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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A Chicken Chok’in Checkpoint on Unhidden Chains

This post checkpoints Royle 17-16774, a rare example in The Hidden Logic of Sudoku of conjugacy chains dealing with a monster cloud of candidates. The checkpoint also catches SudoRules dismembering a “useless” XY nice loop that somehow sneaked into THLS.

If you worked on the tough homework, thanks for being sysudokie about it.   We have to defend our turf sometimes, from those who take Sudoku as a challenge for coding, but come to believe they have found in their computer code the zen of human solving. THLS is a concealed form of this, with its unpublished starting gun number scanning and almost as immediate candidate thinning, its burdensome multiple grid maintenance, and its humanly impractical “logical complexity” order of battle. More generally, sysudokies have to resist all manner of reasons for searches for which only a computer has time.

Need a celebration poster? How about this one!

17-16774-quadsWith the toughest two rows to mark, two naked quads show up. I choose to continue on the rows, with the bean pod quad markers left in place.  Berthier reports these beauties as hidden triples, which are “logically” simpler (3 < 4), but require all candidates, besides just being harder to spot.

Any trouble getting there? Welcome to the club. They slipped right by me, at first.  Usually, I show you the results of my back tracking without admitting it.

17-16774-nice-loopYou have to forgive me, however, when you see this picture of the devastation wreaked by this tight little XY nice loop in the space swept by the quads.  In addition to the normal elimination in the lines of each link, weak or strong, there are the two box/lines

So what did SudoRules do with this “useless” nice loop? No worries, actually. The rule based beast found the XY chain surrounding each wink of the nice loop.

They’re always there. That’s why a nice loop works. When it is allowed  to exist, that is.

17-16774-fick-double-alsIf you’ve found your copy of THLS where it slipped out and landed on the floor beside the bed, then you came upon Denis’  trace error, reporting two XY4 chains instead of three. He did mention the elimination of the missing XY-chain, however, to the consternation of his most diligent readers.

For sysudokies, 16774 just keeps on giving. Gordon Fick noted that the nice loop can be interpreted as a doubly linked ALZ-XZ, with the same toxic sets. The second restricted common locks the two ALS, giving a toxic set for every number. One ALS gets the 1; the other, the 2.

For the record, the basic trace:

17-16774-basic-tr17-16774-x-chainsThe nice loop/ALS-XXZ  leaves it to a couple of c-chain (X-chain) woodsmen to clear cut the forest. The black 4-chain is followed on the X-panels by a red 6-chain. The collapse is triggered by 6r9c7.

My co-conspirator Gordon Fick points out four rarely seen and equally fatal finned fish in  17-16477 on the 4-panel as well.


17-16774-fick-fin-swordThe panels show the blank line tally markings for two alternate finned fish on the 4-panel, and two more on the 6-panel. As always with finned fish, you should satisfy yourself that there are no kraken victims among them.

Next week begins the examination of Berthier’s modified XY-chains, with the xyt chain. These last two innovative offerings of The Hidden Logic of Sudoku involve a continuing Sysudoku issue, arbitrary guesses in human solving.royle-17-2769

I know your New Year’s resolution was not to do a Royle a week, but  you might like the sysudokie basic on Royle 17-2769, the first xyt chain example in THLS XVII. Just skip the unique rectangle that solves the puzzle before you get to the xyt chain. It assumes you know what.

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THLS Limps In On Chains

This post identifies the conjugacy chain, the last fundamental chain type to be introduced in The Hidden Logic of Sudoku, as a weakened X-chain.  Since the last few chapters introduce extensions for XY chains, it becomes clear that THLS is incomplete on the fundamentals of links and chains.  Such deficiencies explain why symmetry spaces are deemed necessary for SudoRules.

17-1020-fick-aicBefore examining c-chains, here’s a checkpoint on Gordon Fick’s AIC on the line marked grid of Royle 17-1002.  If you placed your AIC hinges, it was easy. There were only the same three, and the AIC went through all three. Following the SOB, I didn’t put on my AIC hinge glasses before the X-panel scan of the last post, and this chain sneaked by undetected in front of my clutter distracted eyes.



Sysudokies can live with Denis Berthier’s description of XY chains. His idea that it is cells, rather than candidates, that are linked seems a little quaint, but the mechanism of left candidates and right candidates linking the bv cells does account for the very simple AIC action of the XY chain. Besides, many early Sudoku authors shared this view.

But reading through THLS Chapter XVI on “conjugacy chains” and realizing that the remaining chapters  are about extended XY chains (xyt and xyzt chains), a sysudokie experiences a disquieting sense of loss. The conjugacy chain, the last major category of chains in THLS, is just an X-chain, and a weakened one at that.

Earlier in THLS we read that two cells are c-linked, when they are in the same unit and share a number exclusively in that unit. That makes the c-link is a unit induced strong link, a slink.

Beginning Chapter XVI, the  THLS definition of the conjugacy chain reads:


That is to say, a chain of identical values, with every other link, starting with the first, a slink, and ending with a slink.  No nice loops, no confirming ANL and no grouping..

How peculiar.  In THLS there is no reference to the general concept of an alternate inference chain. Come to think of it, throughout THLS there is no mention of the logical definition of  strong and weak links between candidates. The slink and the wink are confined in narrow, limiting definitions based on cells sharing a unit. Mixed type AIC do not exist. Grouped inference chains do not exist. The AIC anatomy of an XY chain and the bv reversed AIC chain is undefinable. Nice loop coloring and extension is impossible.

Actually, these missing fundamental elements may account for the whole edifice of hidden symmetries. THLS can be interpreted as proposing computer generated hidden logic symmetry spaces as a remedy for a huge blind spot in supposedly expert human solvers, requiring the starting line generation and constant maintenance of extra grid spaces. You can believe this necessary if you lack the candidate linking fundamentals required to identify in nrc space the reverse XY non-hidden counterparts of XY chains in the hidden logic spaces. Berthier invites that very conclusion with this:

“For advanced examples, see chapters XV, XVII and XVIII, where hidden chains of various types are introduced and shown to be irreducible to non-hidden chains.”

In this c-chain chapter, there is little incentive follow the “proof” that X-chains should never be closed. It plays out somehow in super-hidden subset rules, with subsuming of longer chains with loops. On that point, how can one see that the loop can be bypassed, and not see the logical consequences of the loop itself?  THLS misses the nice loop again, here in its simplest form.

Ironically, Berthier includes an observation that regular fish are special cases of the c-chain. Or as THLS states it “the Swordfish(row) and Swordfish(col) rules are subsumed by the c6-chain rule”.

absnasty-iv-11-nice-loopHere is an instructive example from the Absolutely Nasty IV review. The c6-chain on the same candidates as a swordfish is a nice loop!  Not only that, but this nice loop c-chain has a victim the swordfish cannot touch. So much for subsumation.

But sometimes, the fish wins. If row 2 had an 8 candidate in r2c2, the fish would be fine, but the c-chain would lose a vital c-link.


Sysudokie friends, the nice loop is not to be disparaged. You put one together before you even think about all the removals it may cause.

17-57-remote-pairWith the many decisive X-chains appearing in the earlier Sysudoku review puzzles, I expected the same in the chapter on conjugacy chains. The first c-chain example, Royle 17-57, illustrates one reason there are few.

On the freshly line marked grid, the bv field is so rich, there is a 6-cell remote pair that overrides the X-chain (red).  With it’s identical bv, the remote pair is a string of flashing lights.

After the removal, why mess around? The first coloring is an easy wrap.

17-118-wrapOf his second c-chain example, Royle 17-118, Berthier states: “its L4_0+C4 elaboration is equal to its L1_0 elaboration.”

The Sysudoku basic equivalent to L1_0 follows conventions designed to enable readers to generate traces that match the blog checkpoints.  Apparently, the Sudorules  resolution path does reach a state in which a four cell c-chain exists. It doesn’t matter, when immediately after the simple line marking, there is a coloring wrap(r6c9).

To finish the puzzle from this point THLS suggests either a 4-cell XY chain in nrc or a 4-cell XY chain in crn space.  This may explain why there are no pencil marks on THLS grids. It can be embarrassing at times.

17-147-chainsBerthier follows with two more examples of disappearing conjugacy chains, before reaching one of relevance to Sysudoku. It is Royle 17-147, showing alternative resolution paths via c4-chain and XY4-chain.  In a bv field like this there may be more, but these should be among the shortest.

The reason the bv are not marked with green squares?  It’s because the grid was filled in the bypass.  Are you following this, Wayne?

17-147-near-bugThe grid above is also a near BUG (Bi-value Universal Grave). Coloring. Here the arrows show the order of a coloring that  resolves it with the 4 and 8 in r3c2 being the same color. That color is thereby wrapped, and 9r3c2 carries through to the solution.

In The Logic of Sudoku, Andrew Stuart suggested an unproven rule for near BUGs. It was to confirm the 3-cell candidate that appears three times in a line. Unfortunately, in his book, he applied it to a puzzle with three solutions. In THLS, Berthier points this out, and faults Andrew’s rule for assuming uniqueness, i.e. that the puzzle possesses a single solution.  Denis claims that using the rule hides the non-uniqueness of the puzzle.

Andrew gave no rationale for the rule, and certainly no explanation of why it depends on absolute uniqueness throughout the puzzle. The incident may have inspired Berthier to make a  similar claim against unique rectangles, confusing uniqueness among cells on the four corners of a rectangle spanning two boxes, with absolute uniqueness of the whole puzzle.

In coloring, as illustrated above, you are continuing to follow the implications of the givens without assuming the value of any cell. It such a logical train arrives at a solution, it is unique. Coloring is a car on this train, a collective property of a slink net. Coloring can be used to uncover multiplicity of solutions only by assuming one of a cluster’s colors to be true.

Andrew’s unproven rule had no such standing, and he made no claim that it did.

17-16774Let’s do one more, another X-chain steam roller like Royle 17-1020 back in the 1/16/17 post. One that makes up for these logically least complex c-chain softies. Introducing Royle 17-16774,  a line marking monster where bv are few. But also, where the X-chain is the buzz saw that cuts through the trees and moves out the wood.

Then, when line marking is complete, there’s a 4-chain and an XY nice loop that SudoRules never saw. Also, a decisive 6-ANL that SudoRules did see.


And we’ll get to see what fisherman Fick caught from the beach bordering the tropical forest.




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Remaining THLS XV hXY Avoided

Here we complete the report on the puzzles selected in The Hidden Logic of Sudoku to show that hidden XY chains are an “inescapable tool for the advanced player”, and therefore  require the generation and maintenance of symmetry grids. None of the selected puzzles actually require this burdensome effort on the part of a human solver.

17-4167-remote-pairPicking up where we left off in THLS Chapter XV on hXY chains, the elaboration candidates of Royle 17-4167 are restricted to 3,6 and 9, and admit a decisive 6 cell remote pair. Since Berthier reports it to have a 5-cell XY chain in crn space, he can claim a solution of lower logical complexity. The solutions have the very same basis, the hidden XY removing 3 r5c5 and the remote pair removing 69r4c5 by means of five slinks.

But give me a break.

Remote pairs are so easy and require nothing extra.

17-5546-8-chainNext, we have a 6-cell hXY chain in crn space, another pushover in good old nrc space.  It’s Royle 17-5546. There’s not as many bv, but the 8-panel has the slinks for a decisive 8-chain.

I included the line marking fill strings to show what an easy line marking it was.




17-5546-remote-pairIt’s true that the XY railway comes before the X-panels in SOB, but where would you put hidden chains? I take them as very  extreme, and would do symmetry transformations only on a weeded grid.

In the collapse, you wouldn’t want to miss this little four cell remote pair.

But now, watch out!  We’re down to the heavy hitters. Coming up in Sudogen-9617, you’ll need an XYZ wing, a 6-cell hXY chain in crn space, then a 7-cell hXY chain in rcn space,  or alternatively a 5-cell xyzt-chain (details later), followed by the above hXY chains. Are you sure you want to go on?

sudogen0-9617No?  OK,  just assume the puzzle composer would not embarrass herself with an  obvious rectangular multiple placement in the solution (post 12/13/16).

The candidate 3r6c2 has to be true or the composer is  embarrassed.  Adding this guardian triggers an immediate collapse. 


Making this UR assumption has no effect on the risk of a less obvious multiple solution, even if it happens to occur as a part of one.

Finally we get to the end of this series on hidden XY chains, and your homework.  Royle 17-1020 was unreasonably stingy in basic. The trace looks simple but how long did it take you? How about number scanning it first and then eliminating the extra candidates, to see how long that takes?


For those not participating, here’s the line marked grid.


The results aren’t pretty either. All three bv in the same unit.  Let me know if you did better. But you know this doesn’t look that bad to those who actually do number scan hard puzzles. They wade though this kind of swamp regularly.

17-1020-x-chainsOK, where we go next is the X-panel. X-chains can cut through the candidate fog.

The 2- and 3-panels yield three indecisive clues. It’s not unusual for my X-panels to recover some missed line slinks when I do them. The 3-chain removal yields a box slink and box/line clue.




Of course, the x-panel is also my fishing hole, and I pulled out a creature that didn’t make it into THLS, a finned fish.  In the left panel below, you can see that my 2-chain removal actually demoted a jellyfish to a kraken swordfish.


The right panel illustrates another frequent contribution of X-panels. The 2-panel suggests limited patterns. When a coloring cluster is defined, the solution pattern must be consistent with one color or the other. With freeforms left to right, the red freeform starts on green at r5 but cannot cross c5. It must leave the S box, has used r3, and cannot include a blue cell. It is the only chance for 2r3c3 to have a pattern. An orphan is a candidate belonging to no pattern and is removed.

Confirming the orphan is not enough, and we move on to AIC hinges, where I know there are counterparts in standard nrc space to the hidden XY chains in the symmetry spaces. Pattern analysis being rather extreme, I’m leaving the orphan unremoved, but I’m not forgetting it.

Thanks to the reader whose comment linked below prompted me to correct the right panel above, and illustrate coloring restrictions in Sysudoku Limited Pattern Overlay.  Pattern analysis, as explained in Andrew Stuart’s The Logic of Sudoku, was known to Berthier. We know that because of his criticism of Andrew’s near BUG example. But pattern analysis also, is omitted from THLS.

17-1020-aic-anlAn AIC hinge is a cell holding two slink partner candidates. Being in the same cell puts a wink (weak link) between them, so there is a alternative inference chain segment of three links around the hinge. If you can complete a nice loop through a hinge, it removes the other candidates from the hinge. In this case, I found that my chain of slinks and hinges could be completed as an AIC almost nice loop(ANL), eliminating the single candidate between winks. The collapse is immediate.

My Sudoku accomplice Gordon Fick, armed only with THLS, and not knowing that I was selecting this puzzle for the review, sent me two more ways to exploit the same two lines r4 and r5 for the same critical removal 2r5c1. One is an AIC almost nice loop that does not depend on my removal of the kraken swordfish victim. Can you find it? Go back to line marked grid and map the AIC hinges. Checkpoint next time.

17-1020-fick-tricksThe other way is an amazing ALS-XZ. The Almost Locked Sets r4c347 (blue) and r5c34789 (green) have a grouped restricted common 4 in the E box.  That means candidates of every other number in the two sets contain a true one. 2r5c1 sees all 2’s in the two sets. Do you spot things like that?

I wish I could.

As to hidden XY chains, the first 2017 post showed there may be no Sudoku grids solvable only by hidden logic XY chains. Each one seems to convert to a reverse-XY AIC in standard nrc space.

These next two posts have demonstrated  humanly accessible alternatives to hidden XY chains. These are supported by efficient, clutter avoiding basic solving. 

SudoRules doesn’t do finned fish, much less krakens, or ALS, much less ALS-XZ,  or AIC, much less reverse-XY AIC ANL, or patterns, much less orphans, or slinks, much less coloring. Its reliance on hidden logic is understandable. It runs on modern computers, where any number of definable extra grids is acceptable.

Next we look at c-chains, as defined in The Hidden Logic of Sudoku. C-chains are the Berthier counterpart to X-chains, only more complex and less capable.

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No THLS Case for Hidden XY Chains

This post and the next demonstrate that the examples presented in The Hidden Logic of Sudoku as requiring “Hidden” XY Chains actually have alternative, and much less demanding, resolution paths in the Sysudoku repertoire.  The burdensome maintenance of symmetry grids is not justified by this book.

In Chapter XV of THLS, Denis Berthier presents a series of increasingly long hidden XY chain puzzles, with reports on the significant removals found by his rule based solver SudoRules. The implication is that hidden XY chains are required for the solution of these puzzles. These examples are presented in support of Berthier’s contention that hidden chains are “an inescapable tool for the advanced player”.

Having gone through the Chapter XV hidden chains puzzles, I am relieved to report that THLS  fails to show that hidden chains are necessary at all.  These puzzles make a nice collection illustrating a variety of advanced techniques encountered in the Sysudoku  Order of Battle (menu above).  I invite you to compare the difficulty of these solutions with just the preparation and maintenance required to try out hidden XY chains. If you would have a program for that, then just think about searching the extra grids it maintains for you.

I’ll not include the usual basic solving traces in this report, but if you want to work the puzzles to the  point of illustration, you will be able to recover the givens and the THLS elaborations from the grids presented.

17-211-aicIn the first hXY example of THLS XV, Royle 17-211,  I was disappointed to see that the neat little red XY wing I found along the XY railway had no victim. Actually, that’s a good excuse to look for a forcing chain between two given points, a candidate seeing one terminal and the other terminal.

There’s an advantage to having a destination, and being willing to take any AIC route to get there.

THLS reports an hXY chain of four cells in rcn space, but three cells of XY chain and a look at the 7-panel gets you this removal.

17-211-xy-chainThe collapse is immediate, but there is a less decisive XY hugging the same rail. It buys two more bv on the way to a less advanced solution.








17-619-xy-anlThe next example, Royle 17-619 was reported as a four cell XY chain, followed by a four cell hidden rcn chain. The THLS XY chain is shown in red. I happened upon the black one, with an second victim. You could combine them to eliminate 9r5c2, but there’s a shorter route through r5c3. It’s a good example of life along the railway.

So now what is coming in place of the hidden XY chain?

 Would you believe a Death Blossom Lite? 

17-619-db-liteThe generous patch of bv, festooned with slink webbing, is typical of THLS examples, but you never see them in the book. They beg to be colored, and that is an apt finish for 17-619


17-619-coloring-1I add a second red/orange cluster for the uncovered 6 and 9 candidates. In classic bridge fashion, blue and red 6 in the same box means green or orange is true, which merges orange and blue.

Maybe you’d like to verify that the merger dooms blue and green(red) wins.

I know its unfair, but unlike Denis, I just can’t hold anything back.


17-520-i148-wingIn the next example, Royle 17-520, another 4-cell XY and hXY pair of chains is waiting, but I get to an irregular i148-wing first. The 18 wing is attached by a forcing chain wink. The removal of 1r4c2 leaves a box/line on 1r1c3.






17-520-coloringNow I have to give in to the bv patch and add coloring. Row 5 prohibits orange and green, so red or blue is true, or both. Anyway, an unlucky 8 sees both. Now it’s possible for a “shortcut” wink based on color to tie together an XY wing (three cell CXY chain) to trap a green soldier, condemning the whole green army. The blue army overruns 8r4c5 and its over.




17-11212Finally, here is the coloring of the elaboration of Royle 17-11212, with its 5-cell hidden XY chain threading through this near-BUG elaboration. It’s a lot harder to interpret in rcn space, where this hidden chain lurks. Here, two green candidates are forced into r3c8 for a blue solution. I added freeforms to indicate the order of coloring. There is little need to construct any chains for this.

Next time, three more puzzles with even more impressive, but completely hidden, and equally unnecessary, XY chains.

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