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.

17-1020-panelsThe right panel illustrates another frequent contribution of X-panels. The 2-panel suggests limited patterns. The red freeform shows where a left to right freeform fails to make it across the panel and leaves an orphan in 2r3c3. An orphan is a candidate belonging to no pattern. Confirming that suspicion is not enough, and I 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.

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?

17-1020-fick-tricksThe other 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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Hidden XY Chains Are Reverse XY AIC

This post demonstrates that every hidden XY chain has a recognizable AIC counterpart on the normal grid, nrc space. That means it is not necessary, if you are AIC literate,  to generate and maintain the grid spaces in which hXY chains hide.   Besides that, I have a highlight reel for you after dinner, showing some of the delights you would miss if you stay on the SudoRules resolution path with the very puzzles selected to promote hidden XY chains in The Hidden Logic of Sudoku.

First, I have a couple of  17-35802-xy-235802 nrc XY chains to checkpoint for you.

Here is the second, . . .




. . . and the third, for the collapse.

17-35802-xy-3Now,  with all of the THLS evidence of XY success in nrc space, should we spend the effort to be ready to apply the XY chain rules on the symmetry grids?   

Forget it, pilgrim.

Also, don’t expect the same degree of XY chain success in symmetry land.  None of the 35802 chains we just found would be possible on the symmetry grids, because all three depend on weak links with boxes as the “seeing” unit. In the symmetry spaces, remember, there are no boxes.

xy-wing-in-crnAnd as Berthier himself illustrates in THLS, an XY chain in a symmetry space transforms into an AIC in standard nrc space with corresponding eliminations. Here is a properly numbered version of Berthier’s hidden XY wing example, Figure 2 in Chapter XV, with slinks and winks explicitly shown between candidates. In crn space the 1231 XY chain has a victim seeing both end candidates, forming what my mentor Andrew Stuart calls, an almost nice loop, or ANL.

In standard nrc space, the bv are stretched into row slinks and the row winks condensed into bv.

xy-wing-in-nrcThe corresponding 2-candidate is eliminated, but in the process, a corresponding 5-candidate is confirmed.  Perhaps not in THLS, where confirming ANL are politically incorrect.

It’s true that such AIC are more advanced in the SOB, and  may be more logically complex than XY chains. But we manage with the help of AIC hinges.

Bottom line,  we can do without the laborious symmetry grid maintenance burden of hidden XY chains because we have a richer solving repertoire than THLS.

This is demonstrated further in the next post with a report on the Sysudoku resolution of the hidden XY example puzzles.

In the course of preparing that report, I did come across one puzzle for which I found no other solution than the AIC corresponding a hidden XY chain. Your homework Royle 17-7295, diligent sysudokie, makes a concrete, supersymmetry example. Here is your basic trace checkpoint:

17-7295-basic-tr17-7295-aicIn the resulting line marked grid, none of the five UR delivered results. I struck out with the bv scan, the C-panels and coloring as well. Perhaps I wasn’t looking hard enough, because I kept thinking about the hidden XY chain to nrc AIC example above, and its THLS companions. I wrote in my AIC hinges, and the reversed XY AIC came right out and saluted.

The AIC hinge is a cell with more than two candidates, with two or more forming slinks. Such cells allow an alternating chain to pass through. It’s a reversed XY chain, with slinks added on each end. The elimination brings an immediate collapse.

But wait! Is this the transformed XY chain in the THLS trace for 7295? Berthier does say that 7r1n4 is eliminated in crn space by his XY chain, corresponding to 4r1c7 on our planet. Oops, we eliminated 4r7c1. Is that a typo or did he find something different?

I know it sounds ridiculous if you’re not a THLS reader, but Berthier announces the cells of his XY chain in cnr space, with no grid diagram. Remember, he shows no candidates in any space.  I’m not willing to convert the whole grid, or even all remaining candidates, but I can convert chain’s cell list onto a nrc grid and see what we have. I think you should try it and check me out on the following diagram.

17-7295-aic-to-crnIn crn space, my AIC looks like an ANL confirming 7r7n4, but that is something SudoRules appears to know nothing about. This something has the same effect as the THLS trace’s hXY removal of 7r1c4.


The puzzle has some wiggle room and we don’t have an exact match.


17-7295-thls-in-crnIt seems likely that this is the XY ANL that SudoRules came up with, but when I modify my AIC to get that 4r1c7 elimination, I need the NE4m slink below to make it work. That slips an irrelevant 8 into r1n4 here.

Notice the absence of box slinks in both of these hXY’s. They don’t exist in crn space.


17-7295-mod-aicAnyway, I trust you are satisfied that you see what comes with the hidden XY merchandize that Berthier promotes without the price list. I’m not buying any for my human solver friends. Computer solver buddies with sweet routines for XY chains might be interested, but we human solving advocates are not the drudges that  computers are. We’re in it for fun.


royle-17-1020In the next two posts,  I’m showing the alternative sysudokie solution highlights for all the other puzzles that THLS claims to require hidden XY chains.  The purpose is to show that the value of hidden chains lags far below the price in human effort. Here is the capstone puzzle in this series, Royle 17-1020.  Basic solving is tough. You’ll have two weeks for this assignment.  Go!


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THLS Needs a Ride on the XY Railway

This post solves an XY-chain example to illustrate the simplicity and efficiency of XY-chains, as opposed to the rule based overcomplicated view of them in The Hidden Logic of Sudoku.

From the first, this Sysudoku blog emphasized the fundamental nature of strong and weak links, and power of alternating them in chains.  XY-chains were introduced as the most productive form of alternating inference chain, or AIC.  A goal of thoroughly exploiting the easily constructed XY-chains led naturally to the idea of an extended network of them, from which every productive XY-chain can be snipped.

The THLS trace of Royle 17-35802 describes it as solved in L6+XY7. The XY-chain is described as seven linked cells. Among Sudoku writers Berthier is the only one to attach such significance to the length of a chain.

Let’s get started with the basic trace of 35802:


17-35802-basic-gridIncluding the eliminations of the naked pair (black) and the hidden pair (maroon), here is grid on which the only chance for a unique rectangle has just been eliminated.

My record of not finding a Sue de Coq in a THLS Royle 17 remains intact, so we move on to the bv scan.





A rarely seen regular XYZ-wing does show up, a 794-wing.

17-35802-railway-1Now I zoom up on the bv map and draw the XY railway, starting with one long curve for the main line, and adding a few spurs. Stations on the railway are bv.  Starting with 6r3c3, continue on the railway until you find another 6. The first one is 6. If we are leaving the station with 6, it forms a toxic set with the starting 6. But that is not news, is it? At the next matching station we are arriving on 6, but on the next match we do have one of Berthier’s full XY-chains and a toxic set. The trouble is, no victim sees both terminal 6’s. The last 6 is arriving, and the railway is one-way.  We can’t back up to go to more stations

17-35802-l7-xy-chainSo what next? Advance to the next station, where it is 4 arriving and we traverse the railway looking for leaving 4’s. It turns out that the THLS is found on the main line (black), but we can shorten the path by taking the green connecting spur and bypassing the red one, finishing on the black curve.




Unlike THLS, we regard links between candidates, not cells. XY chains are the simplest form of alternating inference chain, the strong links inside bv, and weak links between. Candidates at the ends of an AIC terminal slinks are strongly linked, and therefore toxic. A candidate seeing both of them is false.

Now is a good time to dispose of an embarrassing THLS proclamation that XY-chains should have no loops. You recently saw a nice loop in action on Royle 17-1007.  On our railway above, we could add a spur leaving r6c6 on 7 and arriving on 7 to r6c5, creating a nice loop. It so happens that all adjacent pairs along this loop are slinks in all of their containing units, but any exception to this would bring an elimination of a candidate seeing both ends of a slink.

The legitimate concern behind the proclamation, to put it simply, is that the railway cannot leave and arrive to the same station.  To illustrate the consequences of doing so, THLS gives the reader one of its most confusing head scratchers, Figure 1 on page 199.


In  his “True XY loop” diagram, Berthier has a bv configation for which the only way to construct an “full” XY chain is to go around a loop and use r4c5 twice, arriving on the rail on which we left. Then on the companion “pseudo XY loop” diagram, the numbers are changed to have an ordinary XY wing joining a perfectly fine nice loop. This kind of loop extension does good service in Sydudoku nice loop coloring. What happened? Good and bad are reversed.

Anyway, we can appreciate the THLS practice of solving with the shortest XY chain along the railway. Curves on the bv map are fun to draw, but the real anatomy of the XY-chain is the AIC diagram on the grid. If you’re up to it, I suggest you find the next two XY-chain eliminations that cook 35802’s goose, and compare your artistic AIC curves to mine in the next post.

17-35802-coloring-1Meanwhile, I’d like to illustrate the advantage of opting for coloring when you find yourself in a field of bv clover.

Going back to the railway grid, I had two fine clusters, but they don’t see each other.

But the L7 XY elimination of 4r3c4 extends the red/orange cluster to trap 5r3c5, and wrap red, and then blue, for a complete solution. 

Review your coloring and make it happen.

royle-17-7295Or maybe you’d rather spend your sysudokie time on basic preview of a supposedly hidden XY chain puzzle to be examined next. It’s Royle 17-7295

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The Limited XYZ-wings of THLS

The XYZ-wing is limited in The Hidden Logic of Sudoku, as it generally is in Sudoku literature, by  a limited conception of the weak link.  This post presents a resolution of Royle 17-12407 using an iXYZ-wing, a form of the XYZ-wing utilizing the forcing chain as a wink, a means of “seeing”. This is one of the puzzles that THLS claims “can only be solved” by hidden logic.

In Sue de Coq’s celebrated post of October 24, 2005, titled  “Two-Sector Disjoint Subsets”, he predicted that the technique he was introducing would soon be forgotten, because Sudoku solving was soon to be dominated by forcing chains. That didn’t happen, but perhaps the viral rush to demonstrate forcing chains as an advanced method explains why they are largely overlooked today as a form of “seeing”, a weak link in which two linked candidates cannot both be true, because if one is true, the other is false.

Sysudoku uses forcing chains as winks interchangeable with winks due to the candidates sharing a unit. The XYZ-wing sparked my interest in forcing chain winks, when I realized how unit shared “seeing” limited the ability of outside candidates to see the three candidates of the wing’s toxic set. Shortly after came the realization that the winks of the wing itself could also be forcing chains. I already had a platform for a focused search for these irregular XYZ-wings , a table of bv cells.

17-12407-lm-gridLet’s begin the unhidden conquest of Royle 17-12407 with a checkpoint on its basic solving. A remote pair jumps out of the line marked grid to remove 7 and 9 from r1c8.






17-12407-basic-trThe bypass completes the THLS elaboration.  The concentration on numbers 36789 makes box marking and line marking easy, but it creates a host of “almost Sue de Coq” situations.

17-12407-xyz-mapThe number concentration also produces a host of potential XYZ-wings. To analyze these, I copy my bv map and add potential XYZ and WXYZ hinge cells. Placing this panel alongside the grid, I look for forcing chain wing attachment and toxic set victims for each hinge cell, crossing out the unsuccessful ones. Here is the resulting XYZ map for 17-12407.

17-12407-i789-wingThe prize for this systematic search is a gem of an i789-wing, with a remote wing attached by a grouped 8-chain ER.  The single victim, 9r6c8, sees the remote wing 9 via an ER style 9-chain.  What a charmer!


17-12407-trap-1The removal appears to be indecisive, but when we color the 79 slinks we have a grouped blue node of the blue/green cluster that traps a neighboring 7, . . .







17-12407-trap-2expanding the cluster to trap another 7, generating a box/line SE7m removing 7r1c8, which is green, confirming that blue is true, and collapsing Royle 17-12407 immediately.

No hidden chains required.




17-35802Now we go to XY-chains, the next topic in THLS and the SOB as well. This covers the XY-wing, the shortest possible XY-chain, but instead of that, we’ll  look at Berthier’s longest XY chain, in Royle 17-35802, to show how the Sudoku XY railway resolves his difficulties with XY loops and “pseudo loops”. As usual, you get a shot at it first.



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THLS vs. Unique Rectangle

This post updates the UR chart, with credit to Denis Berthier, then debunks another unfortunate THLS pronouncement. The homework puzzle illustrates another penalty a human solver pays for interpreting  The Hidden Logic of Sudoku as solving advice.

Now in our THLS review, we cross our line between finding candidates and exploiting relationships among them, that is to say, between basic and advanced solving. First in our order of battle are the bv scan methods, those methods requiring nothing but the line marked grid as it is. Although it took a while to decide this, I concluded that the unique rectangle comes first among advanced methods. The characteristic aligned naked pair is a red flag, and after finding a pair of cells for the other two corners, the search is on for removal of any candidate forcing the dirty rectangle, or the confirmation of any candidate absolutely required for its prevention.

With attention focused on this goal, I look for any means, forcing chains included. Then I haul out my UR Types chart to either classify my result as a known type, or to look ways to meet one of the chart conditions.


My chart was compiled from the Hodoku site (9/8/15) during a review, and includes very brief reminders of justification, and the corresponding types from Andrew Stuart and the defunct Sudocue site.  Berthier’s descriptions of the UR, in his concluding Miscellanea chapter, includes versions of Types 1 – 4 above.

Berthier’s UR descriptions are hard to be read without diagrams, and are overly specific, but deserve credit for an improvement in my chart.

Type 1 is expanded to include multiple extras in one  corner. A single extra is confirmed. Both UR candidates are removed in all cases. Berthier’s Type 3 needlessly limits the form of the naked subset. His Type 4 and Type 4b are both covered by the requirement for a slink. In his Type 4 it’s a box slink; in his Type 4b, the line slink. Sysudokies can interpret the UR Chart to include the possibility of forcing chain “seeing”, a concept totally missing in THLS.

17-6526-urThe first UR example in THLS is Royle 17-6526, shown here immediately after line marking. The naked pair along one side makes eliminations. The other side creates a box/line, removing 8r9c7. Your homework is to determine the UR type, and the UR removals. If you peek at your copy of THLS, you still won’t have the complete answer.

What rolls my eyes in THLS UR coverage is Berthier’s deferral of the unique rectangle to his last resort, on the grounds that it  “assumes uniqueness”.  In Miscellanea (XXIII.3.4., p.361) Denis says:

–          “A resolution path provides a stronger result when it does not use rules based on this assumption: it does [then] prove uniqueness;

–          Rules based on the assumption of uniqueness should therefore not be applied before rules … that do not use it.”

With regard to the first point above, it is more accurate to say that uniqueness is proved when a solution is reached with no assumption having been made about the truth of any candidate. In that case every derived clue is proven, leaving no possibility of a second solution. Assumptions have nothing to do with it.

Berthier’s second point is an empty one.  Its comes after self serving argument that none of the THLS resolution methods assume uniqueness. Therefore UR methods should come last, not first, in advanced methods.

No, Denis. Unique Rectangle methods don’t assume uniqueness.  Instead they assume there is no uniqueness failure of this simple deadly rectangle form. That’s all. The same common sense defense can be offered for extended rectangles and other defined patterns of multiple solution. Unique rectangle is simply not based on the general assumption of uniqueness.

In fact, methods based on such an assumption cannot be defined and do not exist. The reason is evident in the multiple solution cases encountered in the Sysudoku collection reviews, reported in the posts of US Air12/25/12, Ferocious 7/02/13, and Fiendish 11/18/14. There is no predictable pattern in these multiple solutions on which to build any logic on such an assumption.

No, even the trials I’ve used to reveal multiple solutions do not assume uniqueness. They organize candidates into strongly linked armies of candidates, one necessarily true, and the other, false. They offer a humanly practical means of demonstrating multiple solutions, when the failure of advanced methods raises questions of uniqueness.

Getting to the homework, here is a basic trace for Royle 17-692. The THLS elaboration was completed in the bypass. Did you find the unique rectangle?


royle-17-692-urIt’s a Type 4. One of the 9-slink parners must be true, therefore both of the corner 3’s must be false.  This easy logic collapses the puzzle, before any other advanced methods are attempted.

Berthier puts UR last, and works out two xyzt-chains (reviewed later) to remove 7r1c5 and 5r3c5, only to use the same UR in an unwarranted admission of desperation.


17-12407Next week we continue our evaluation of THLS treatment of advanced techniques, in SOB order, with the XYZ-wing.  Regretfully, we have to skip Sue de Coq, whose existence Berthier doesn’t acknowledge. The THLS XYZ-wing examples are plain vanilla ones, too insignificant for sysudokie homework. No, your mission – should you choose to accept it – is to find an irregular XYZ-wing, with one wing attached by forcing chain, in Royle 17-12407.  This puzzle is claimed to require no less that four of the aforementioned  xyzt chains, two of them hidden, plus two xy chains! I guess you’d better find the i-XYZ wing. I’ll go on from there to a solution with coloring.

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Suset Jelly in nrc Space

Here we work through an advanced supersymmetry example from The Hidden Logic of Sudoku, without the symmetries.  The humanly practical tools of Sysudoku are entirely adequate to the task. The theme of this review is that the burdensome emulation SudoRules, Berthier’s seeming emulation of human solving, is just unnecessary.


The basic solving trace of Royle 17-1007 shows a stingy bypass. I take it to the first highlight, a naked quad. Or did you note it as a hidden pair c8hp17?

Did you notice that 5 and 9 nquad candidates are entirely within the SE box?  One 5 and one 9 must be true.

17-1007-nquad-1Then, on the first 7f: coming up, on r1, a naked quad 1789, or perhaps the hidden triple 246 soon to emerge.








This trims the free cells back to four, and we eventually get a third naked quad in r3, with two numbers confined to the NW box.

17-1007-finish-trThe finishing basic trace shows the second and third nquads plus a hidden pair in between, and only one line left for the close.





17-1007-urFollowing the Sysudoku Order of Battle into advanced, one of the aligned pairs leads to unique rectangle removals. Either 9 forces 7r3c3, and a deadly rectangle.

Next post is a review of  Berthier’s position on unique rectangles and multiple solutions. In THLS,  UR is in the last, “miscellaneous” chapter and must be applied after regular fish.

On the homework assignment, you were to skip the bv scan methods and go directly to X-panels. I have a particular one for you.

17-1007-9-panelLet’s say you wanted to find every regular fish  on the rows. Suset enumeration does it. Start with a list of Susets, the row number/9 positions along the row. Then build larger susets by taking the union of numbers with the union of positions of every pair of susets. Since there are two blank rows, there is no need to go any further than four positions. Three rows and three positions define a swordfish; four of each, a jellyfish.

For the full story on limits on regular fish, go to What Regular Fish to Fish For (3/27/12).


See if you can discern the systematic sequence, that skips no possibilities, here:

The jelly suset is 1247/2469. Get it?

Less systematically, and less comprehensively, you get there faster making combinations of the lines with the least positions and the most overlap in positions.

Of course, none of this is usually necessary. We can find most fish readily with a blank line tally on the panel. For row fish, mark (-) a set of lines on a blank column in the same direction. Then on a blank line, mark(+) the covered positions(columns). This is done on the left panel below. On the right panel is the blank line tally for the complementary swordfish along columns, that has to be there.


The jelly and swordfish have the same victims. In fact they are subsets of a different form, different views of the same thing, like the complementary naked and hidden subsets of a unit. That is why susets apply equally to both subsets and fish. Regular fish are subsets.

17-1007-xy-chainAfter these removals, the puzzle is finished quickly by an XY wing, extending to a long XY chain wiping out every other 7 and confirming every 9 along the chain.







17-1007-xy-railAlthough Berthier rates every chain link as an added level of logical complexity, XY chains are extended to such length effortlessly. In fact, it pays to extend XY rails as far as possible, because multiple chains can be systematically whacked out of a common rail curve. Here is the rail underlying the two adjoined chains above.


17-1007-nice-loopNow I must confess to the reason for my request that  you skip the bv scan in the Royle 17-1007 in your homework. It is this XY nice loop, found on the same bv map before the SOB calls for X-panels. It removes the same candidates, and destroys the fish, and along with them, the illustration of supersymmetry for which it was chosen.



When I first took up XY chains in Sysudoku, I disputed Denis’ statement that “XY loops are useless”.  Of course this position completely overlooks nice loops like this one, but it comes from the fact that a long chain with a loop can be broken up into shorter chains with the same victims. To Berthier,  longer chains, being of higher logical complexity, are therefore useless.

In the sense of his words, Berthier is right, even here.  The above loop is a series of two XY chains.  But to those not adhering to his unusual definitions of those words, some of his proclamations are ridiculous. More on AIC loops later.

royle-17-692Next we examine the unique rectangle methods Berthier recommends, and his conclusion that they should only be employed as a last resort, because they assume uniqueness. Do you believe that?  Here is Royle 17-692, a puzzle that demonstrates that this conclusion comes with a cost. Maybe you’d like to fill in the candidates, and see what you would do with it, beforehand.

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