Sudtyro wrote:Thanks to Rep’nA for a clear and detailed explanation. I believe you are right about the limits on where the cells in A and B can lie. If you allow the cells of an ALS to be in different houses, then your analysis very neatly eliminates both the extra 6’s and the 8’s in this problem using a single ALS-XZ rule, and that seems fairly significant.

My only heartburn is that the Sudopedia definition of an ALS requires its cells to lie in the SAME house (and only truly makes sense to me that way). As a beginner, I rely on basic definitions like that in order to build consistent solution strategies. Your strategy clearly modifies this definition, and that leaves me feeling a bit uneasy.

Perhaps I am wrong. The ALS xz-rule is not how I think about these issues and I am bound to mess up. As I mentioned in my first post, I think the way to go is with Subset Counting. With that there is no ambiguity. If you aren't familiar with the technique, let me explain it in this situation.

We have four cells r4c49, r5c59 that can contain the following candidates {3,6,8,9} where,

a priori, we allow repetition. Looking at the grid, we see that there can never be more than one 3, two 6's, one 8, and one 9. So the four cells will each, if you like, choose a number out of a hat containing [3,6,6,8,9]. If we somehow removed two of those numbers ahead of time, the four cells would only be choosing from a hat containing three numbers, an impossibility. So, for instance, if r4c7 = 6 or r4c8 = 6, then none of our four cells could be a 6. This is tantamount to removing both 6's from the hat, leaving us with only three numbers for four cells, a contradiction.

The exclusion of 8's is only slightly more complicated. An 8 in box or column 9, but not in r45c9, will immediately remove 8 from the hat and force r4c9=6 which removes another 6 from the hat since the only way to get two 6's is for r4c4=6 and r5c9=6. Left with only two numbers for the remaining three cells, we get a contradiction.

Of course, this may seem rather glib and complicated compared with the ALS rules, but in practice it is (at least when given the cells) much easier to apply. Note that there is no concern here about houses or even any decision to be made about how to break up the cells into the sets A and B, etc.