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.