A Colorful Ending to World’s Hardest 200

Here we translate into sysudokie notation the XY-chains and coloring that drag off World’s Hardest Sudoku 200, to its sudowiki solver solution. Hopefully you were better able to arrive at your own solution.

First are the new developments on the bv map, where a XY-chain ANL triggers boxline Nr4, which triggers boxline SEr9.







This brings in a naked triple and more XY railway, a 2 eliminating ANL, .  .  .








.  .  . and a 3 eliminating ANL.

The follow up




opens it up for another cluster, and r5:

not(blue and orange) => green or red

sends us looking for a bridge elimination.

Instead, yet another XY-chain expands both clusters, to trap 6r4c5  and wrap orange on r3.

Red leaves a single blue 1 on r4, and there’s nothing left to say.

Except that the review continues, without solver assistance, and with Worlds Hardest 36, with ratings falling to 0.91.



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Andrew’s Solver Tames Worlds 200

This post walks through the “very  hard” part of the solution path taken by  Andrew Stuart’s Solver through puzzle 200 of The World’s Hardest Sudoku Book by authors unknown. This is a remarkable turn of events for a blog devoted to manual solving, but there are reasons, and the results are good.

My interest in Andrew Stuart’s The Logic of Sudoku, now out of print, is renewed as I work on topics for the Sysudoku Guide pages. Although we differ on some topics, Andrew’s is a solver that can be confined to humanly practical methods.  It can serve, therefore, as backup for human solving.

Having made no progress on 200, I reluctantly turned to Andrew’s Solver as a way to include 200’s difficulty in the ongoing review, but also with an eye towards expanding my understanding and gathering telling examples of methods I couldn’t spot.

First in the SysOB is APE r8c12, where 3r8c2 is denied a partner by the 39 bv, and the 13 combination in ALS1356 in r8c567. I missed class the day unaligned APE was covered in The Logic.

Then a BARN or BNS1, a bent 4-set 1 2356 in the Sc4 bent region. The common remainder value is 6. Solver notes still have it as a “WXYZ wing”. There is still room on the grid to show the next in order, an XY-wing ANL.

Also from the XYZ-map, a classic, but dead  Death Blossom Lite (11/29/11) hinged at r6c6.

From the X-panel, a 3-chain ANL (black) extends into a 3-chain ANL confirming SW3.





Even on the 4-panel, the grouping on these almost nice loops is very difficult to see. You might need to bone up on Sudoku group theory.







Consider the next unaligned pair exclusion a reaction to the grouped ANL removals, as the bv r5c5 becomes available as a target cell.  Combinations 45 and 46 are denied the target cells. The removal’s Cr5 box/line terminates 4r6c6.






The solver followed the Sysudoku phasing to this point, but now does the AIC scan prior to coloring. I confirmed that coloring is not effective just yet. The solver does a confirming AIC with a jaw dropping ALS node.

The ALS node

is based on the slinks between candidate groups in an ALS.

The removal of 3r9c9 brings a third ALS to bear on APE target cells r78c9, and 3r7c9 is allowed no partners. The blue/green cluster expands to trap two 4’s. An AIC confirming ANL is cobbled from 1-chain, 4-chain and XY-chain segments.



The solver continues with XY-chains, coloring, and a naked triple. In case you’d like to take it from here, the finish is reported next post.  The parade of diverse advanced methods in the solver solution, many weakly signaled and difficult to spot, justifies an extreme rating.  And it’s not over yet.

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World’s 36 and 200 Basic, and More on 25

In this post, checkpoint traces on World’s Hardest 36 and 200, and a direct comparison with  two Strmckr solutions to Worlds 25. Also, a Gordon Fick contribution to the cause.

Here is Strmckr’s first solution, right out of line marking. It’s an almost nice loop AIC bringing the adjacent winks together to remove 2r1c7.

It looks easy, but it’s not, considering what you have to be looking for, to spot it. There are two multiple value cells and two AIC hinges involved. Strmckr’s “singles” collapse is rapid and complete.

At the point in the SysOB where AIC  hinges are  applied, I have two coloring clusters.

Instead of the pattern analysis and trial, I should have noticed the decisive trap. It’s that both blue and green remove 2r1c7, the same removal as Strmckr’s AIC ANL.


Coloring hastens the collapse as well. The blue/green cluster expands before 6r7c3 wraps green.

The approach towards a BUG explains why 25 can be very hard.

I’ve now released Strmckr’s 4/24/18 comment with his second solution.


It involves a cleverly selected ALS pair. Strmckr takes no credit for that, but instead, ascribes the  removals to “transport”, a forum concept whose definition does not respond to my search.

No worries. With the pair of 2’s as a restricted common, 4r9c7, and 6r7c7 qualify as ALS_XZ victims. Shortly after, 6r4c2 and 2r1c7 follow.

But there’s more.  Sysudoku’s own ALS_XZ expert Gordon Fick responded to the homework challenge and the ALS Partnering idea with an apt alternative.

Working with the bv 28r1c8, try the possible RC with 8r8c8. The r8 row has extra 8’s, but there is an SE ALS with 2 and a single 8.





Now for the checkpoint basic traces. First the basic rout of Very Hard 80, one of two rated second highest by Sudokubooks. Obviously not the monster.

If you’ve never read a trace, filling in the grid as you go, try it out. It took me several tries to get the trace right, and it is very useful. Just give yourself time to know exactly what each move is, and why it is known to work. Read Sysudoku Traces as necessary.

The second trace then, is for a near monster, World’s Hardest 200.

Here is the line marked grid, with fill strings. If you can solve it, tell me how you got it started. I’m sure sysudokie readers will be interested.

Next week, the World’s Hardest review continues with 200.

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The World’s Most Over-Titled Sudoku Book

This post starts the review of World’s Hardest Sudoku Book, tracing Very Hard 25, the only one rated a 1.00. The review will cover the highest rated 10, taking the rating down to 0.90. The lowest of the 200 is 0.76. Again, I know nothing about the rating, or the publisher, Sudokubooks. There’s no background information, or even a copyright notice, in the book.

Another reason to be prepared for surprises is that Strmckr, a Sudoku forum contributing expert, and an interested reader, has already sent in his solution notes on Worlds 25 in a comment. I can’t publish, or even look at, his note until this post is out, but an exchange may appear later, if only as 4/24/18 comments.

As for the my sysudokie trace on 25, it started in a scary way with a bypass of the bypass, and no clues from box marking, either.

But the outlook brightened with line marking, and on the first candidate grid, with an XY-wing generating an Ec9 boxline removal. Also, it was good to spot a BARN, even though its locked 4’s claim  no victims.




Then on the X-panel, a 4-chain. In the XY-wing area.





The bv field in the East tower encourages coloring, and a 6-chain enables an expansion for a trap of 8r6c1.



A 6-chain/XY AIC almost nice loop and a second cluster don’t break the ice, but with the removal, . . .



. . . the 6 and 8-panel freeforms reveal single and compatible blue patterns.


 Blue quickly confirms orange, and the solution.

The heavy reliance on XY-chains, the X-panel, coloring and patterns (templates) is typical sysudokie. The Strmckr approach is probably different and sure to be interesting.

For next week, the next two review puzzles are rated 0.92.  One falls in the bypass and the other is a slippery monster.




You have a week to decide which is which, and to solve one in sysudokie basic. The basic traces of both puzzles will be posted next week.


Then we’ll continue on with the tougher, and perhaps, more on 25 solution paths.



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Strmckr’s Double Common ALS_XZ

This post translates an instructive example from “strmckr”, an expert forum contributor, and adds some observations about double common ALS_XZ.  Then Sudent comments on some Sudoku history that Strmckr included with his example.

In his comments on the April 10 post on a possible WXYZ vs. BARN settlement, Strmckr supplies a very helpful example of a double common ALS_XZ. Forum participants, through a long history of text communication, transmit the grids to each other in unformatted text.

Here is Strmckr’s example, more accessibly marked for Sysudoku readers.

Many ALS_XZ are BARNs, but is this double common one a “double barn”? No, this isn’t a “barn size 4” at all.

But it does work.  It’s a valid ALS_XZ, and also a BNS0 Robert Hanson’s Bent n-Set 0.  The 4-set remainders are disjoint, and every value is therefore locked. That itself guarantees the removals, including 8r3c8.

Strmckr was especially considerate to point out that his example is also a Sue de Coq, evaluated as we do in Sysudoku, with remainder ALS. The 3 or 6 not in SEc8 must come from 36r8c7, and the 7 or 8 not in SEc8 must come from 78r1c8.

But getting back to the double common ALS_XZ above, Strmckr states for us the fact that all values in both ALS become locked. That’s true of all ALS_XZ. He could add that with double commons, the uncommon values in each ALS are locked as well. Each ALS loses a common value and both are therefore individually locked in each uncommon value.

But that’s not all. The locked sets includes the double common values, but unlike the uncommon values, common values are locked across both ALS; not  within each ALS. That’s because we don’t know which ALS gets which common value, so a common value victim must see all common values in both ALS.

Thinking about this, I realize that I was overly restrictive in ALS Partnering, my lens prescription for spotting most ALS_XZ. Every member of a restricted common must see each other. That guarantees that when one of the ALS gets a common value, the other doesn’t. But the wink between common values in the two ALS can be an AIC. That can open the butterfly box for some amazing ALS_XZ. On the downside, it makes solver coding and comprehensive human spotting even harder.

My reply to Strmckr’s first comment asks his opinion on inference chains as weak links in XYZ. We missed out on that, because he took it as question on ALS nodes in AIC, which are covered in Sysudoku.  His statement that “xyz-wings are always constructed as a barn” prompted the question.  Apparently, Strmckr and the forum remain unaware of Sysudoku irregular XYZ-wings, and irregular victims of toxic sets.

Or perhaps, it’s another instance of Sudoku coders avoiding the weak linking AIC, a.k.a. forcing chain, as a subject of discussion. It’s burdensome to program them for a solver. Line and column “seeing” does encompass a simpler and more easily coded world.

Strmckr’s comment claims agreement with the subject post, but actually he disagrees. There are two disagreements in his statement introducing the double common ALS_XS rule:

“ALS-xz double linked rule does apply to wxyz-wings, aka ‘barn size 4’ “.

Strmckr does not explain, or give any reference to explain, how the ALS-XZ double common rule applies to the WXYZ wing that is not also an ALS-XZ.  And the statement ends by describing BARN 4 as a commonly agreed synonym for WXYZ-wing. It certainly is not that .

Strmckr believed he was rightfully extending the WXYZ in his original BARN post.  But actually his “single bent value” BARN is equivalent to Bob Hanson’s Bent 4-set 1, leaving no doubt that Strmckr was inadvertently mislabeing Hanson’s BNS1 as a particular form of WXYZ.

The BARN and the BNS1 are two means of spotting the same thing. If a single value is bent, then the bent area remainders share a single value, and vice-versa.

In his comment, Strmckr takes credit for dismembering the 4-candidate hinge of the WXYZ.  That abandons the primary feature that characterized the wings. The forum should not have agreed so readily.

Strmckr comments that his WXYZ modifications were for “opening up its very narrow search window”. The very narrow search window is due to the increasing size of the toxic set with increasing wings, aggravated by an unnecessary limiting of weak links to house membership. “Seeing” the entire toxic set gets harder very rapidly.

But this is a logical consequence of the XY, XYZ, WXYZ, . . . definitions, and there is no advantage gained by compromising the WXYZ identity to relieve it.

Programmers have a better way to extend and maintain interest in higher wings. It’s AIC weak links.  Humans don’t find them all, but they do have fun with them.  Have you seen what they do for the XYZ?

Next post starts a new review, with a twist. The book is inaccurately titled “World’s Hardest Sudoku Book” from Sudokubooks, the unidentified publisher who is, by my guess, Moito, Rebecca Bean, and A.D. Ardson as well. The review puzzles are the hardest 10 by the rating assigned by Sudoku books, which is the same format as the others just listed, but not explained by any of them.

Your homework is the only 1.00 rated, #25. The twist is to use the review to decide where the ALS Partnering and APE Type 2 should be placed in the Sysudoku order of battle, right after unique rectangles, or just before AIC hinges. I’ll try for easy ones in both places, and after reviewing checkpoints, you’re invited to find some I missed.

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An ALS Toxic Set Update

This post reviews Sysudoku history on the problem of spotting ALS toxic sets, and presents the updated  recommendation in the ALS Toxic Set page just completed for the Guide.  This spotting process, to be referred to here as ALS partnering, will be applied in future reviews, and updates

In sysudokie speak, a restricted set of like valued candidates guaranteed to contain a true candidate, a solution placement, is a toxic set.  “Toxic”, because any outside candidate seeing all members of the set is removed.

In a pair of ALS sharing a restricted common value, the like valued candidates of a different value in both ALS are a toxic set. The expert community uses the term ALS_XZ for this toxic set, understandably because almost all of the ALS toxic sets actually spotted by human solvers are ALS partnered with bi-value cells.  The XZ is a reference to the bv wing of XYZ or WXYZ wings.

The logic of the ALS toxic set, that one of the ALS loses the common value, and becomes locked, is best explained in those general terms.  On the other hand, Sysudoku has struggled with the obvious fact that the complexity of detecting a general ALS toxic set is too much for human solvers to undertake, especially those with the knowledge that methods easily spotted with template aids, like XY chains, fish and coloring, come next.

On my first encounter in July 2012, the Sysudoku remedy was to graphically display all ALS, then comparing all promising pairs. By October 2015,  I was ready to admit defeat, with a compromise, updating the 2012 post to generate one ALS at a time, looking forward in the sequence for promising partners. As an algorithm that saves half the ALS vs. ALS comparisons. Also it means that only the most promising, as a partner of a given ALS, are marked on the grid for testing.

It didn’t work. Over the intervening 2 ½ years and 150 posts, I have found very few ALS toxic set eliminations. My talented friend Gordan Fick, on the other hand, has spotted many for me.  My problem is, being neither a computer or Gordon Fick, I need an explicit signpost procedure, something I can recognize, that leads me to the crucial test, and will lead other ordinaries to find enough ALS_XZ to inspire them.

So now Sysudoku takes a third swing, an ALS_XZ spotting technique, is available in the Guide. I’m just hoping to get on base with ALS Partnering.

This is applying the construction vs. search principle to ALS toxic set.  Rather than looking for two ALS that fit together in the right way, look for possible first ALS components. With each one, you are now engaged in one or more guided constructions that will arrive at an ALS_XZ if it exists.

This prescription was tested with the blog cases, all of which it covers. Then to test evaluate the scanning effort and spotting improvement, the selected review puzzles from Castillo’s Only Extreme Sudoku were scanned immediately after line marking. This confirmed that ALS partnering is a large effort.

As to results, In Only Extreme 218, a regular 671-wing,


is duplicated by an ALS_XZ.



A more significant result occurs in Only Extreme 347. The 6’s are locked in the 3 value BARN, but there are no victims.  Also, this is not a BNS1, the remainder values being 569 and 569. But it’s undeniable that the victim 5 is looking at all 5’s in the two ALS.

Missing this removal in the review is significant as well. It enables a Sue de Coq and avoids a coloring trial.

Although ALS partnering does provide a clearly marked trail to follow, it still takes considerable time per productive ALS_XZ. In the Sysudoku order of battle, it is placed with other bv scan methods, to be  completed  prior to template constructing methods. That avoids the appearance of unnecessary scaffold construction. Aside from that consideration, ALS partnering might be better positioned after coloring.  The time it takes might be better spent on a puzzle actually requiring it.

Next week, a sysudokie version of the case study puzzle in Strmckr’s comment on the April 10 post, and some insights it brings. To bring up comments on a past post, click Comments at the bottom.  Also in the next post, a little more on the WXYZ, and some accounting for our honest differences.

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A WXYZ vs. BARN Settlement

In this post, progress is reported on the Sysudoku Guide, and on  a possible territorial agreement between the WXYZ wing and the Bent family.

The Guide, that link up there on the menu bar, is taken shape. The Basic pages are complete, with all the elements of Sysudoku Basic explained and demonstrated. That includes the bypass, 3-fills, slink marking, box making, line marking, subsets, and box/lines.  On the Advanced Side, there are pages now on unique rectangles,  Sue de Coq, APE, Bent n-sets, ALS toxic sets and XYZ-wings.

That advanced sequence may not fit your order of battle, but it is meant to delay the construction of visual aids until they are needed.  With XYZ Wings, the sysudokie beginner encounters the first of these, the XYZ map. Later will come aids for XY chains, X-chains, fish, coloring and patterns. My grid and aids are ©PowerPoint and ©Word template files.

Writing up the XYZ page led me back to the friendly argument I enjoyed with Andrew Stuart and the EnjoySudoku forum writer Strmckr, about the overlap between the WXYZ-Wing and a discovery by Strmckr.

It is basically a labelling controversy. Labels help us sort out the interplay of ideas and we all know that misapplied labels get in the way. What happened in this case is that the WXYZ wing, starting life as a logical extension of the XYZ wing was already compromised and losing its label to overlapping methods, when Strmckr noticed this property of an overlapped type of WXYZ-wing.

One of the WXYZ  overlappers is the ALS_XZ.  A lesser known one is something labeled here as a bent n-set, a collection n cells containing only n values (hence naked), sharing one box and one crossing line. I learned about it from Robert Hanson’s Sudoku Assistant report.

Here is the sysudokie view of the WXYZ -wing and one of its victims. The dashed lines are weak links (winks) between like-valued candidates in the hinge and the wings. OK, the wings can also be in larger ALS packages, but the winks remain. Any outside Z that “sees” (winks at) all four Z’s is toast, because its looking at a true candidate. If Z is not placed in any wing, it goes into the hinge. That’s identical to the XYZ-wing rationale.

The thing is, if weak links are limited to candidates sharing a box or line, its hard to find a sysudokie WXYZ wing, and even harder to find a victim. Instead, you need split hinge and a crimped wing arrangement to make it work, and those arrangements are bent naked 4-sets within the box and line of the hinge. Also, they are often ALS-XZ, but let’s leave that out of it here.

Bob Hanson stated an easily checked condition for the bent n-set to have a locked, and therefore toxic value. That value has to be the only n-set value shared by the n-set cells in the bent region’s remainders. In sysudokie speak, the box/line intersection is a chute, and the units minus the chute,  remainders. An n-set meeting this Hanson condition a Bent n-Set 1, or BSN1. That’s almost Hanson’s term, but with “n-set” where Bob said “naked subset”, which it isn’t. You know, labels.

By the way, if the remainders have no n-set value in common, you have a BSN0, with n toxic sets. The candidates of every n-set value see each other. It’s a subset, not of a box or line, but the bent region.

Now we get to the heart of the matter.  By diligent observation, Strmckr discovered another, equally easy condition for a BNS1:I  If the n-set candidates of only one value are bent, i.e. not “restricted”, i.e in both remainders, then these candidates are locked, i.e. must include a true candidate, i.e. are a toxic set. The “almost restricted” condition guarantees the BNS1 condition, and therefore has a basis in proof as well as observation. Hanson explains why the BSN1 works. My latest attempt at a simpler explanation is on the Bent n-Set guide page.

I had thought that, at the time of Strmckr’s “almost restricted WXYZ” introduction on the EnjoySudoku forum, the WXYZ had already been compromised with a distributed hinge, and BNS1 with n = 4 were being mislabeled as WXYZ, and that Strmckr had little choice but to  label his discovery as a property of the WXYZ wing. In his comment below on this post, however, he points out that the dissolution of the WXYZ hinge was his proposal, made to add flexibility to the WXYZ. Now his forum thread on the alternative condition for BNS1 is properly titled Bent Almost Restricted n-Set, the sysudokie amalgamation of Strmckr description and Hanson terminology, and he freely describes it as “the barn” without caps.

All of this is actually relevant to your homework, to solve Andrew Stuart’s example 4, In his WXYZ page under ‘Bent’ Sets.  When Andrew challenged his Sudokuwiki readers to find four WXYZ -wings, I thought there might be one of those hinge cell, unit seeing, classic WXYZ wings among them. Did you find one?

Here is the basic trace, with an active and interesting bypass to make the remaining stages routine and easy.






Prior to the first WXYZ, here are two of Andrew’s Y-wings (XY wings), and a regular XYZ wing. No iXYZ or or XYZi were available.






An AIC almost nice loop is enabled, and leaves the 4-set shown in Stuart’s WXYZ example 4.



The 4-set 1289 is in the Wr6 bent region. It is a BNS1, with remainder values 18 and 289. The single common remainder value 8 is locked. The 4-set is also a BARN, with single unrestricted value 8.





The removals leave a 4-set in bent region Cr5, with 4-set remainder values 349 and 89, and common value 9. The toxic set 9 removals in r6 leave a Cr4 boxline triggering the collapse.

Oops. It takes only two “WXYZ wings”, instead of four.


Maybe somebody will show me the solver’s other two. I have to recuse myself from using any solver. Hopefully, Andrew Stuart can now agree that these two “WXYZ wings”  should be classified as BNS1’s or BARN’s, depending on how you identified them.

By the way, all of the sudokuwiki WXYZ examples are also ALS_XZ. My next post updates sysudokie readers on that.

Seriously, there is really no good reason why the classic form of WXYZ should not be given the exclusive use of the name. That would create space for iWXYZ, WXYZi, and iWXYZi pursued with inference chains as weak links. In concession to the bent side, n-sets are not limited to n = 4, and should not be tied to a label that is.

Also, the bent family rationale is logically worthy, and needs to be respected and passed on to Sudoku enthusiasts without asssociation with an entirely different rationale. The work of Robert Hanson and his mentors, and Strmckr’s BARN perceptions, need not be muted by prior inaccurate labeling.

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