Two Solvers Play With UHC 265

Heine’s left page ultrahardcore 265 is solved in two posts. This post follows the alternative methods of the review solvers Beeby and Sudokuwiki until they converge on a long series of AIC in the second post.

Here is the grid after the bypass clues and subsets and box marking strong links defined by boxes.

We always have the basic trace, showing the sequence of moves. The Guide explains them, but the trace conventions become clear when you see the moves and and the trace notation that reports them at the same time.

In the line marked grid, line slinks and bv are marked, and fill strings are shown. The trace records the order of line marking, in increasing order of unplaced cells. The Sysudoku grid places candidates on corners and edges to distinguish slinks in boxes, rows and columns.

The order recommended here for DIY solving is spelled out in the Guide: obvious UR and remote pairs first, then a scan over boxes for bv, SdC and BARNs, then make a bv map and find the bv railways for XY ANL, then scan a XYZ map for regular and irregular XYZ. Then make the X-panels and scan value by value for X-chains and fish, and the edges for easy pattern analysis. As bv accumulate, build colorng clusters. If still necessary, explore all AIC, including ALS nodes.  All else failing, map ALS and and mark singles and aligned value sets for ALS_XZ  

On the ultrahardcore review, I’m applying two human oriented solvers to this order, mixing  Sudokwiki’s fixed order of methods with Beeby’s menu of options for fish, ALS, and AIC  methods. The result is characteristic alternatives along a consistent solving path.

For UHC 265 the review path starts with two X-panel results, a finned 1-wing from Beeby and a 6-chain ANL by Sudokuwiki.

The fin is 1r1c9, and the victim 1r2c7 is in the fin box. None of the potential kraken victims 1r2c1, 1r8c1,  and 1r8c7 see the fin via forcing chains.

Boxline NEr1 removes 6r1c6.

Next is Beeby’s finned 6-wing removing 8r7c5 and a Sudokuwiki ANL confirming 6r2c5, to the same effect.

The removal of 9r7c8 brings a hidden dublex of 9’s in c79 and

A third finned X-wing or 4-chain removes 4r1c4, and 8r2c8 is removed by Sudokuwiki’s 348-wing or Beeby’s equivalent ALS_48.

The 4r1c1 removal allows another finned 4-wing removing 4r2c7, but Sudokuwiki leaps ahead with a grouped 4 – ANL and a green extension to remove two more.

Beeby pulls ahead with this pair of hidden UR’s on the left,  a short one for a go-ahead 9 clue and a long one to get one of the NE 4’s.

That leaves a 4 – ANL for the other one.

A Sudokuwiki boomer from 9r5c7 duplicates the E9 clue and the solvers are on the same grid again.

Next time we’ll report a long series of Beeby’s simple chain AIC that get us to a third hidden UR and a coloring wrap of UHC 265. Want to do it first?

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An ultrahardcore 221 ALS AIC Clinic

Continuing a solution path with a finned starfish (5 lines), Beeby carries us through an unlikely  series of ALS aided AIC, dancing around two stalled coloring clusters. The contest ends with regular AIC finally expanding and wrapping the clusters.

Here is the finned starfish, with a coloring “observation” as a possible start.  It went nowhere.

The starfish on columns occurs if fin 6r3c4 is removed. It’s victim, 6r1c5 sees the fin. So if the victim is true, it removes the fin, and is eaten by the starfish.

The starfish has victims r125c1 as well, but if any one of these is true, that would confirm the fin and prevent the starfish.

A second cluster has a better start.

The next Beeby find, an unlikely looking ALS chain , starts as a 1-way from 7r7c2. To spot this ANL, however, you need to have an ALS map with 6r9c3 identified as an ALS single.

Next is a second ALS chain grouped ANL which first uses a bv a7c2without regard for the ALS, then takes a  shortcut through the red/orange cluster to enter the ALS at 1r4c2, using its 6 value set as an ANL terminal group.

Again, you have to recognize ALS c2 1457/13467 to spot this possibility.

Going for a third strike? It won’t win the game, but on this shot at the 7’s seen by 7r1c5, we must have tried every other red candidate before 8r5c1, then seen the route to the c7 ALS with single 6 and 7 value set.

Having run through the available ALS node AIC, Beeby responds to a request for simple ALS with this ALS_62. 

I remembered the SW ALS from the ALS_26 I recognized as duplicating the Sudokuwiki APE. Both of the ALS here have been available since 6r3c9 was removed by the complex chain just before the finned starfish.  

However, the removal expands the blue green cluster to reach another ALS single with a 7 value set. What’s missing from this report is the three part ALS map you and I would need to ever spot this. Beeby reconstructs it on the fly every time we ask for an ALS AIC.

Running this option dry, I ask for an ALS wing, which is a chain of ALS, including bv.

This one combines a boxALS and a row ALS, tied together with a bv. One uses a value set for an AIC starting slink, the bv does its normal node job, and the third uses its value set as a pointing group for an ANL.

The removal leaves a clue and a naked triple in r1.

Now two more simple chains. First, a new 1-way turned ANL, which enables

a second, decisive ANL. As traced below, it expands both clusters,

trapping 7r7c6 to wrap orange.

Then, after the red confirmation   follow up,

a 7-chain finishes UHC 221.

Next week we start left page ultrahardcore 265. 

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UHC 221 Vies for Right Page ultrahardcore Status

In this post, an unbalanced line marked grid is chiseled down some by an unlikely series of 1-ways and ALS chains by the two DIY oriented solvers.

Starting with the bypass grid, with the bypass clues and fill strings in the Bradley Hand ITC script, and the givens in more formal Calibri. The bypass is a game of finding all clues and subsets possible before strong link marking. Two 3-fills are marked with fill strings.

It’s a good result for a Stefan Heine ultrahardcore. Note no givens for value 7.

The basic trace gives a step by step account of the three stages of Sysudoku basic: bypass, box marking and line marking.

The Close completes the marking of lines not completed when the crossing lines were completed.

The Guide (menu bar) explains the goals of each stage. Trace conventions are described there, but experienced solvers can interpret the trace moves when they follow them on their own grid. If that fails, see the Guide for details.

The line marked grid shows the order in which lines were marked with candidates and line slinks. It also shows the fill strings added in line marking.

In UHC 221, candidate values are very unbalanced, with values 1, 2, 4, 6 and 7 prevailing.

There was a naked single for 7 on the third line marked. By following the trace, you can see the state of the grid when the naked single is found.

To start, Beeby finds what it labels a sashimi 1-wing. If either victim is true, columns 1 and 6 require the row 6 position to place a 1. Sudokuwiki sees it as a simple 1 – ANL.

Next, a  Sudokuwiki APE  is Beeby’s  ALS_26.

And one  dfc simple 1-way.

If 1r7c6 is true, 1r7c2 is false, but if it is false, the  the 1-way AIC makes 1r7c2 false.

Then another 1-way. 7r4c3 is false if 7r1c3 is true or false.

As Sudokuwiki signs off, Beeby does an ALS ANL

And a complex 1-way from 1r2c4 removes 1r2c7, with branching winks along the 1-way AIC removing candidates preventing two slinks in the 1-way AIC.  A grouped  extension in red extends the 1-way to a second victim. In a complex 1-way, the AIC branches off from candidates the 1-way direction assumes are true if the starting candidate is false.

This AIC is marked by the complex branch as a 1-way rather than an ANL, because the AIC only works in the branching direction. It can start in 7r5c6, but not 7r1c5.

The removal allows . . .

a grouped slink in c5 that could start a boomer eliminating 1 or 6 in r1c5.  Then the wink into 7r1c1 makes it a Beeby simple discontinuous loop. What isn’t continuous in this loop is the alternating inference. It’s an almost nice loop (ANL) but a 1-way because in AIC building, we start them on slink partners. The 1-way AIC shows that if 7r1c5 is false, 7r1c1 is false. That along with if 7r1c5 is  true, 7r1c1 is false(the wink), proves that 7r1c1 is false.

Let’s end this post with a third complex 1-way.  Where does it start? It’s a 1-way aimed at the 6’s seen by 6r3c4.  7r3c1 is suppressed to enable the final slink, and 2r9c3 is suppressed to allow the 2 slink in c3 needed to get there.

The SEc9 boxline follows the removal.

This intense session leaves a lot more for next week, with a rare finned starfish, and a series of ALS chains to be found.

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UHC 177 Stretches It Out

In this post, Heine’s left page ultrahardcore 177 gets from one cluster wrap to two more in a long series of AIC and boxlines.

Continuing from the green wrap of last week, blue SW3 takes the last r9 5 outside of the South box, for an Sr9 boxline in the 5’s.

There’s enough bv to start a another cluster, but we continue in AIC building. This looks like a 1-way starting from 2r6c5 that happens on an ANL. Or you can put in that last wink into 2r6c6 and call it a simple 1-way.

Now after C2 => NE2

creates a tight little sequence of three events. First a 6-5-6 AIC ANL removes 6r3c12 for a pointer group 6r2c13 removing  6r2c6. This removal invokes a pointer group 6r13c5, or if you prefer,  boxline Sc6, removing 6r79c5.

Finally, an XY ANL.

A simple chain gets 5r2c6, and an extension gets 5r2c4, together invoking the boxline NWr2 removing 5r3c1.

Beebys’s next AIC ANL brings a follow up

allowing two more simple chains that wrap red.

The first is an XY ANL expanding the cluster.

The second is an XY ANL trapping red 4r4c5 and 4r7c4.

In what’s left, we have a pink and tan cluster. It traps 2r2c2 and wraps pink when two pink 8’s are forced in c4.

It’s a blue, orange and tan solution,

Next is left page ultrahardcore 221.

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UHC 177 Introduces a Beeby Option

This post gets left page ultrahardcore 177 to coloring with an interesting Sysudoku Basic and a kraken X-wing accessed with a recently added AIC option.

Stefan Heine’s ultrahardcore 177 makes many left page concessions, but makes a fight of it. The basic trace shows a moderately productive bypass, and a naked triple in line marking.

The naked triple is unusually productive, and

the full grid is graced with two hidden unique rectangles. To unravel them, use the slinks to trace what happens to the rectangles if the removal victims are true.

Beeby follows with a simple ANL out of AIC building. It has three winks in multicell nodes, a string of slinks.

On a rerun looking for alternatives, a necessity for analysis with Beeby, I requested a recently added  AIC option, an AIC with X-wing, and was rewarded with this kraken X-wing.

On the 9-panel the X-wing and fin are easily found, but “kraken” denotes a finned fish where the victim is not in the fin box. The AIC is required here to verify that the victim sees the fin, making it a victim.  Figure the logic.

On the first run, I didn’t ask for, and didn’t get, the kraken removal.

But its result was duplicated. Instead, a second Simple AIC produces a decisive ANL. In DIY mode, the ANL could have been started as a boomer from 8r4c1 or a 1-way from 8r5c4.  The four removals generate four new bv cells, opening the middle for coloring, and generating clue W8 and boxline Wc1.

The cluster traps 6r5c3, and the boxline removes three 9’s.

The blue/green expansion and and cluster exit AIC from green 3 finds an ANL with blue 1r5c2, wrapping green.

Do the follow up for the first grid of next week’s post:

It exposes a moderate puzzle within ultrahardcore 177.

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Closing Out ultrahardcore 133

Resuming on ultrahardcore 133 after two 1-ways, and a r1 naked pair 16 removing 16r1c3, and the resulting NW naked pair 48 removing 4r3c3.  Solver Beeby finds this ALS node AIC.

Starting from 6r2c9, it was a possible exit from the cluster for an ANL on 6. Instead, the 4-chain reaches a single in the C238 ALS 2689 with internal slink to single 6 for the ANL.

Beeby diagrams this as an ALS_46 as well, possibly because ALS chains were recently added to the AIC request options.

Next, a grouped 1-way through the cluster from 6r6c5 (black), is also extended in the cluster(red) to the 6 value set in r127c4 ALS 1469 seen by ANL victim 6r6c4. This is an  example of expanded possibilities of AIC building via ALS chains.

The follow up is modest, but we’ll take it.

After this simple chain,

Beeby describes a discontinuous loop 2 with a wink that doesn’t work:

(1)r7c4 = (1)r1c4 – (1)r1c1 = (1)r3c13 – (1)r89c3 = (1)r9c1 – (1)r9c5  => -1 r9c5.    

Here’s what that looks like. A grouped 1-chain wants to go to a second group to reach the opposite terminal of an ANL and remove 1r9c5. But what about the red weak link? It’s saying that if group 1r3c13 is true then group 1r89c3 is false. Is that a fact?

Not exactly, if the group 1r13c3 is true, then 1r3c1 or 1r3c3 or both are true. If it’s  1r3c3, group 1r89c3 is false, and the red wink works. If it’s 1r3c1, then 1r9c1  is false and that makes group r89c3 true. Oops.  

Fortunately, the removal is duplicated later before it is needed.

Beeby does bring an instructive example of ALS node AIC building. Starting on 6r3c3, you’re looking for a wink back into r3c3 but get lead away to 1r7c8. Are you going to see the possibility of an r8 ALS with single 1 and  a 6 group skipping 6r8c3? That would lead you to the effort to construct r8c2789 ALS 12346.

Now having the ALS, would the 4 value set skipping 4r8c3, and the 4-chain  from 4r9c3 back into the 6r3c3 lead you to this one?

If so, it leads to a simple chain, and a follow up trap.

Leading to an expanding cluster wrapping blue in c6. The solution is green.

Next week, we start on ultrahardcore 133+44 = 177.

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Ultra 133 “Simple” Chains and Discontinuous Loops

After an productive naked triple in line marking, and two bv scan methods, left page ultrahardcore 133 starts AIC building with Beeby “simple” chains. Then we take in two “discontinuous loops, i.e. simple 1-ways. The decisive ALS AIC’s and colorful finish are held for the following post.

Here’s the basic trace,

and the grid after line marking, but before the removals. Beeby does the naked triple silently, but it’s worth a picture in my book.

On the grid is a hidden UR, and a BARN. In the unique rectangle, if 1r7c2 is true, 2r7c8 is, 1r8c8 is, and since 2r8c8 is not, 2r8c2 is, and a second rectangle solution is made by interchanging 1 and 2.

Sudokuwiki notes label the BARN a WXYZ, and Beeby duplicates the removal with a short AIC ANL.

The follow up

brings the first of many Beeby simple chains. The first two, a 9-chain and  its extension.


we get a  grid spanning 4-ANL, an AIC ANL removing 8r5c6, and extended AIC ANL removing 8r5c3.

Beeby’s simple chain series ends with this quick 1-way, after a long search fails to find a “Continuous” AIC. The 1-way removal happens to trigger a Wr6 boxline netting two more 6 candidates.

Why are  continuous searches long? Beeby’s continuous category includes any nice loop, without reference to a starting point, The search is expected to capture  any wink back to any candidate assumed false or slink back to any candidate assumed true,  when starting on a slink partner. If the search were limited to a wink back to the starting point, it would be much quicker, having a single target for loop back. But that’s not what users want. Users can accept the limited targets of the Boomerang and the 1-way. And the search can end when the last target is bypassed.

After one more simple chain, an AIC ANL, and it’s follow up

Beeby slips in another basic subset without comment, a West box naked triple.

With the cluster expansion trap of 9r7c6, which Beebie does (grey) by taking cluster slinks as winks, Beeby gains a slink into a simple  chain ANL.

Let’s wind up with this “discontinuous loop 2” (1-way) removing 1r1c2, which also functions as a boomer, coming back to wink into the starting cell. Since Beeby gave that label to the previous 1-way with extra removals by boxline, it must mean simple 1-way plus anything else. OK.

After the naked pair removals following this “type 2”,  ultrahardcore 133 continues with ALS AIC and a Beeby move you may be able to straighten me out on.

We’ll bring it in without a trial next week, Tuesday at 2.

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Wrapping Up UHC 89

This post completes a coloring solution of Stefan Heine’s left page ultrahardcore 89.  Two examples of cluster exit AIC building bring a second finned swordfish, and a cluster exit wrap of the large cluster.

After this follow up of last week’s finned swordfish,

Beeby finds a 7 link XY chain through the cluster to 1r3c4.  Beeby doesn’t stoop to Medusa coloring, but we do. Think of it this way. Build an AIC out of a large cluster to an outside candidate. Now any cluster candidate of the same value, and color opposite the exit color is a possible ANL partner, and there is a cluster AIC to it. Just draw in the corresponding strong link.

As a cluster exit from orange 1r3c4, this XY ANL removing 6r3c5 suggests we also look for another red 6 ANL terminal.  One red 6 claims two victims, one claimed by the second. When we play the other end as an exit from red 9r5c4, we’re looking for an orange 6 ANL partner for 6r4c5. 

The r3c7 removal allows a short XY ANL with green terminals, wrapping blue.

The red/orange expansion

lines up another finned swordfish, we didn’t really need, because

an easy cluster exit wraps orange.

Next time, Stephan Heine’s left page ultrahardcore 89 + 44 = 133.

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A Box ALS in Left Page UHC 89

In this post, a large box ALS with many outside candidates becomes a starter in AIC building, and a second cluster becomes a AIC target. 

Restoring 2r8c7, we continue with an box ALS node AIC chain, using its 5 value group in an  ANL. It could be started as  that same 1-way with 8 targets of the previous post, and continuing to 2r9c3, when we notice the possibility of an ALS with 2 and 5 value groups for the grouped ANL. Instead of enumerating ALS, this is recognizing them as needed, certainly in the spirit of AIC building.

This could lead to another chain going back along the chain to another 5 seeing 5r6c2 and the resulting boxline.

After a small follow-up

I finally get a XY AIC offer from Beeby, making it a promising time to start a new coloring cluster.

The ANL produces a pair of 2’s guaranteed to include the true one, and completing a hidden dublex 2-wing.

The pair also remove 2r9c3, converting the SW five cell ALS into a naked quin.

A second XY ANL escapes the red/orange cluster to expand the blue/green cluster,

trapping 5r2c8 to expand red/orange to trap 9r89c9.

We also get a sweeping shortcut 3 – ANL.

Note that 3r1c9 seeing orange means that any red 3 in the cluster is a possible ANL partner terminal.

The AIC building party continues with a nice loop making very little use of the clusters. The removals are based on the fact that every link in a nice loop includes a true candidate.

Here is the 3-panel showing the 3 removals from the nice loop. The panel is marked in an unused row to show the finned swordfish in columns 2, 4 and 8. The fins are the 3s marked f. If the fins are removed, columns 2, 4, and 8 will share row positions 3, 6,  and 9 marked by + signs in an unused column, forming a swordfish. The swordfish removes candidates in other columns in its three rows.

If 3r9c1 is true, the fins are removed, the swordfish forms, and removes 3r9c1.

On the grid, the row swordfish icons show the column positions sweeping other columns. Small rounded squares mark the fin candidates, and the diamond marks the removal.

After a short follow up, it takes a few AIC and another finned swordfish to wrap ultrahardcore 89. You might like to do it ahead of the next post, which will show added possibilities of AIC building around a cluster.

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UHC 89 Breaks Out from a Monster Basic

This post coordinates the two DIY solvers used in the ultrahardcore review, choosing and interpreting solver methods and ignoring some removals to match the Sysudoku DIY order of battle. A series of early moves breaks the ice for AIC building that continues in the following post.

In Sysudoku Basic, what  do you get when the bypass zeroes out? Maybe, a good reason to move on to the next ultrahardcore? Not this time.

With the box marked grid alone, you get an example of box marking notation. It’s a set of slink lists, one for each value, with cause and effect expressed by string indenting.  S => SWt. N => S. You figure out what S, SWt, and N mean by examining grid results.

As you fill out your grid, what would you do next. Check the trace, did sudent do that? It beats reading explanations.

Now work your way through line marking. It’s about getting this grid constructed, one line at a time. What do those digit strings on the side and bottom represent? What is the order of processing lines in the trace? Why that order? It matters what corner or side a candidate is placed on. What does the cell position tell about strong links?  What is different about Close lines? Need the Guide? This line marked grid has a few encouraging signs, once you get it constructed. And every move is recorded in order in that grey box trace above.  How can that be?

Starting the bv scan, Andrew Stuart’s Sudokuwiki makes a find. It’s a BARN, a bent region quad with a single value in both box and line. One candidate of each quad  value is true, removing any outside candidate seeing them all.

Working with two very different solvers, a single solving pat requires  keeping  them synchronized on removals.

Neither solver does Bent Almost Restricted n-set (BARN), but Sudokuwiki interprets BARN as a form of WXYZ, and Beeby often duplicates them as ALS_XZ. It does this matching removal by ALS_53.

The first Beeby AIC is this 1-way from 3r6c4. In the 1-way, 3r5c5, which is false if 3r6c4 is true (the wink), is found to be false (the AIC wink) if 3r6c4 is false. In AIC building, this is also a boomerang starting from 8r5c5 and returning to the starting cell.

Beeby follows with a second 1-way or  boomer.

As a 1-way, it starts from 5r3c2, a bv partner that  sees 8 5 candidates cells and starts an AIC  looking to wink into any of them. It finds r3c2. You could branch off the AIC with slink or wink, as needed, to see if there are others. Or starting a possible boomer from 3r3c2, you have a more focused search with one cell as a target.

Next, a Beeby complex 1-way. 8r91c is a plausible start for a simple 1-way, given the starting chain just seen, but in the search for complex 1-way, when we get to 6r2c2, and don’t have a wink-to-slink combination to go further, we look back along the 1-way AIC for a wink out to a slink denying candidate. That would be a one of two candidates of the same value in the same box or line.

That process is so much more complex, that to reflect practical DIY solving, I ignore the removal until it is shown to be essential. It may get itself removed first.

I do the same with the next, and last practical DIY move, a grouped common ALS_98.

Luckily Sudokuwiki is aligned enough to step in with a couple of its digit forcing chains.

I don’t go along with the solver’s spotting rationale for digit forcing chains, or dfc, but they  can be interpreted as Sysudoku AIC building repertoire. On your left, and the puzzle’s right, is a grouped boomer from 7r6c8.

Then another dfc,  a twin grouped boomer from 3r6c7.

Looking at the full grid, the 2 removal from the triple makes a hidden pair E27 removing 4r7c7 and the boxline Er7 carries away four  more 4-candidates.

This leads Sudokuwiki to a coloring trap  on 4r7c3, and an APE that may be hiding something.

 It rightfully discards 26r9c8, whose combinations with all r2c8 candidates see two value sets in one of the ALS. So why couldn’t any candidates of r2c8 be discarded?

Because they all form a combination with 4r1c8.

These two removals prove to be essential, but APE by cell combinations or by ALS_XZ  both require exhaustive search of cell combinations and their ALS sets. The problem for DIY solving is the number of possible APE cell pairs, and ALS aligned around each of them.

Another resource delayed for  exhaustive ALS analysis is ALS wings. Here is one available at this point, involving a box, line and a bv. The removal can be ignored, because it is duplicated before it is needed.

Next week continues on ultrahardcore 89 with a series of eliminations by a single box ALS working with bv. These  AIC with simple ALS nodes are favored above APE or ALS_XZ for DIY solving, because there is no searching. ALS are constructed as a means of continuing an AIC.

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