Ultrahardcore Review ALS-wings II


The  ultrahardcore ALS-wing survey continues with examples from UHC 47. The survey illustrates how ALS-wings are found as ALS are added to ALS maps.

Continuing the survey into a second review puzzle UHC 47, the reward is this well laid out, classic ALS-wing. It’s another rows vs columns building assignment and an ANL with value group terminals.

The orange ALS is added to the row map first, then red on the column map.  Orange goes on red’s match list.

We did acknowledge that a column vs row scan is necessary. We get 4 as X, but no Z.

Now when the green ALS arrives, and a singles match occurs with red, we have 4: r9 orange on red’s match list.  Having that, add a copy of the three ALS  on your grid and mark the victim.

My two human oriented solvers have shown that many Death Blossoms (Sudokuwiki) to be ALS_XZ (Beeby).  UHC 47 has an interesting case.

In this Death Blossom, all three of the stem cell candidates 679 are seen by value groups in two ALS. The victim 5r8c9 sees value groups in both ALS, so if the victim were true, it would lock all other value groups in the ALS, removing all three candidates from the stem cell.

Now to get back to the subject at hand, the Death Blossom has an ALS-wing cover. When the c9 green ALS is added, your scan matches the  c5 red ALS on single 9. Your match list then connects the c5 red ALS to the c6 blue ALS and get to the exit on 9, where a fourth ALS has a value group matching a last added value group, and 5r8c9 sees the matching groups.

We just learned that an ALS-wing be part of a longer ALS chain, and that the match list is useful even if all ALS are on the same map.   

But Stefan Heine gets the last word on this one. Besides the Death Blossom and the ALS-wing cover, there’s a third way to grab 5r8c9. It comes come with the addition of ALS  c9 16/569, when it is found to be an ALS on a grouped AIC and its 5 value group gets to be a terminal of an ANL.

This suggests, that we follow up each new ALS addition to the map with a little mental AIC building. Take the ALS exit wink from every value group, and build towards the background AIC net, using the value group as a starter slink.  And even when that succeeds, leave the ALS on the map, and go for the ALS_XZ and the ALS-wing.

Next week, we skip to ultrahardcore 179 in the ALS-wing survey. The survey continues to refine a systematic approach to ALS overload in DIY sudoku solving.

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Building the ALS-wings of UHC 3


This post begins a review of examples of ALS-wings from the ultrahardcore review. The review will illustrate a DIY strategy for constructing them, that is compatible with the concurrent ALS mapping and ALS_XZ building processes of the previous posts. Ultrahardcore 3 starts us off.

An ALS-wing is an AIC with ALS nodes only. The previous posts describe an ALS mapping process, starting with the calculation of all ALS in suset tables, then as these are drawn in row, column and box maps, copying those with promise as AIC extension nodes onto a current grid with AIC segments as a background. Also described was a ALS_XZ scanning process in which, as an ALS is drawn on its map, it is matched against ALS already drawn to form ALS_XZ for immediate removals. An ALS-wing is a special AIC  made up of ALS only. It is complete in itself, and independent of existing AIC. To build it as its ALS are added to the ALS map, we leave information on the ALS map, which allows the wing to be constructed when its last ALS is added to the map.  From the review we have a host of examples that illustrate how it’s done.

For perspective on ALS-wings, here’s a tally of ways ALS are used in the review puzzles. For ALS_XZ we considered  bv or box or line ALS partners for the ALS entering the map. Line vs line, bv and box partners are counted separately, as are singles only vs. alignment restricted commons. Of 12 review puzzles, only one double ALS turned up, and one had no ALS_XZ before trial. There were many AIC with one or two ALS nodes. Of the ALS-wings, more had a bv ALS than didn’t.

Many situations arise as a new ALS is drawn on one of the three maps. Let’s review them by considering how each ALS-wing of the ultrahardcore review gets put together.

Here is the first ALS-wing, from UHC 3. It is an ANL. One terminal is an ALS value 7 group slinked to a 6 value group winked to a single 6 in the same box. Internal slink to 2, box wink to 2 in the yellow ALS and internal slink to 7.

Which ALS  is last added?  Orange. Looking at the column map as the orange ALS is added, we see the scan of column ALS added before it would pick up the 2 singles in red and the scan back from the red single 6 would find the 6 value group in the green ALS that slinks to the 7 group, the ALS terminal working with the single 7 for the ANL.

The green ALS is on the  column map because it is a column ALS as well as a box one. It’s also on the box map, but the yellow ALS column scan comes first.

Now having followed all that, can you see why, in our scanning order, the ALS-wing would be bypassed by a two ALS chain?

I’ll give other readers that answer after the next one.

This next one came a few moves later.  Is it different?

Yes, a blue ALS overlaps the green one. The ALS chain

(9=6)green – (6=2)red – (2=9)blue is an ANL confirming the green 9 group, and the chain

(7=6)green – (6-2)red – (2=7)blue confirms the green 7 group.

To account for finding this, notice first that this is two ALS wings. Figure out the black removal first.  The red ALS is the last added. There’s the singles match with blue and the wink to the 6 group and slink to 9. The red removal is the same.

Now look back if you need to, to see what would happen if you were a computer code doing our wing finding process. By the way, that’s how it’s done. You create the data first, and then imagine the actions and translate. The orange ALS is not on the map yet. Red is last added. Scanning from the single green 2, you see the single blue 2 and the coloring slink to the green 7.  Its an ANL with two ALS nodes.

Many moves later, with very little change in the c78 columns, two of the ALS are reused and a third is expanded by one cell for a fourth ALS-wing from the ALS column map.  The green ALS gained one cell and the value 4. In fact, both of the ALS were on the c7 column in the suset table and on the map. The switch places 4 in the green ALS, where it becomes a single ANL terminal, along with single terminal 4r7c2. But why does this occur now, and not immediately after the wing above? It’s because in Beeby code, the completion of an ALS-wing does not prompt a repeated attempt to build ALS-wings from neighboring ALS. It did prompt a repeat of the build, in case the change leads to another removal, and it did.

For what happened to trigger a rebuild from a different ALS, look at what Beeby did just before this ALS-wing. It was the only double ALS_XZ in the review, and it made  removals  that prompted ALS revisions, ALS_XZ, and ALS-wing rebuilds, including one from ALS c7 1234/34679.  

Finally, a challenge for the three part ALS map, a row vs. column scan. Without the green ALS, it’s a grouped ALS_93, but adding in the green internal 34 slink, you get a nice loop, made up of ALS internal slinks and ALS to ALS winks.

Solver Beeby does credit the 3r1c2 removal, and that means it recognizes the nice loop despite 3r1c2 not seeing both ends of the green internal slink, or both ALS_93 ANL terminals. And  Beeby does see both ends of the wink between 3r4c2 and the green ALS value group 3. And by the way, that is the only single value nice loop link that can be seen by outside candidates. Both victims see that wink.

That was fun, but the challenge was how would you construct this thing as the last ALS comes in on the column map. Without the green ALS, there’s no slink between the 3 and 4 value groups in the Southwest box. For this one, you would have to scan for box or aligned connections to ALS on the maps, and scan for a common connection between them for a nice loop. Quite a burden.

On the other hand, if these appear frequently enough, you might , for every ALS on its map having the X with the added ALS, but failing the Z, draw a curve on the map connecting them, to represent the chain to be supplied by a later ALS. You could even draw the curve to a transfer terminal for the other map, and continue it to an ALS on the other map.

We’ve burned the oxygen for this post, and will come up for air next time on UHC 47 ass our survey of ultrahardcore ALS-wings continues.

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An ALS_XZ Wrap Up of UHC 311


This post shows what it means to maintain three three ALS maps. ALS map updates are followed through a series of  ALS_XZ until a decisive coloring wrap on Stefan Heine’s ultrahardcore 311.

We are exploring ALS mapping as ALS_XZ scans are interleaved with AIC building and other sysudoku tools.  When AIC building stalls, the ALS maps are constructed with the aid of suset tables. Then the maps and tables are maintained, move by move.

Beeby’s ALS_XZ series begins with the SW to S scan with the alignment matched ALS_61. The 1r9c6 removal adds a SWr9 boxline.

By this time, significant updates have occurred, so we’ll start with the maps as updated after ALS_61 and walk through the updates during the series.

Comparing this ALS box map with with the earlier one, we’re seeing removals generating more and smaller ALS faster than the larger ALS disappear.

Beeby’s next ALS_XZ is made possible by the 1 removals in c2

That means that ALS_65 would be found on the bv vs box scan as part of the ALS_61 box map update. The 5 r9c2 removal now affects all three maps. We see above that it doesn’t  affect the values of the SW ALS or its box ALS match ups. Same with the row and column maps.

The next Beeby  ALS_89, however, would also be found in the ALS_61 update. When ALS c2 5678/56789 . . .

was added to the ALS column map.

Now let’s follow what’s been happening on the row ALS map. After ALS_61 we had this, as 5r9c3 is being removed by ALS_65 above.

In the suset row table, we see that the 5 removal will allow cell 3 into the 12467/124569 suset, making it 6 cells and 6 values and destroying the black ALS.

Now when ALS_89 removes 9r9c2, we get two ALS to join ALS_29.

On the grid it looks like this, with the update ALS W 5689/25679 and two removals from the Wr5 boxline.

In the 9 removals follow up, the bv vs box scan hits this ALS_14 and the removal in r4 produces another speedy delivery ALS from the row suset table.

The row vs row scan then produces the final ALS_42.  It’s final because the 2 removal brings a quick blue wrap:

Next time, some attention on fitting Beeby’s ALS wings into ALS mapping.

Following up on this example, these seem to be the elements of ALS map maintenance:

Have copies of row, column and box maps together for updates. 

On each removal, check each enclosing ALS for a new single or alignment. Scan the other two maps with each new single or alignment for new ALS_XZ.

If the removal was a single, the ALS is now a subset. Remove candidates seeing its locked value groups.  Remove the ALS boundary

If the removal was not a single, watch for the opportunity to drop the cell from the ALS, along with the removal, for a smaller ALS. 

Update the suset tables for each unit containing the removal. Check for new ALS and singles.

For a newly confirmed clue or subset, follow up each of its removals as above.

Next we  will survey the ALS-wings of the ultrahardcore review to envision how ALS-wings can be generated, along with AIC extensions and ALS are added to the ALS maps.  

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Continuing the Box ALS Scans for UHC 311


This post continues with the box to box, row and column scans for ALS_XZ.

 The C and S ALS fit into one red/orange cluster with a trap. The clusters meet in cell r7c8, showing that blue or orange or both are true.

Box to box requires a plan to cover all pairings. Here’s one:

NW vs. N and NE, then NW vs. W and SW, Including W vs. SW, then

N vs C and S, including C vs S, then NE vs E and SE, including E vs. SE, then

W vs. C and E, including C vs E, then SW vs S and SE, including S vs SE.

It’s just single and lines matching singles and lines.

On my background grid with the new cluster installed, the C vs S scan picks up this ALS_46. The  restricted common curve was already there.

That’s not exactly what the Beeby solver found, and accounting for that teaches a new trick. The 8’s are so arranged, that removing cell r4c4 from the ALS drops exactly one value, or another ALS. In ALS W 45689/245678. we can remove 6r4c6 as well.

Moving on to box vs line, my biggest obstacle  is having to compare two maps at a time.  In ©PowerPoint, you can click View and pull up an extra window. Everything is harder on paper.

On boxes vs rows, we run into the same candidate being a single in two overlapping ALS. Here, with NW 246/2358 vs r1 25/256, a restricted common curve points out the problem, and the solution.

In the ALS_XZ , two 5 value groups would share a 5 value truly belonging to one of them because value 2 belongs to one of the ALS. Here that doesn’t happen. In the solution, candidate 2 belongs to both or neither of them. There is no ALS_25 here.

OK, let’s see if that answer gets us out of this ALS_XZ maze. It does. There are six possible ALS_XZ, but each is a pair of ALS with a common value group, just like the above. How about the 5 single  in red ALS? It can’t really see either of the two cell 5 groups in the two partner ALS.

That frequent occurrence dealt with, we’re into box vs line ALS_XZ. Nothing happens on r123 vs NW or c123 vs NW, or 123 vs. NE. Or c78 SE. Then ALS overgrown NE  159/2369 and r7 19/359 match on singles 3 we have ALS_39.

The removal leaves red and green candidates strongly linked. The clusters merge, and we choose to make red blue, keeping the expanded blue/green.

In the big cluster, we find a lite coloring trap. If blue, 2r4c2 is true and the c1 group is not, making r9c1 blue lite.

Next, an XY ANL or an ALs_52 captures three onlookers.

Next, hitting the Beeby simple ALS button repeatedly brings a series  of five ALS, expanding the ALS to a wrap. Map wise it’s a box to box, a bv to box, a column to column, a row to row, and a bv to box, and another row to row to a coloring wrap.

 Starting with the changes made to the ALS maps so far, you could follow map updates and get a feel for the effort involved.

Next time, we’ll see the play by play, and look into the place of Beeby’s ALS wings in Sysudoku with ALS maps.  

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Scanning the Column ALS and Some ALS Coloring


This post displays the column and box ALS maps, and scans for AIC the columns. The box scans are started, but get interrupted by an ALS coloring expansion.

We start with the ALS column map on ultrahardcore 311 . It was drawn at the first ALS_XZ found by the Beeby solver after ALS building.

Matching columns as we matched rows, we add column ALS left to right, and scan left from each column for a match on any circled singles or bv, starting with column 2.

The long ALS with singles produce many long internal slinks between different values, but surprisingly, no significant connections occur between the leftover AIC segments.

In c3 ALS 2345/25678, the 6 value group is marked. It’s because it’s confined within West box. That means it can form a grouped wink with single or value group confined within the same box. The 6 group teams with the single 6 in the red c2 ALS we just added.

Now copying the matching ALS to the AIC grid, we can see  that that 2r5c2 sees the box confined 2 group of the c3 ALS and all 2’s in the c2 ALS.

The mapping and scanning process has led us to Beeby’s ALS_62 we saw two posts ago.

We might have marked the 9 group in SWc1 for possible X matches in c2 and c3, and removed it when we got to c4.

There were no further matches in the column scans. Moving on to box ALS, did you draw some?  Box susets are different. Cell positions are numbered in keypad order.

If you got past that, your box ALS map looks something like this.

Marking for the ALS_XZ partnering scan has a new element. Lines are added to signal when a value group has line alignment. 

A Center box ALS starts a cluster can you spread it?

When partnering box to box, we can follow the keypad order, west to east, and north to south. First though, we watch for ALS extension nodes from each box.

In Northwest, another unaligned ALS expanding the cluster.  The internal 26/258 slink turns 8r2c3 blue, and the cell turns 5 green. 5r2c1 gets trapped.

The removal makes ALS r2 13/358 an ALS in the NW box, but the coloring expansion shown here leads to the wrong conclusions. Alignment with r2 is the reason. In r2 ALS NW r2 13/258 is in conflict with ALS r2 1379/35789. We have to forego the color expansion and 8 trap.

Next time we’ll update ALS ALS maps and continue with ALS nodes and ALS_XZ scans.

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Building ALS Maps With Suset Tables


This post displays the ALS map for rows of UHC 311, at the moment when the Beeby solver returned the first ALS_XZ. The suset ALS marking and building aid is introduced, and suset tables for rows, columns and boxes are shown.

In systematic ALS_XZ searching, pairs of ALS form an ALS_XZ when there is a group weak link X between one of their value groups, and when an outside candidate sees a group of another  value in each ALS.

It became clear that the ALS map could be made much more workable by dividing it into three maps, one for row ALS, one for column ALS, and one for box ALS.  Here is the row ALS grid.

It’s surprising that there are that many ALS, just along rows. But you  build this map once, then bring  updated copies along with you.

Calculating, listing, and updating ALS are more easily performed using a numerical representation introduced much earlier in the blog, the suset.  In the suset, two digit strings identify cell positions within the unit and value groups within the ALS. The three ALS in the third row above are described by susets 79/238,  379/1238, and 279/1238. Positions and values are listed in increasing order to make string comparisons easier.

Here is the table of susets for the row ALS of UHC 311.

Each row is represented by two lines. The lighter line lists the cell positions and values. The second, darker line lists the ALS. ALS are built up by combining values and cell positions. For example, in that third line, we notice the common values and combine two cells with three total values for one ALS 79/238. Then we seek to add one cell and only one value. We find two ways to do it, adding 3 and separately, 3.  Adding more cells will cover all values and cells, with our suset not representing an ALS. We’ve left several susets in the table with lines through them to indicate this has happened.

Map drawing starts with the suset table in place, and with a copy of the current AIC grid available. With the table in place, we now concentrate on drawing the ALS on the map As we add an ALS, if it looks promising as an ALS node, we can place a copy on the current AIC grid and check it out.

 For example, here is a  row 2 ALS turning  9r2c8 green. Internal value groups and an internal slink turn the 9 group blue, and a box slink finishes the expansion.

Although bv cells are ALS in themselves, being one position and two values, we place them in the suset tables on the light lines only, not on the darker lines. wo of them in a box, but not a single line, form an ALS. That showed up last post as a pair of bv doing just that, another way to do the cluster expansion above.

Note that it’s the internal slink between 2 and 9 single values that extends the coloring, turning 6r2c8 blue and trapping 6r2c7.

As you build the ALS maps, you can scan for ALS_XZ partners, starting with bv cells. Even before that, you might scan the boxes for unaligned naked pairs.

On the Row ALS map, as you add ALS r8 579/1456, light blue in the map above, and add the single value circles, check for matching bv, then scan previous rows for matching circles. The match of 6 on row 1 gives you a restricted common (X) with ALS r1 27/256. The other single (Z=5) in that ALS is seen by the ALS_65 victim, 5r8c8, which also sees single 5r8c9.

We’re not quite finished with the ALS_XZ scans from r8. Scanning down to r5, we get a singles match on 1 but then no 4 or 5 candidate or group sees both 4 groups or both 5 groups. No victim, no ALS_1Z.

No matches with the r9 single 9 and we’re done.

Let’s finish here with the column and box ALS tables, and invite you to draw some ALS.  Next time, we’ll add the maps and fill in some details on ALS_XZ building, and imagine what it was like to construct those found by the Beeby solver on the way to a new UHC 311 solution path.

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AIC Building Trial Follow Up for UHC 311


This post returns to AIC building after an encounter with a spontaneous ALS led to expanded coloring and a color trial. That will be normal under a plan of incremental ALS mapping.

The confirmation of red had the follow up of (N7, N2, S8) , and the blue/green gets an easy ALS to expand the cluster.

ALS building picks up with a grouped ANL.

What follows is a sequence of 1-way AIC, almost a Philip Beeby exclusive form of discontinuous loop.

For DIY performance, you start this one with an AIC from 6r1c7 because  it sees other 6 candidates. They are off when 6r1c7 is on. Your AIC from there assumes it is off, and if it winks at any of those 6’s, they are off regardless. The AIC is effective in only that direction.

A second 1-way starts from 1r5c7.

Then one cell gets robbed by 1-ways 3 and 4.

Next post introduces the next Sysudoku step after AIC building in human solving of extremely hard Sudoku. It’s the building of ALS maps for rows, columns and boxes. These completed maps become an updated solving resource, like X-panels. Building the maps is a solving process in itself, with every added ALS possibly being a new ALS node extending the AIC network or a new ALS_XZ partner, making a removal.  

Let’s review Almost Locked Sets and the ALS_XZ. An ALS is a set of n cells in a unit(house) containing candidates of n + 1 values. Candidates of the same value within an ALS we call value groups. ALS value groups of a single candidate we call single values, or singles. Group strong links exist between the n + 1 value groups of an ALS. If any value set is removed, the n remaining groups are locked, that is, each remaining group contains a true (solution) candidate of its value.

Group weak links can exist between value groups and outside candidates or groups. There are  strong links between the ALS value groups. If one group contains no solution candidate all other value groups do. The group wink into the ALS, the internal group slink to a group of another value, and the group wink out, these form the  ALS node on an AIC.

Here is the Beeby solver’s first ALS_XZ for UHC 311. We build ALS maps to find all ALS_XZ in the unlikely event that we need them all. The ALZ_XZ is based on the above mentioned locking property of Almost Locked Sets. A pair of X value groups see each other. Here one is a 6 group and the other is a single 6. The group wink (dashed line) is generally known as a restricted common. Only one of the two ALS will have X = 6 in the solution. The other ALS’s value sets, including it’s 2 group, will be locked.

A 2 candidate victim sees (group winks) both 2 groups, so it does see a true candidate. The victim doesn’t have to be in either ALS. , and it can be a group.

ALS_XZ is a consistent source of candidate removals, but spotting them is difficult. Finding them exhaustively seems overwhelmingly so, because ALS are so numerous, and because you have to see the X and Z connections between two ALS for each one. But like AIC building, there is a practical way for human solvers to do it, when you’re not lucky enough to spot the right one, the ALS maps.

First time through, we tried all ALS on a single map. Once was enough. The practical alternative is three maps, for rows, columns and boxes. Binary value cells, our bv, are ALS and can partner to form ALS_XZ. They’re already on all three maps, so we just include them as we scan the map for ALS_XZ partners.

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Coloring ALS Value Groups in UHC 311


This post introduces the expansion of coloring clusters by coloring groups by grouped strong links, such as the those between value groups in ALS. This advance is especially significant as we address whether or not to exhaustively construct ALS and ALS_XZ in Sysudoku.

On the grid following a first phase AIC building without ALS, I superimposed an ALS with a value group aligned with 2r3c4. If only I could find a wink from the long AIC from 2r2c6 into the ALS value groups 4, 5, 6 or 8, the almost nice loop removing 2r3c4 would be closed

I thought I had one, but discovered three weekly posts later, I had not.

That concentrated my attention, and after adding a blue/green cluster to the flock of bv, I found the red/orange cluster connecting ALS value groups 2 and 8 with the naked pair N27. It even delivered an extra trap outside the ALS, and tied in another ALS from my later ALS building.

The red side offered  the result I wanted, and more.  I tried the orange side, hoping for an observable contradiction. Instead, I got an immediate collapse, with with a blue wrap and large numbers of naked pairs along the way, and a reversible rectangle near the end. It reveals a puzzle of almost two solutions, or almost three, one hard and two easy.

Next week we introduce the exhaustive generation of an ALS map, with a ©PowerPoint template to aid the process.   Exhaustive ALS building takes time, but can be organized and kept in bounds. The task fits in systematic order of battle against very difficult puzzles.

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Recycle AIC Building on UHC 311


This post shows how the AIC building done by the review solvers on ultrahardcore 311 can be systematically duplicated by a human solver. The technique is a graphic one, very amenable to ©PowerPoint graphics, and slink marking of Sysudoku Basic. This and other examples will be the basis for AIC building recommendations in the Sysudoku Guide.

AIC building is a phase of Sysudoku Advanced which follows the bv scan  for Sue de Coq, WXY-wings, and XY-chains, and the X-panel analysis for X-chains, grouped X-chains, regular and finned fish, and edge limited freeform pattern analysis. After these systematic and exhaustive elimination methods, the human solver can reasonably begin AIC building, a systematic search for Alternate Inference Chains, with its many possible starting points, continuations, and terminations. Let’s look back to the post of last November 3, where solvers Sudokuwiki and Beeby immediately followed line marking.

A series of four AIC’s resulted, before the first ALS_XZ. With solvers you never see the many trial chains that do not lead to any result. Is it an overwhelming number, though? Maybe not.  Consider how the chains generated on a grid are interdependent,  restricted as they are by Sudoku placement rules. In the review, three of the four solver AIC start on the same slink.  Many segments are shared among these four.

If we set out to find every grouped, fully extended AIC without ALS nodes or 1-way branches, how many would there be?   We’ll do our top down, left to right scan for starting slinks. Eligible slinks have a wink directly to a next slink, and we can go off each end of our starting slink. Record your guess on how many distinct AIC, counting extensions and not counting extensions.

Starting off with the r1 5-slink,  we can’t leave from 5r1c2 because it doesn’t see another slink partner.  In that sense, AIC building is like lite tree building. It’s a surprise how far the AIC goes, getting into a loop but branching out to get to two possible slink terminals, but on 3, not 5.

Veteran readers know that a loop can often start a coloring cluster.

Here we realize the AIC contains  a 1-way from 3r1c2. Either terminal 3 sees the victim, and if either is false, the chain makes 5 true in the victim’s cell. In the review, Beeby made the same removal with a complex 1-way, later.

Now we shift the 3’s to mark the new slinks in NW and r1, and move on to the r1c4 slink for an AIC starter.  From the 3 it gets into the same loop, but going in the wrong direction to slink out on the 9. From 2r1c4  there’s a long branching AIC, with one branch reaching an ANL.

The second elimination in r1 generates N3, which destroys an ALS_XZ partnering ALS in the review.

Now moving on to the N 4-slink and 4r3c4, we’ll blue out past segments to put them in the background, going to black for new extensions. We get an ANL removal with each extension.

The right branch brings us to an extension to 4r8c9 which teams with our staring 4r3c6 in an ANL removing 4r8c6. Btanching off at 8r8c6 to 4rc5 removes 4r4c6.

Recycling AIC is good.

We get to do it again with a grouped extension from 5r8c1. The prior removals provide that closing slink for a confirming ANL and N4.

 A benefit of systematic AIC building is to have many AIC segments on a background layer, for  possible use later.

We’ve accounted for all the solver AIC findings. How many AIC did we have to generate? I’d say a reasonably small number.

Next week, we start a similar reverse engineering on the solvers’ ALS findings, introducing a graphics tool for systematic searching. That too, is a complexity nightmare for human searching that can be kept in bounds. The prospect is a systematic order of battle against very difficult puzzles.

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Dov Introduces Lite Coloring of ultrahardcore 311


This post finishes the lite coloring revisit of past trials with Dov Mittelman’s lite introduction. It leads to a solution of Stefan Heine’s UHC 311 without trial. This puzzle occupied 3 posts in the ultrahardcore review, starting with November 3. Dov’s solution begins at the first grid of A Coloring Trial Weakens UHC 311 of November 17, at the grid before the coloring trials.  I’ve added arrows for the lite trees, and the immediate traps and confirmation. I’m quoting Dov’s email of November 23. You can tell he’s been reading Sysudoku altogether too much.

“UHC 311 wishes to settle before going to trial.”

“Starting from the toxic orphan, in c2, 8(6,2) is light orange. 2(6,2) immediately vanishes in a puff of logic.”

Yes, 2r6c2 goes, because either red or orange is true.

“Also, we have a bridge between orange and green. Red and blue make a toxic pair. 5(9,3) storms off in a huff. “

“Light red can also grow, and 6(7,8) goes light red, bringing with it, in order, 1(4,8) and 1(5,5).”

The bridge at work in a lite trap. Note how the SE 6 group carries the red lite tree. Oops, my bad. I didn’t mark the c8 6 slink.

Now E1 => C1, for:

““The red-blue trap kills 4(5,5) and 1(8,5), establishing the 1-clues and expanding green-blue.”

Adding a lite blue tree from 5r8c1, blue light 4r8c9 and green 4 trap 4r8c7 to leave full blue 4r8c9, green 5r8c9 and blue 5c7c9.

As 5r8c6 is trapped, both 5’s in the South box become blue lite and blue is wrapped. Does that have consequence for red/orange? Yes, if orange is true, orange 8r6c2 sees green 8r6c5, its last candidate. So orange is wrapped. We have the clues of the color trial, without the trial.

Deleting the blue and orange candidates and trees, and promoting the green and red trees to full colors, we carefully pick out the second right page trial free solution among the 12 ultrahardcore review puzzles.

Deleting the blue and orange candidates and trees, and promoting the green and red trees to full colors, we carefully pick out the second right page trial free solution among the 12 ultrahardcore review puzzles.

Next time, we’re staying with this puzzle UHC 311 to revisit its AIC Building phase, and to show how AIC building can be organized for systematic, exhaustive exploration, without the human-like solvers.

And for the readers that have been looking at my early posts on Susets(10/25/11),  and ALS Toxic Sets(7/17,25/12), I’ll soon be updating and applying susets to generate the ALS map, and to duplicate by hand the review’s impressive solver ALS findings of 11/3/20 systematically and exhaustively.

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