aware of how they work, you might stumble on a promising one before that, but systematic search is not practical.

17-33442-basic-trAs a reminder, Sysudoku basic derives its candidates with a minimum of unnecessary additions and eliminations. A slink marking bypass uses unwritten slinks to derive clues, then adds pencil marks for box slinks, then adds line slinks, finding subsets in the process.

Berthier number scans first, then removes candidates by subsets, mostly hidden, and box/line interactions similar to slink marking.

The basic solving of 17-33442 took some effort, but not as much as number scanning. And its product is much more ready for advanced solving. That is a major theme of Sysudoku.

17-33442-lm-grid-fake-8On the line marked grid, here is the path of SudoRules first advanced move, as reported in Berthier’s THLS trace. It is supposed to be a c-chain, eliminating 8r4c3.

But it isn’t that. Although THLS displays no candidates, we know that 8r4c4 to 8r2c2 is not a strong link, because the victim 8r4c3 is there to be removed.

17-33442-grouped-anlTHLS does not acknowledge what this actually is,  but it solves the puzzle, without help from xyt chains. It is a grouped 8-chain ANL, confirming 8r2c4 or eliminating 8r1c56, depending on which slink you put in at the top. It does eliminate 8r4c3, as required for Berthier’s xyt-chain, but then you have to stop it.




17-33442-xytNow if we go along with the removal of 8r4c3 without the rest of them, we do get to the same collapse by means of a longer xyt-chain. The xyt chain logic: starting in r1c3, if 8r1c3 is false, the chain removes 8r1c56, as interfering candidates 9r2c2, 9r4c2, 3r4c3,  9r4c1, and finally, r2c4 are removed.  And if 8r1c3 is true, the same two are removed, so they are indeed eliminated, giving us the same decimation of 8’s.


17-33442-collapse-trOne conclusion we can draw is that, not only do longer xyt become tedious and difficult, as Berthier acknowledges, their ultimate success or failure becomes impossible to predict. If you think an xyt is necessary, and you don’t see its completion before starting, you’re making an arbitrary choice, and flirting with trial-and-error.

We’ll never see it happen in advocate examples, but the repairing XY chain under xyt revision may just peter out. Or worse, the chain starting assumption may be contradicted without completing the full chain. In that case, assumption then becomes a confirmed guess. If the xyt chain from this assumption is not simple enough to be readily predictable, it’s a type of trial. Defer the search and find something less nebulous if you can.

royle-17-9373This THLS account on behalf of the real human solvers winds up with Berthier’s xyzt chains, a derivative of the xyt chain with family ties to the Sysudoku iXYZ-wing. I think regular readers will understand why I’m skipping the hidden hxyt-chain.

If you’d like to do sysudokie basic on the first xyzt-chain example in the THLS chapter introducing it, here it is, Royle 17-9373. I didn’t find the xyzt-chain when I got there, but I did find an easier alternative in the regular advanced repertoire.

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An xyt Color Wrap

In the spirit of Valentine’s Day, I come across a coloring resource in Berthier’s xyt chain.

The elaboration of Royle 17-20565 in Chapter XVII of The Hidden Logic of Sudoku is a challenge in itself. By the theory of looking for stuff where the light is better, I  look for line subsets as the line is marked, and the lines in closure don’t get examined until – you guessed it – closure.

17-20565-nq-1So with this one, I was in closure on column 5 when I spotted this naked quad. Of course the hidden triple is just as available at this point in Sysudoku basic.

Even if you were too busy sending Valentine cards, and declined the homework, you can appreciate how tough the line marking was, and can imagine how much worse number scanning and hidden logic transformations would be.

17-20565-nt-2So with some relief, I followed up, and got the next cleanup bonus.

If you haven’t gone further already, avoid looking at the grid below. Instead, mark the follow up on the naked triple on your own grid, and contemplate where you would start xyt-chains. 

The ticket to ride is a pair of linked bv, with a potential victim within sight of the starting candidate.

17-20565-two-xytThe SudoRules first xyt (black) makes a single cell repair in an XY eliminating 4r3c8.

This enables an xyt making two XY repairs and rounding two corners, for another elimination, and a hidden single in r6.

The follow up celebration ends with a skyscraper in the 7’s that you can easily spot (slink-wink-slink), then Berthier reports a hidden xy chain for the collapse.

17-20565-coloringInstead, we choose to invest the bv dividends in Medusa coloring, and manage to color most of the bv with two clusters.

The bridging logic is

Not(orange and green) =>     Red or blue.

And also,

Not( red and green) =>

   Orange or blue.

Oops, that means blue is true, because

(red or blue)& (orange or blue) => (red or orange) and blue => blue.

In trying the obvious, we stumble upon an insight: The xyt version of forcing chain logic is a recourse for coloring.

17-20565-xyt-wrapStarting on one of the few remaining uncolored bv, the xyt assumption “if 9 is false” then includes that red is false and orange is true, and in two ways, 9r5c3 is false. No need to go further. 9r5c3 is false, regardless of 9r5c7.  That wraps red, with two red 4’s in c3.

Remember the shortcut wink? Coloring applies in all the bv clover examples the Royle 17 series exhibits in THLS.

royle-17-33442We dig a bit deeper into the xyt chain next post. This time, our Royle foil is Royle 17-33442. Its another “solved in half the numbers” wonder.

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Berthier’s xyt-chains in THLS

Next we review the introduction of xyt-chains in The Hidden Logic of Sudoku. These are modified XY-chains which allow extra candidates in the chain cells, and unlike regular XY-chains, make eliminations regardless of whether the chain starting candidate is true or false. This post displays several of the shorter xyt-chain examples in THLS.

Denis Berthier invented something worthwhile, but dangerous, with his modification of the XY -chain into the xyt-chain.  The xyt chain is an XY chain with “extra” candidates embedded in its node cells.  Unlike the XY-chain, the xyt-chain has a direction, and one “starting” candidate.  The embedded chain is built with an assumption that the starting candidate is false. Under this assumption, every “right” candidate with the exit wink it true, and every “left” candidate with an incoming wink is also false.

Extra candidates do not stop the embedded XY chain AIC action as long as they “see” a prior right candidate in the chain. The “full” xyt-chain ends with a right candidate matching the starting candidate.  Any outside candidate seeing both terminal candidates is false, because either the starting candidate is true, or it forces the ending candidate to be true via the xyt-chain.

In THLS the xyt-chain is described as a more general XY-chain, of which the “pure” xy-chain is a special case. That is incorrect, because the designated direction of the xyt-chain, allowing branching of the chain, is a special condition applied nowhere else in the AIC family.  The mistaken idea that AIC are somehow not lot logical may come from similar confusion with branched forcing chains.

After we actually look at some of these chains on the grid, perhaps you will agree with me that, while innovative, and something to be aware of, a comprehensive search for them should be among last resort measures.

17-2769-ur-gridLet’s check it out the UR you were to skip in the Royle 17-2769 homework. Here is the unique rectangle grid, with the very easy decision force a 9 into the rectangle.



Following the THLS policy to save the UR for a last resort, SudoRules comes across the following xyt chain.

17-2769-xytLook at 9r9c7 as a possible starting candidate. Assuming it false, we see an XY-chain moving to r8c2, but for the extra candidate 7. That assumption would make 7r8c8 true, removing that value from r8c2 (red wink) and allowing the red slink.  So, not 9r9c7 => not 9r9c2 by the chain. But clearly,  9r9c7 => not 9r9c2 as well. 9r9c7 has to be true or false, making 9r9c7 false. Same result, fatal in both cases.

The start of an xyt chain does look like an arbitrary guess, but its construction is logical, if unpredictable. In a comment on this post, Mittleman points out the ALS-XZ in this chain. The r8c23 ALS 379 and the r89c78 ALS679 have a 7 restricted common, and the chain’s victim sees all 9’s in both ALS. It’s worth noting that the starting bv pair of an xyt-chain usually form a promising first ALS with two singles. Not always, as we see next.

17-5105-x-chainThe next example, Royle-5105 requires a 5-chain to prepare the grid for the xyt-chain. Berthier does not include it in the trace and the elaboration, though the xyt example requires it.






17-5105-xytAn XY-chain starting with 9r1c3 is completed in r1c4 by erasing 4r1c3 by means of the assumption value 4r1c3.


17-5105-xyHowever, the Sysudoku OB gives a more easily spotted alternative.

The simple XY chain to the right removes three 4-candidates, setting up a regular XYZ-wing below with two ER victims, and 17-5101 is done.








Our final example is an illustration of an xyt-chain that might be spotted as a needed modification of a normal XY chain.

It is a coloring scenario in Royle 17-1365. Again, an abundance of bv encourages us to build two clusters.

17-1365-no-bridgeBridging logic applies with

 not(orange and green) => red or blue,

but 9 is the only number common to both clusters, and no red 9 means no bridge.

Then we see an “almost” XY loop spoiled by an extra 9 in one cell. 9r2c4 would erase it. Take 3 3r2c4 as the starting candidate. If false, the XY-chain moves around to 23r2c7 and 3r2c2 is removed.

17-1365-color-xyzt-color-trapThe extended cluster places a red 2 in r3c8, where two 2 candidates see red and blue. One of them wraps orange, and shortly thereafter, red forces blue.







17-1365-293-wingOn the other hand, I might have followed the normal SOB course, and pulled out this i293 wing, which confirms 3r4c7 and permits the remote pair that confirms 3r2c4 and collapses 17-1365.

My conclusion,  after these few examples, is that xyt-chains are harder to spot than regular Sysudoku alternatives.


While they are constructed straightforwardly enough, a comprehensive search for them would take a too many unproductive constructions, and should be deferred at least until the advanced SOB methods are given attention.

royle-17-20565Next we look for examples of  longer xyt chains in THLS XVII, and encounter surprises. The first one is an even stronger tie-in with coloring, perfect for Valentine’s Day. If you  would like to be there with your own slides and have mislaid your THLS again, here’s my next target, Royle 17-20565.  You’re welcome.

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