Ultrahardcore Review ALS-wings V


This post completes a survey of ALS-wings in the review of Stefan Heine’s ultrahardcore collection. The study was to determine how ALS-wings can be constructed as ALS maps are built. This post covers five more examples in UHC 443 found by Phillip Beeby’s solver. The survey shows that all of the ultrahardcore ALS-wings are constructed in the systematic ALS mapping process.

Ultrahardcore 443 and 487 were added to the review when the 400 puzzles I based my selection on turned out to be 500. It turned out well.

In 443, this severely overlapping ALS-wing defies graphical display. My rendering of Beeby’s notation is

(5=1)blue – (1=8)red – (8=5)green => -5 r6c1

A possible remedy for the congestion is a modified grid giving the congested row an extra line. Ahhh! That reads better.

The next column vs box ALS-wing had to be redrawn to fit in one column on the map. It differs from the 311 column vs row case above in that the bv is linked to the column ALS is already on the map and  and is roughly confined to one column. A possible remedy is to have the wink to the small green square to signal a bv attachment. Then the scanner can refer to the current grid to determine the matching value and its location for the ALS-wing.

This case is also covered by the preliminary scan for wink out bv and the match list. The preliminary scan puts bv 27 on the c1 red match list, and black ALS finds it there.

A second 443 example in which the row ALS entry on the ALS row has a small green square winking at 5r2c4 awaiting the scan from the red ALS being entered on the ALS box map.

Alternatively bv 59 is put on the blue ALS match list on its preliminary bv scan, the red ALS row scan includes matching bv partners on the ALS being scanned, in this case 9 on the r2 blue match list.

Continuing on 443, a bv ALS-wing, and another ANL with a value group at one terminal and one of its members on the other. The victim 7r6c1 sees both.

Why is it not a confirming ANL making 7r6c8 a clue? Because the value group 7 is a terminal, acting as a candidate. It is not an internal slink.

The wing happens with the blue ALS on the row map and as we draw the red one on the box map. On the way down the map, we already examined row ALS matching the bv partners of red single 9. Now we have to recognize this one as a value group  aligned match with red single 7. Then we can see the AIC around to it.

Here’s an easily read ANL. At construction time, blue and green ALS are on the row map when the red ALS  is added. Blue matches the red 8 first and goes on the c8 red match list. Then green’s 5 group matches the red 5, we note the match of blue and green value groups, and see the victim before we even copy the blue and red ALS to the grid.

One more. An ANL with bv 8 on one terminal and value group 8, including bv 8, on the other. The red ALS pulls up the blue one by either the match on exit 5 or the match on bv 8. Both matches are required

UHC 487, the last puzzle of the expanded ultrahardcore review, provided Beeby no ALS results prior to trial. This ends the review survey..

Have I made the case that ALS-wings, like ALS_XZ can be constructed with ALS maps? The will and patience to construct suset tables and ALS maps is a large order.

Next week we consider what  Beeby’s complex 1-way is and what a DIY version of it looks like.   

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Ultrahardcore Review ALS-wings IV


The survey of ALS-wings in the ultrahardcore review continues, with new wing building situations, another ALS version of the Death Blossom, and an amazing ALS.

In this UHC 267 ALS-wing it’s a two box ALS , connected by a column ALS. A natural spotting scenario on three ALS maps would be column first, looking for wink out connections on each end. But if we’re drawing the column ALS first, it’s not until we are drawing the East box ALS and looking box to box. In fact, it’s when we draw the East box and scan for lines to match it that we see it connect to the red ALS, and the match list reveals that c9 red connects to the NE blue.  On the victim hunt, we only have the chain ends to worry about, because red and blue ALS_3Z victims would be removed.  That leaves a green 1,7,8 vs. blue 2,8 to match up for the grouped ANL on 8.

In two Death Blossoms with stem r2c6, each 9 victim sees 9 value groups in two ALS petals. The value groups slink to singles seeing both candidates in the stem. If either victim is true the stem is stripped of candidates. Replace the stem with a bv with internal slink and it becomes a bv ALS-wing ANL.

Add a cell and value to one ALS and get another the ALS-wing, or Death Blossom, removing 3r6c4.

What happens here in our ALS-wing building process? It’s column vs box with the red ALS added before the blue one. Actually, if you are AIC building from each wink exit as the  red ALS is added, you’re at the 9’s in the Center box and at 5r2c6, so you could make the ALS connection with both blue ALS before they are drawn from the suset table.

That brings us to 311, and the ALS-wing that gave me pause about row vs column ALS matching.

Adding the red ALS after some AIC building, we might be scanning previous rows for matches for 6 in c6 as well as 1 in c5. As we add the column ALS, if we scan for matching bv for each single and aligned value group, we would not require the saved little green square in the red ALS row that we thought earlier  that we needed.

Instead, the blue scan for bv would trigger a row scan for 1 in c6, the match on the red single would call for a match of value groups, yielding the intersecting 8 groups and 8 victim..

This takes advantage of bv being marked on all three ALS maps.

Going on to UHC 355, here is a box ALS being scanned vs. the column ALS. With the new single marked on the column ALS, and the aligned value groups marked in the box ALS. The ALS chain defining the ANL is not hard to see.

UHC 399 brings another bv ALS-wing where the box ALS with attached bv is scanned over the row ALS. The scan matches the 6 from the preliminary scan of the red ALS for bv. This example exposes the fact that a line match should include a group within the box of a bv partner, here the blue single 6 in the SE box.  In addition to that 6, the red ALS is prepared to match row ALS values 1, 3, 4, and 7. I had to verify the blue ALS by writing out the suset r7  123579/1345678 !

Next week we finish the ALS-wing examples with five from UHC 443. Each post has brought some new insight on the DIY construction of ALS-wings. The next one is post 496.

I’m thinking of pausing the weekly posts, letting the 500th become the home page with navigation information and current news. The pause will allow long delayed blog improvements and the Guide to be completed. Perhaps later, a few puzzles of different levels of difficulty can be taken through the entire Sysudoku Order of Battle, as envisioned in recent posts.

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Ultrahardcore Review ALS-wings III


This post’s examples from UHC 179 include ALS-wings with bv cells, and uncover two essential facts about ALS value groups.

This post’s examples include ALS-wings with bv cells, and uncover two essential facts about ALS value groups.

Instead of following up on its own, the Beeby solver comes back after recording the result to get a prompt from the user. We got curious about the new bv r1c6 on our bv map and its link to the blue ALS single 3.

On the left, we took the wink out from the single 3 and connected it to 3r1c6, then used the bv internal slink to wink back into the aligned 6 group. The internal slink converges on the 9 group, along with the internal slink from single 3, and we appear to have an ANL confirming the 9 group. That is not something that Beeby would report, but it would report the S6 and N3 clues as an ALS result.

Then I noticed that I was one slink away from a nice loop, with the 9 group seeing the 6 group and the single 3, the two ends of the added slink. So which is it? Is the 9 value group there or not?

To resolve this, let’s go back to what we know about ALS, and more specifically about value groups, and more generally, about groups. All groups, including value groups in ALS, are sets of candidates of a single value.  A group is true, when one of its candidate members is in the solution, i.e. is true. A group is false, therefore, when none of its members is true. A slink between groups means that if no candidate is true in one group, some candidate is true in the other. That works for ALS value groups.

That’s enough to refute the confirming ANL on the left, in which internal group slinks from groups 6 and 3 converge on the 9 group. That’s saying the 9 group is true because value groups 6 and 3 are not true. Well, in an ALS, one group is false and the rest are true. Two value groups can’t be true.  Converging value group slinks don’t make confirming ANL.

OK, does this mean the nice loop on the right removes the ALS 9 group?  Actually, no. ALS value groups slink, but they don’t wink. Value 9 group can’t “see“ the 6 and 3 groups. But we do have a nice loop, where ordinary unit based seeing makes removals. In fact, within a unit, any two groups of the same value see each other. So 6r9c6 is removed because it sees both ends of the 6 group wink in c6.  S9 becomes a clue and the 9 group is the false one in the ALS.

This ALS-wing is about as crowded on the ALS columns map as it is here.  Anyway, starting at the c9 green 5 group, the ANL is (5=8)green- (8=2)red- (2=5)blue.  Here’s another one with two ALS in the same unit.

On the DIY mission, you need to trace this out as you add the third ALS from the suset table.  Let’s say the blue ALS is the last added, and you still  have the red and green ALS on the grid. Now the blue ALS brings a second 5 value group in c9. The aligned value groups leave out one victim. The blue ALS can join the chain with its single 2 in the SE box wink or the c9 aligned group wink. Red is in the middle of the chain either way.

If the green ALS were last added, you might have looked at blue and red and found that no 5 sees both groups.

Oddly enough, my other review solver, Andrew Stuart’s Sudokuwiki, gets the removal with one of the ALS. It’s a boomerang starting in 8r2c9 and carried by the ALS 5 group and exit wink back into the starting cell. It’s also a 1-way starting from the 5 group. True or false, the group creams 5r2c9. This is OK, because it doesn’t use the ALS value group as a column 9 group, which it isn’t.

It is another hint that a try out for each new ALS as an AIC starter is in order. What if you’d constructed the ALS-wing we began with, and found no victim. Beeby wouldn’t have reported it either.

Next week, we continue the ALS-wing survey with ultrahardcore 267, but I shouldn’t pass 223 without mentioning the post of October 5, 2020 in which, after an Single Alternate Sue de Coq trial removes 9 candidates, Beeby finds two row vs column ALS. The second one is the review’s only double ALS, removing 6 candidates.  You might want to estimate your chances of spotting these with separate row and column ALS maps.

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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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