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.

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.

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:

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.

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.

After

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.

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.

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.

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.

Heine’s left page ultrahardcore 45 offers a host of examples illustrating when a Beeby “discontinuous nice loop” is and isn’t a 1-way. Coloring closes the class.

UHC 45 has a remarkable number of bv and line slinks, given the monster-like cloud of candidates on the line marked grid.

The basic trace shows a good number of box slinks as well, to go with the two bypass clues.

The solvers start teaching early. Sudokuwiki finds an APE new to me. I once thought “aligned” in “Aligned Pair Exclusion” meant the elimination cells are in the same line or box, and had never seen a counter example. Actually, APE “aligned” means the elimination cells are both aligned with a set of ALS that collectively exclude at least one combination of candidates in the pair.ypass clues.

So you look for ALS aligned with two cells, with values sets matching the cell values. Then look for a value in one cell whose every combination with values in the other cell is contained in one of the aligned ALS. You can ignore 4 and 5 combinations, they are absent from both ALS.

Beeby duplicates the Sudokuwiki APE in an ALS_72. The ALS has common value X = 7 and the victim sees all Z = 2 candidates in both ALS.

Beeby next finds this X, XZ based ANL, which I would get to in AIC building.

The removal allows an XY ANL onto the railway.

Next is a series of “discontinuous nice loops”. Some end with 1-way logic, but many become 2-way almost nice loops, or ANL. The Beeby’s notes describe this grouped ANL as a discontinuous loop, type 2, starting at 7r6c6. In AIC building, I would have started at the 7 slink in r8, seeing the c6 7 group as a link to 7r6c6 to complete the ANL. The removal brings a host of naked pair removals within the box.

In Beeby’s next, starting from 7r5c3, the 7-group is the end a 7-slink, and an ANL terminal seen by the victim. Or is it one of several ways the AIC started by assuming 7r5c3 is false can prove 7r5c3 is false? That’s a simple 1-way.

In both of the last two examples, a group is taken as a slink partner. That is, a slink combined with a group functions as a slink.

The next “discontinuous” is also a simple grouped 9-chain ANL.

OK, here is an authentic simple 1-way. You start with a possible target which is false if the starting candidate is true, and the AIC started with a false starting candidate makes the target false, not true, and not by making a third candidate seeing the target true.

Now two true 1-ways starting on 8r4c4.

The first winks in to 1-way target. That is extended to another target. The extended AIC AIC makes the target false by making another candidate in the target cell true.

The second 1-way triggers the Nc5 boxline.

With the next boomer from 6r9c5, we add a blue green cluster.

Then we see the red orange 1-cluster wrapping blue.

Red orange expands, and SW9 => S9 => orange,

and we have a green and orange solution, without trial.

Coloring can be combined with freeform pattern analysis to determine a small set of pattern trials for Heine’s ultrahardcore 1. Typical choices for this decision are reviewed. A single trial produces the solution. For this we go back to the starting grid of the previous post, just after the simple 1-way of August 3.

Here, a pattern is a set of candidates of a single value with one member in each row and column having no placement of that value. Given a cluster of that value, the true pattern can include candidates of only one of the cluster colors. This fact can be used to limit possible patterns for trials. Looking at clusters on the UHC 1 panels, . . .

In the North to South freeform patterns of 7, we have two compatible with green and two compatible with blue.

A coloring trial of blue will use two placed 7-cells, but a trial of one of the two blue patterns will use 6 7-cells, and will be much more decisive. On the other hand, the failure will not determine color, and will leave up to two more failures before pattern and color are known.

In the 4-panel, North to South, we get one red pattern and four orange.

In this case, the red trial enlists 5 cells determines color, but failure leaves four orange patterns to be tested.

Looking further, the 5-panel doesn’t have slinks interposed in two directions, and therefore imposes no color restrictions on the number of patterns.

The better trial plan comes from the 6- panel with a cluster of 6 cells, also having two line slinks in each direction. The only feasible direction is East to West, and there is one pattern compatible with each color.

A failure of one trial guarantees the success of the other.

Before we explore the trial of one of these patterns, consider that the Single Alternate Sue de Coq of the previous post is also a very decisive trial. This leads us to ask, how far would we have to take that trial to know which of these two patterns is true? The answer is striking. In the trace of the initial follow up, before the double hidden unique rectangle result, W56 of the trace marks the naked pair that reveals the red pattern to be false. We could add the orange pattern to the SASdC trial, making it more decisive, but it’s still a trial. We can’t add the (NW3, NW4) of the SASdC trial to the orange pattern trial until we know that the pattern implies these two placements.

Here is the set up for the orange pattern trial. Knowing from the trial of the last post that the red pattern is false, let’s see if the orange pattern trial reaches the solution, how it compares to the SASdC trial.

The pattern follow up leaves a naked quad in r7 and a 2-chain ANL.

The ANL removal brings another quad in c2.

Followed by an effective ANL.

And after E2 =>SE1=>SE2, a naked triple and Sr7 boxline,

a simple AIC places NW4, removing 4r8c1 and removing 13r8c3. Then right next door, a hidden unique rectangle removes 4r9c4. Trace what happens if 4r9c4 is true.

After S4 => SW2, we need two more colors for the extensive cluster. The trap expands blue green enough to remove 9r6c6, and rouge/tan enough to remove 1r13c6.

Then a simple ANL removal allows

A red wrapping ANL that happens to trap 3r1c6 as well, and tan takes over

The pattern trial was easier, but it could have required two of them.

Next time, we continue 0n left side with ultrahardcore 1 + 44 = 45.

This post reports a rare outcome in an SASdC trial, the extra term carrying through to a solution.

Here is the grid with NWc1 marked as a Single Alternate Sue de Coq chute. The contents of the wall can be described as

NWc1 =

(5+6)(3+4) + 342.

The bv 56 says 5 and 6 cannot both be in the chute, so one of them is present, requiring 3 or 4 to also be there., unless 5 and 6 are both missing. The SASdC trial is testing for exactly that.

If the extra term fails, then the NWc1 becomes a normal Sue de Coq removing extra 5 and 6 candidates from the NW box.

The trial gets off to a great start, with a double hidden unique rectangle.

Each row slink in the rectangle produces a reversible rectangle solution if the opposite side 5 is present. Beebe sees one, and deletes a 5, but that destroys the hidden UR on the other side. We don’t expect a solver AI to hold the effect of one 5 until it tests the effect of the other one, but we oversee the removal of both.

Going back for c1s67 => SW5, Beebe begins AIC building with this remarkable assembly of parts. Continuing the XY chain from r1c5, Beebe recognizes the aligned pair as a slink partner in the ANL, one it can “see” in the C box.

Now the XY chains get easier and build more bv. An ANL with an Er6 boxline.

They keep coming, another easy XY ANL.

Another bv means another XY node, but it also means more resources for coloring.

One more.

Now a red/orange cluster stops the drip, drip, drip.

r6c6 => not

(orange and green)

and

r5c9 => not

(red and green)

So (orange or red)

=> not green

Now following the arrows, blue 1r1c5 => 2r1c9 => orange .

Instead of a contradiction, the trial produces the solution.

Next week’s post demonstrates another type of trial with ultrahardcore 1, combining coloring and pattern analysis.