This post corrects an error on the post of December 8, that prevents a lite coloring trap in ultrahardcore 399. Then there’s follow up on the trap, with the lite coloring marking in place. Trial is not avoided, but the lite coloring hastens the final wrap by contradiction.
Here’s the correction on a goof by yours truly. On the APE by Sudokuwiki and ALS_87 by Beeby, I deleted 7r3c8, instead of 7r2c8. In the post, that error is carried forward without incident, not affecting any eliminations or the coloring trial which solves 399.
Now coloring lite has a trap, on that very 7r3c8 spared by my error.
7r3c1 is blue lite, 7r6c8 is green lite.
The removal does little damage to UHC 399. Realizing that, I enlisted Beeby by stepping through its former finds, and using its REMOVE feature, to take out the lite trap’s 7r3c8.
Beeby came through with one more removal, Take it as a boomer from 7r7c8 or a 1-way from 3r4c8, depending on your starting point.
Beeby runs out of human options now, and I put in some time on winning one for lite coloring, but couldn’t. Maybe you did.
There are alternatives to the green coloring trial of the review.
Cc4 contains 6(1+8)(4+5), so orange comes with a C45 pair. The SASdC in Nr1 requires a trial of Nr1=572.
Also there’s a pattern situation on the 7-panel left by the lite removal. You could test the three patterns containing 7r7c9, 7r6c8, 7r8c3 by including them as clues and dropping the 6 orphans. If that trial fails, you would know that 7r2c9 and 7r3c1 are true, and that 7r2c5 and 7r2c7 are orphans.
However, since we get a boost on the green trial as 7 green lite candidates turn green, we’ll take that. Comparing to patterns, if the trial fails, we get 6 clues and 6 removals.
The point is that lite coloring also enhances trials.
This time, compared to the green trial in the review, the green clues change little, because we have already incorporated most of their effects. The color expansion is in the red/orange cluster, which grows in a distinctive way, by adding naked pairs.
looks like I may have to try the red or orange army. Then I notice that in r2, orange will enable a now naked triple to confirm 4r2c5 and then red 1r2c1. No, Virginia, orange doesn’t get to confirm red. It gets tossed for trying, and collapse follows.
Compared to the review post, the color solution has the same reds, but a few more greens.
There’s now a lite coloring page in The Guide, mostly based on last week’s post. It’s a child of the coloring page (drag off right). Guide pages on Sysudoku trials are coming soon.
As to posts, the next one will look back to a previous attempt to extend coloring. Suggestions on collections to review or topics to explore are welcome, either in comments on posts or at email@example.com. Please attach comments to relevant posts.
This post explains lite coloring as I understand it from comments and emails from its apparent inventor, Dov Mittelman. I only know Dov as a reader of this blog. He (or she) does his analysis in his head and on his cell phone, and is not given to definitions or proofs, but is a very talented Sudoku solver. All this doesn’t help much in making things clear for ordinary solvers like me. On the other hand, I’ve gone much further than most Sudoku writers with the interpretation of strong link networks by coloring, so I can understand why, seeing this, Dov had the patience and fortitude to explain it to me.
Although I can tell you how lite coloring works, I hadn’t been blessed with good examples until another reader friend Gordon Fick put me onto the “Red Russel special”. Here’s why examples are scarce: A lot of hard review puzzles finish with coloring, but it’s pick, pick , pick, candidate by candidate until a cluster or two can be started, and then it ends quickly. Lite coloring is needed most when the clusters are there but aren’t working.
The puzzle I name here the Red Russel Special is a Sudoku presented long ago on the EnjoySudoku forum by contributor “Red” Russell. Its givens and prospects for Sysudoku solvers appeared at the end of the previous post, the last one of 2020.
The basic trace shows where the four X-wings and hidden quad occur. Yes, nada on the bypass. Beginners, you can learn all about traces in The Guide, linked above.
Now as we stop on that line marking 4-wing, the grid shows why, as we mark the r8 slink in its southwest cell corners, it’s so easy to spot the matching slink in r3. Of course we need to leave the fish icons on to pick up more 4-wing victims as we line mark more rows.
Pardon me for dwelling so long on Basic Sysudoku, but the Red special illustrates line marking X-wing spotting so well. If you don’t actually do it this way, take the time to follow the line marking as line fill strings get longer and longer. We line mark the easiest first. If it’s confusing, then read the line marking page in the Guide along with the grid pictures.
I missed one X-wing at first, but here is all four of them together, the first two on rows. and the second two on columns.
Yes, Virginia, X-wings are fish. For what makes a fish a fish, go through Advanced to fish in the Guide.
Let’s leave the hidden quad for later, and recognize that each X-wing is a coloring cluster that could grow. Coloring? Better see
The clusters are small and bear no apparent relation to each other. This “Red “ Special shows how lite coloring connects the clusters, extending their capability to trap and remove outside candidates.
Here is a definition of lite coloring:
The definition itself explains how lite color extensions create new traps. If an outside candidate sees two opposing cluster colors, lite or regular, it is seeing a true candidate, because all candidates of a true color are true.
The definition doesn’t tell us how to identify the lite color candidates. They come in a sequence. With any color candidate, full or lite, identify slink partners seen by that candidate. They are false when the starting candidate is true. And in each case, the slink partner is color lite in the starting color. The color lite procedure is, generate a branching tree of lite candidates of the starting color. Naturally, the root of the tree is a full color candidate. The sequence does not include candidates of the opposing color, which generate branching sequences of lite candidates of the opposing color.
Let’s walk through the marking of a tree of lite candidates of the Red Special clusters.
Starting with blue 4r3c1. If true it erases 5, 6, and 7, and no other 4’s, except the green ones. Only 6 is a slink partner, and its slink partner is blue lite. It is blue when blue is known to be true. The cell corner slink marking directly assists lite marking.
In Sysudoku, we draw arrows to mark lite candidates. The arrow stands for a wink to a candidate erased by blue and slink to the lite blue candidate. You can use either the arrow or the wink/slink sequence for marking. It doesn’t matter which blue candidate the arrow comes from. Any candidate on any arrow path from a blue candidate is blue lite. What matters is blue becoming true.
Lite coloring inventor Mittelman uses lighter shades of the same color for lite candidates, but I use arrows. For one reason, a candidate can be lite in more than one color. Also, in PowerPoint, my colors are already light shaded to make values visible, so it’s hard to make them lighter.
Continuing the lite sequence from blue 4r3c1, it’s OK for blue lite 8 to claim yellow r5c2 is also blue lite. But that doesn’t mean yellow is blue. It becomes blue only when blue is confirmed to be true.
The other blue candidate 4r8c5 claims red 6r7c9 as blue lite! A red candidate can act as blue lite in a trap. It does that, making 4r9c9 blue lite. Since 1r9c1 is green lite, 4r9c3 which sees blue lite and green lite in r9, is false. Red 6r7c9 is red, but blue lite 4r9c9 is not necessarily blue, but for sure, blue or green is true, so blue lite 4r9c3 or green lite 4r9c3 is true. The trap works.
But there is something else. Blue makes 3r7c5 true and that forces orange candidate to be false when blue becomes true. But orange and red are opposing cluster colors. If any orange candidate is false, they all are, and all red candidates are true. So now, red is true when blue is true. How about that? Red is blue. A bridge between clusters.
We don’t see it here, but a lite coloring wrap occurs when lite coloring chains from opposing cluster colors meet on a candidate of another full color. The full color is true, and its opposite is false. No waiting for a chain color to become confirmed.
Let’s merge the red/orange cluster with the blue/green cluster and re-mark the red/orange lite sequences as new blue/green lite sequences.
1r3c6 is blue lite and green lite. It wins either way .
6r1c7 sees green in the NE box and lite blue 6r1c1.
3r6c7 sees green lite 3r1c7 in c7 and blue lite in the East box.
It’s a short follow up to blue 3r7c5 and a green wrap.
Now green lite sequences disappear, and blue lites become blue clues. Including yellow 7r5c2. Yellow 7’s are confirmed as blue, and the purple 7’s they see are removed.
The investment in lite coloring pays off with the additional blue clues, and collapse is immediate. Just be careful picking through the clutter.
Here’s the lite coloring solution with the surviving blue color.
Way back on the Basic trace, the hidden quad appears as X-wings and removals are removed, on the closing scan of c5. The subset candidates are encircled in light blue and the blue diamonds are the removals.
One more line marking fact: although all the candidates of the quad are identified when r5 is marked, the columns have not been examined for subsets. Rows or columns, whatever set finishes first, the unmarked “other” lines are examined for slinks and subsets, i.e. they are “closed”.
The quad removals bring a sharp collapse. Here is the Sysudoku trace, down roughly to the green wrap above.
The Red Russel Special gained attention for several reasons. It’s posted here, and in the Guide, because it illustrates so well the technique and possible results of Dove Mittelman’s lite coloring.
Next week, we correct an error in the December 8, 2020 post and make another attempt at a trial free solution to ultrahardcore 399 with lite coloring. The 7 in r2c8 belongs in r3c8. I’ll leave it uncorrected for now. Aw go ahead and DIY it, you’ll see where the error applies.
The ultrahardcore review ends, with Stefan Heine’s 487 giving up a naked single and two skyscrapers, and a jaw dropping complex 1-way, shuts down the solver team. Slicing at it with three onion peeling pattern trials leaves very little for the team to chew on. The post ends with a few words on how the 2021 Sysudoku blog will be a little different.
In line marking UHC 487, r6 gives you a naked single W7, free of charge.
Then in r1, there it is, another skyscraper in the 2’s. This time, the ANL victims are slink partners. We get two clues, and we don’t need to leave the chain behind.
But that’s not all. The S2 clue leaves a 6-slink in r9 that matches a slink partner in r7. This skyscraper we leave on for another victim in r4.
After this promising start, my solver team gets nothing, except a truly impossible Beeby complex 1-way for one more removal, plus a SEc8 boxline removal.
Readers who despair over my diagrams should see the corresponding Beeby chain notation below.
The special characters mark the departure of winks to erase an interfering candidate. The symbol repeats at the slink created by the erasure. These erasures are valid only in the chosen direction of the chain.
To avoid unnecessary notation reading headaches, ignore the special characters at first, simply accepting the (=) slinks. Then draw the branches from the character marked candidate whose erasure enables the slink started at the second symbol.
On the 6-panel, there are only two South to North freeforms containing 6r9c1. Starting the trial on the first 5 of 7 candidates, the trial itself will probably select the right path, if there is one. If not, then the failure confirms 6r9c8 as a clue, removing it from the 6-panel.
The trace is long, but there is a steep collapse to a contradiction as two 3’s are forced inton c7. The clue and two removals do not wake up the solvers.
Next I look at other edges. From East to West, there is only one pattern containing 6r3c9. If the freeform fails, 6r4c7 is an orphan as well, because c9 must be included.
Again this test of 6 candidates fails quickly.
Now we have two sets of East to West freeforms, three from r4c9 and and two from r6c9. The red one fails to reach c1.
In a trial of patterns containing r4c9, an initial coloring yields two traps.
gets to a 497-wing
followed by an XY ANL
and a green wrap.
So what pattern was the true one? Now we know.
That winds up our review of Stefan Heine’s ultrahardcore right page collection. Let’s keep trying for ‘no trial’ humanly practical solutions to these gems.
So what happens now?
There being no urgent reader requests for another collection review, early 2021 posts will be about expansions and revisions of The Guide, that link on the bar menu. You will be in on explorations, independent of a review schedule, of current and prospective Sysudoku Guide topics, prior to its publication in book formats.
Posts will continue to be weekly, Tuesday noon EST, but I expect the blog will be much more amenable to meaty comments and very careful replies. As many readers know, I keep comments of interest only to myself to myself. Comment about your ideas. Put them on a post or a page.
The first 2021 Guide topic is lite coloring, to my knowledge, an invention of reader Dov Mittelman.
Thanks to an alert from another expert friend Gordon Fick, here’s a puzzle recently highlighted on the EnjoySudoku forum that provides an excellent example, when you overlook the hidden quad that destroys it. Next week, you’ll get my definition of lite coloring. If you’re into Sudoku Basic, try to corral three X-wings in line marking, then the quad. The puzzle is attributed to “Red”, a.k.a. Ed Russell, an early regular forum contributor. The forum offers “Pattern Game” solutions for the Ed Russell special, which could be one of those Guide topics later.
After the hidden unique rectangle hinted last week, a pair of Single Alternate Sue de Coq trials gets the review solver Beeby moving again. ALS-wings and a finned X-wing get us to a colorful finish.
The hidden unique rectangle left on grid is signaled by the bv 6 and 7 repeated in three more corners. The resolution method is marked by the three side slinks in the UR value 6. If true, 7r8c9 generates a reversable rectangle in a double solution.
Both solvers now quiet, my choice for a trial is the Single Alternate Sue de Coq Cr5, with contents described by 7(5+9)(1+8) +817.
That’s because it can’t contain 5 and 9. It must be 5 or 9 or neither. We test neither, i.e. 817.
The test reaches a contradiction on the 9th breadth first level, leaving the Sue de Coq
Cr5 = 7(5+9)(1+8).
Now the (5+9) and the bv 59 sweep other 5’s and 9’s from r5, and the 9r5c9 removal brings the boxline removing 9r3c7.
This leaves Sue tugging on our sleaves with two Single Alternate SdC’s
Er4 =1(2+8)(7+9)+971 and
Wr4 = 5(7+9)(2+8)+582.
Both Er4 =971 and Wr4 =582 produce the same contradiction as before, and both remove 7r4c8, leaving the pair E28 to remove 8r6c8.
After a 2-chain ANL, Beeby does a pair of ALS-wing, the first, depending on singles values,
The second, sharing an ALS with the first. The 7r4c2 removal allows
a finned 7-wing. The 7r7c9 fin sees the victims, so if it’s true, they’re toast and if it’s false, the 7-wing toasts them.
Next, after a 7-chain ANL,
and NE3 => N3,
Beeby’s next overlapped ALS-wing.
We sit back and watch as r5 naked triple generates W1 and the West box naked quad.
Now what is that thing?
It’s a classic boomerang with an ALS providing the closing slink.
It’s also a 1-way. If 8r7c4 is true, 2r7c4 is not, and if 8r7c4 is false, the ALS AIC promotes another 2 in the South box.
The removal adds a critical slink for the hidden unique rectangle.
Seeing an XY chain,
And an easy AIC, I think that closing this down with coloring may be overdue.
On first coloring, the bridge not(green and orange) becomes mute as green is wrapped by a trap induced boxline.
In the follow up and red/orange expansion,
1, 2 and 4 squeeze out 7 and 9 and the Sc5 boxline and trap removing 7r8c4 confirms orange.
Here’s the colored solution.
One more ultrahardcore to do, UHC 487. The site and the portable versions of the Guide need attention, and the only reason for doing another collection is to confirm any reader opinion that it is likely to reveal more about human solving. Stefan Heine’s ultrahardcores has certainly done that.
This post reports a series of Beeby ALS-XZ featuring alignments in Z and X, almost to the point where a trial is needed.
A simple basic trace means a hard start, with a crowded grid.
Sudokuwiki’s 579-wing is a Beeby ALS _97. Both solvers do it twice, because nobody codes a second look for victims.
Next we get a digit forcing chain boomer from 1r6c1.
Rendered by Beeby as a overlapped ALS-wing. Can you follow the money?
Now Beeby continues the ALS clinic with an ALS 4_3 with grouped X and grouped Z,
an improbably aligned ALS-wing,
then another one equally improbable.
Using Phillip Beeby’s solver, you choose the method category on each step, so the order of methods found can be subjective. Sudokuwiki imposes its own order.
Now after a series of ALS, Beeby runs to exhaustion with AIC.
First is a relatively simple complex 1-way starting a 6 r2c6 with a single branch to manufacture a 1-way slink. It’s ending 6r3c8 happens to see two victims. This isn’t done twice. Beeby notes both in the chain description drawn for you. You’re welcome.
Then you get a more complex 1-way, starting at 1r6c1 and sporting three branches.
Looking at this, you realize there is no definable boundary between practical and impractical human human solving.
Sysudoku readers know exactly how to find such a thing, but they have a life, so its likely that nobody will.
Let’s leave the rest for next post. Start with the hidden UR that leaves the techniques of this review exhausted. Then decide on a trial from the repertoire of this review and go for it. Or meet that challenge we started the review with, an ultrahardcore right page solution without trial.
Or do some homework on the last review puzzle, ultrahardcore 487.
In one post, the 10th ultrahardcore review puzzle is wrapped, but not without an introduction of something new in Systematic Sudoku. Ordinarily advanced steps begin in line marking, and lite coloring is introduced.
In line marking Stefan Heine’s ultrahardcore 399, the Sysudoku marking of line slinks pays off on the third line.
On r3, the 6-slink doesn’t match the one on r9 for a 6-wing, but the alignment on one column signals a 6-chain known as a skyscraper. The weak link doesn’t depend on column 5 being filled.
The X-chain can’t identify all victims until their rows are filled. So, like we do with X-wings, we leave the skyscraper on the grid until the remaining rows are filled.
The skyscraper claims a victim 6r7c8 when c8 is finally marked,
This time it didn’t matter, because the 6-wing on columns c6 and c7 removes that victim before r7 is filled on the close
Sudokuwiki and Beeby, like most solvers, do X-wings and X-chains after basic because they number scan all candidates around the givens first, before searching for anything. Beeby finds the skyscraper first, and notes it as a Sashimi 6-wing. Now in advanced mode, they continue to fight it out in the notes on what they’re finding.
Sudokuwiki’s APE becomes Beeby’s ALS_87.
Then Sudokuwiki digit forcing chain Sysudoku boomer from 9r7c8 is duplicated by an eyebrow raising Beeby ALS_31.
Sudokuwiki’s dfc grouped ANL is Beeby’s grouped 1-way.
And a Beeby grouped ALS-wing is described as another digit forcing chain.
The solvers disagree on how to start the boomer, and Beeby takes the longer path
In this Sudokuwiki APE, the blue ALS excludes 41 and 43; the green ALS, 47 and 48.
Beeby’s ALS_34 mixes the ALS a little differently.
Both solvers hang up after Beeby’s second ALS ANL
But an extensive slink network is left behind to support two clusters.
Sysudoku reader Dove Mittelman suggests the marking of candidates in a “lite” color if they are true when the color is true. These “lite” candidates work in coloring traps. A candidate that sees a lite candidate or group and the opposite color is false regardless of the true color.
I mark lite candidates with arrows because I already use lighter shades for visibility.
Note above that lite candidates are those slinked with full colored candidates of a different value. Lite traps require seeing opposite color candidate, full or lite, of the same value. Lite green 3, 5, and 9 candidates have no blue trap partners. Lite green 7r6c8 does have a matching lite blue 7r3c1, but no 7 candidate sees them both.
I’ll come back after the review when I have examples where lite coloring is productive.
Although lite coloring fails in this case, when green generates many lite candidates, that makes it a good choice for a color trial. Full color traps and an orange wrap in c2 come quickly, confirming green and red.
The solution in colors:
Next, we start on the two extra ultras, with UHC 443.
In this post a SASdC trial plays like a regular hard puzzle from other collections.
The trial amounts to adding two clues, and seeing what happens. But it isn’t guessing, and instead of making a small gain, the trial solves the puzzle. Having given that away, here is the last ultrahardcore, UHC 399, for your homework.
Except it’s not the last. I slipped up on the spacing of the ultrahardcore review puzzles and did not cover the last 100, so I’ll add right page ultrahardcore 443 and 487. After that, I may just halt collection reviews and turn to refinement and publication of the Sysudoku Guide. I can be tempted to do another review if you can come up with another tough collection I can use as an excuse to keep drawing grids. I’m only 83.
On UHC 355, Beeby gets the nod to go to work on the Single Alternate trial after the follow up and reload of the previous post. As in all trials, new clues are marked as centered pencil marks. The first result is ALS_74.
Human ALS_XZ spotting depends on limiting attention to ALS with two singles as value groups, or a single plus group in a box for Z or aligned outside for X.
Next is an AIC ANL with two AIC nodes. It shows why the almost nice loop is more powerful than the logically equivalent 1-way or boomerang from 4r8c9. The latter two don’t remove 4r7c8.
An XY ANL wraps green! That’s remarkable in an ultrahardcore.
Then a hidden unique rectangle of 25 at r19c56,which depends on the corner slinks in one UR value and the opposite side slink in the other. The removal gets us to
Beeby’s “simple” AIC ANL. The 1-way starter is 7r8c4, where you’re looking at three 7 candidates to finish a chain. For a boomer, the ANL is a happy accident, because you’re looking for a closing confirming ANL.
I’m noting this as prep for the Guide, which is for human solving, not solver interpretation.
Another AIC ANL and an rasily spotted Type 1 UR.
The previous remarks apply to the AIC.
The blue ALS is eligible for spotting the overlapped ALS_42, because the 2 group aligns with an outside 2-candidate.
Somehow the human solver has to bring these two ALS together from her collection of eligibles. When you can do that, you’re wasting your valuable time solving Sudoku puzzles.
Just before the end, Beeby does the wildly inhuman. It’s a boomer from 6r7c2. When it reaches 4r7c1, that candidate is recognized somehow as the single value group in the weirdly constructed ALS 346259 in r23567c1 slinked to the 9 value group seeing 9 in the starter cell r7c2,
Then an ironic trigger for the collapse, a short AIC ANL.
One post late, the 10th Heine review puzzle, ultrahardcore 399, starts next week.
Sudokuwiki starts with an APE so unusual as to change my working definition of the APE. APE stands for Aligned Pair Elimination. From every example so far, I thought it was the two elimination cells that were aligned in a row or column. No, it’s that two cells are simultaneously aligned with ALS that can prohibit any combination of two candidates, one from each cell, that see two of the ALS’ value groups. If every combination of a candidate in one of the cells is so prohibited, it is false.
Here, every combination of 7r9c7 with 2, 4 and 9 in r7c1 is prohibited. The blue ALS stops 27 and 47. The red ALS stops 27 and 29. One way to avoid repeating work is to list every combination of the two cells, then cross out combinations stopped by the ALS seeing both cells. There are 4 combinations of 2 with other values, and 10 combinations of other values with other values. A policy for a reasonable human search for APE is complicated.
Sudokuwiki, the computer code, has no such problem with APE. Four combinations to filter through the ALS this time.
It may be the number of ALS and cell pairs that’s more of a problem. How many combinations this time? Only 5.
Next, Sudokuwiki adds this AIC ANL with an ALS node.
Then it identifies 3r8c6 as a pattern orphan, without explaining why. We analyze the 3-panel, looking for a humanly possible way to do this.
The 3-panel reveals a limited number of West to East freeforms crossing r8c6: 3 from r2c1 and one from r3c1. Looking for a toxic orphan. Note that those visiting r8c6 must also visit r9c7. Among the X-panels, we don’t see another whose every pattern must visit these two cells.
But Beeby’s note identifies 9 as the restraining value. Of the five East to West freeform enumerated 9 patterns, three are stopped by 3r6c5 and the other two by 3r5c9. So 3r8c6 is a toxic orphan. After announcing the orphan, Sudokuwiki gives up. Switching to Beeby, we get the same result, but with the APE’s handled differently. You can’t prescribe candidates to either solver, only given clues, so we get to see Beeby redo the APE eliminations.
First, the 7r7c4 is an ALS_37.
Then 2r7c7 disappears via a grouped ALS_42.
The 7r2c4 removal takes an ALS wing, which can also be interpreted as an ANL with the slink chain with two or three ALS nodes.
Then after identifying the toxic orphan 3r8c6, all the Beeby methods fit for human computers are exhausted, and its trial time.
The Sysudoku bv scan reveals Single Alternate Sue de Coq NW3 = 9(6+7)(2+5) + 952. If the trial fails, the remaining SdC NWc3 = 9(6+7)(2+5) eliminates 6r2c1 and 6r3c1 in the NW box. We add a small cluster.
After the brief follow up
and a reload, Beeby continues the trial next week. The reload misses APE elimination 2r7c8, but that has no effect on the trial.
This post gets us into coloring and and a look at the Beeby ALS-wing with an ALS_XZ map. Also the solvers illustrate a new pattern analysis resource.
Picking up from last post, Beeby now comes in with a complex 1-way. That’s how it starts, with a bv slink partner seeing other 3’s. It works out as a 1-way ANL with a branch coming off to whack 4r4c9 and create the necessary r4 slink.
Of course, human solvers don’t “spot” these. They spot possible beginnings and keep chains going, looking back for branches to create slinks.
The removal erases the r1 ALS 2356 on the ALS-map, but it leaves all the AS required for the ALS- wing that Beeby finds next. This gives you an opportunity to search for that ALS-wing on that map before looking. Go back and do that while I create a little diversion by discussing the trade-off between marking and not marking the multi-candidate value groups on the map. The markings would definitely aid you in building the inference chain, but would would add considerable unproductive clutter to the map.
Next is a Beeby ALS_wing, a 3 node ALS chain ANL. You would probably start this chain at the bv r5c5, noticing the winks to single value groups in two ALS on the map. That would lead you to discover that the shared value 8 and the slinks with the 8-groups and the entry candidates. One group is a box group, and the other is a line group.
Were there any other ALS wings?
Then Beeby uses a branch enhanced AIC for three eliminations: a complex 1-way from 8r8c1 to confirm 4r4c1, and extending it to confirm 3r2c1, and having removed two c1 8’s by 1-way logic, invoking the boxline SWc1 to remove 8r7c2.
Parts of that structure show up again in this complex 1-way from 2r2c6 which, true or false, extinguishes 2r2c1.
We hopefully start a second cluster as a simple 1-way removes 2r4c2 whether 2r1c2 is true or false.
Then comes the very complex 1-way, the AIC starting from 3r2c1 and sprouting four slink making branches.
Its removal of 3r2c9 allows this overlapped ALS_62.
Beeby now continues with a puzzling “complex” 1-way with an unnecessary forward branch. The Beeby annotation shows an ending slink to 7r3c7, which is valid, but unnecessary. The forward branch from 7r7c4 could be used to reinforce that the chain assumes 7r3c4 is false. It might be what Beeby does when the starting slink is a line slink, rather than a bv slink.
Sudokuwiki records this one as a digit forcing chain.
Now consider the labeling over abundance. This ALS_36 is also an ALS boomer from 3r7c7, and a unbranched ALS 1-way from 6r1c7. It depends on what you were looking for when you found it.
Wouldn’t you know it? Both ALS were added to the map.
If you can wade through one more ultra complex 1-way, we can pick off another 3 candidate. It starts at 3r2c1 again, and uses much or the same structure as the previous one.
Now the pattern analysis twist. The red/orange cluster expands with the 3 clue, and Sudokuwiki claims that 2r2c3 is an orphan.
Checking it out on the 2-panel, the North to South freeforms are restricted, but there are two patterns including 2r2c3. Note that an attempted freeform through r3c9 is stymied, because it leaves no entry into the East box. Now we know that patterns including 2r2c3 must also visit r3c4, r4c6 and r6c9.
If any combination of these three cells exclude all patterns of any value other than 2, 2r2c3 is an orphan!
Sudokuwiki notes don’t explain that reasoning, and enumerating all patterns of all other values is not normally feasible, but here we notice that 7-patterns are quite restricted, and that delivers the goods.
Going North, all 7 patterns visit r3c4, with one exception, the short-dashed one in the right panel, from r8c2. That pattern visits r6c9. So 2r2c3 excludes all 7 patterns, an unforgivable offense.
Beeby is silent here, but in a coming ultrahardcore in the review, it does acknowledge this kind of rationale for removing an orphan candidate. For future reference, we’ll call this removal method the toxic orphan.
Looking at the diagram above, there is a much easier way to eliminate the “orphan”, by a combination of coloring and the Sue de Coq. NEr2 contains 89(2+7) if red is true or 69(2+7) if it is orange. In either case, the matching 2 or 7 in the r2 remainder must come from r2c6.
We continue from here next week to a solution. That means you may need a preview of 355, the next review ultrahardcore.