ALS-Wings

ALS-wings are AIC made entirely of ALS nodes, and offered as a next step option by Philip Beeby’s PhilsFolly solver.  ANL terminals of these chains are therefore ALS value sets, both singles and groups.  Hodoku offers the same option, labeled ALS-chains, as reported 1/5/16.

ALS-Wings are Alternate Inference Chains, and as such, are best done DIY as explorational construction in AIC building. They’re not something to be spotted. On the other hand, in most cases, they can be constructed, or have their results duplicated by simpler findings. To demonstrate, we examine some ALS-wings of right page ultrahardcores on this blog, beginning with uhc 3, posted 6/9/20.

On the right is the first ALS-wing  Beeby found in uhc 3. In Beeby chain notation:  (7=6)green – (6=2)red – (2=7)orange.

The DIY spotting challenge is identifying the three 3-cell ALS nodes. Only then comes any realization that there’s an ANL victim of the green and orange terminals on value 7.

The post’s boomer alternative is started on 5r1c9 and the 5 box slink marked in basic.

Shortly after, on the next selection of the ALS category, Beeby recombines two of these ALS with a third one. On the left, the 7 removal is also a coloring trap from cluster expansion, and 9r5c7 is the ALS-wing victim.

The Beeby chain description is (9=6)green – (6=2)red – (2=9)blue =>  -9 r5c7.  The outside victim 9 sees a 9 group in the green ALS, and an overlapping 9 group in the blue ALS.

The 7 value ALS-wing is shown separately on the right.

But again, AIC building bypasses the ALS-wings. 9r5c2 is removed by a boomer from 2r5c7 and another for 7 from the same start. But even easier, 7 is removed by either green or blue. It’s the boyfriend trap:  blue 7 sees her boyfriend 7 in the same cell with a green candidate.

In ultrahardcore 91 after a trial, this ALS-18, a 148-wing, or a Bent Almost Restricted 3-set ( 3-BARN) is found, on the right,  to be an ALS-wing as well. It requires the red single 4 to see the blue 4 group it is a member of. An ALS-wing curiosity, if that is possible.

Another form of ALS-wing duplication appears in uhc 135 posted 9/1/20. I was suspicious that an overlapped ALS_XZ might map into an equally simple ALS-wing.

Not exactly. The 2 and 5 cell ALS become 1, 2 and 3 cell ALS and the ANL terminals are a single candidate and a group of three, including that candidate.

As I scanned the 2020 posts for instructive ALS wings, the next one selected was accompanied by this 1-way turned ANL by an ALS value group. This is expected when ALS nodes are included in AIC building.

Then it struck me. That very thing is buried in the ALS-wing above. If we start a potential 1-way on 2r2c5, its an XY chain reaching into the ALS for the 2 value group. We’re in the driver’s seat, verifying a 3 cell ALS to get the ANL terminal group.

In the same post of 9/15/20 is a more typical ALS-wing.  The bv r9c6 is a possible starting point for a 1-way AIC, and using the red ALS to keep the chain going, we get to the single 3r8c6. Now you can imagine constructing the 4-cell ALS to incorporate that 9 group and duplicate the removal,

but if you notice that 3 is signaling by its position that it is a slink partner, you would be completing the 1-way without that ALS. Right?

If starting 9 is true, the top 9 goes. If not, the chain puts a 3 in the top 9’s cell.

Not to be outdone, Beeby comes up immediately with a second ALS-wing, with  cell overlap in c9. It’s two 3-cell ALS and a 2-cell ALS make  ALS mapping necessary for DIY solving.

But in solver investigations, a second opinion is often helpful, and in this case, Sudokuwiki supplies an ALS assisted boomer for the same result.

The 2-slink in c8 leads the DIY AIC builder to the 2 single and 5 value group of the ALS.

Another example of ALS AIC building comes from this Beeby ALS-wing from the ultrahardcore 221  post of 11/2/21.  You might expect an AIC started on a bv to become an ANL when corner turning ALS like these are found to provide ANL terminals.

But its easier than the 4-cell ALS in c1 makes it look.

While you’re assembling that ALS for the 6 value group, you would probably notice the 3 slink right into r1c1, which finishes a 1-way starting on 5r1c7.

But blame me, not Beeby. I asked for an ALS-wing.

To finish on AIC building with ALS, here’s one that does build into an ALS-wing. It’s from uhc 355,  posted 11/24/20.  Same circumstances, a possible 1-way or ANL starting from bv r2c2, with a promising crossing line ALS for a value 7 ANL.

We get through the West Box for the ANL because crossing aligned pairs 6 and 5 form grouped box slinks.

The ALS-wing shows up when we recognize the 5-cell box ALS that houses the crossing value groups.

My conclusion from the review of ultrahardcore ALS-wings, is that, as a response to the complexity of ALS, finding ALS as needed in AIC building is more satisfying than the ALS mapping I attempted to define in the latter blog posts. And for ALS_XZ, set a practical DIY limit with ALS Partnering.