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/ does not contain X or Y
. may optionally contain X or Y or both
.-------------------.------------.----------.
| . . . | XY / / | / / / |
| XY *-XY *-XY | . . . | . . . |
| *-XY *-XY *-XY | . . . | . . . |
.-------------------.------------.----------.
These assumptions can be loosened quite a bit. The fact that r1c4 is an ALS in this example is unfortunate, as that fact diverts attention from the key point. It is important that r2c1 be an ALS to make this work. What's important about r1c4 is that it's a weak ALS which contains both X and Y. Readers not familiar with the "weak ALS" term or concept don't really need to understand it to follow most of this post, but here's a quick synopsis. I'm using Myth Jellies' term for the concept as he explained it in this thread and in more detail here. A set of N cells with N-1 digits locked into those cells is a weak ALS. The complementary set of unsolved cells in the house is always an ALS. In this case, r1c4 is the simplest form of a weak ALS, a single cell containing 0 locked digits. (The next step up would be a pair of cells containing 1 locked digit, which we normally call a conjugate pair for that digit).
Myth Jellies was kind enough to respond to my first significant post to this forum with some words of encouragement and a gentle suggestion that I begin using some formal notation and begin expressing my reasoning in terms of Alternating Inference Chains (AIC's) whenever possible. It took me a while to get started with that, but once I did, it was amazing how things started "clicking" for me. That was good advice.
So let's see if we can write an AIC which expresses what's going on here. The assumption that the slashed cells do not contain either X or Y means that we have some strong positional links:
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(X)r1c123 = (X)r1c4 and (Y)r1c123 = (Y)r1c4
We can use these links in an AIC as follows:
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(X)r1c123 = (X-Y)r1c4 = (Y)r1c123 - (Y=X)r2c1
Now, we can argue that since everything is symmetrical with respect to X and Y in the chain, we can interchange X and Y in it to produce an AIC from which the eliminations of Y in the starred cells in the diagram follow.
However, alert readers will have already noticed that the above chain can be continued:
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(X)r1c123 = (X-Y)r1c4 = (Y)r1c123 - (Y=X)r2c1 - (X)r1c123
Now, I previously commented that the above loop would still hold true if r2c1 were replaced by any ALS in box 1 containing X and Y (but disjoint from the cells in the line/box intersection containing X or Y) since there is a strong internal link between any two digits in an ALS. So next we might want to consider how we can generalize the role of r1c4. What's important, of course, is that there be a weak link between X and Y in some set. So in what type of sets can we say that such a weak link exists? Well, it just so happens that the weak ALS fits the bill, which makes Myth Jellies' name for the concept imminently sensible (no surprise there, of course). It is sufficient to assume that X and Y exist as candidates in the weak ALS, but that they are not among those digits which are locked in the set. (Actually, to be strictly precise, again we must also assume that the weak ALS is disjoint from the cells in the line/box intersection containing X or Y.) Since the weak ALS has room for exactly one non-locked digit, there must then be a weak link between X and Y in the weak ALS (they cannot both be present in it).
So to generalize the conditions which are sufficient for an ALC type elimination, one step forward might be something like this:
Suppose C is a set containing some or all of the cells lying in the intersection of a line and a box, D is a Weak ALS disjoint from C lying in one of those houses (call it house CD), and E is an ALS disjoint from C lying in the other house (call it house CE), and that there are two digits u, v such that
- both u and v are candidates in each of sets C, D, and E
- AND
Note that the assumptions that u and v exist in C, and that C and D are disjoint, means that u and v are not locked in D. I could continue with the conclusions to be drawn, and could give some examples, and will do so if anyone requests. (I won't promise to be timely about it, necessarily. If anyone feels they can and wants to continue with this, please feel free to do so. Just reply back here with your intent to do so.) I'm wondering if it's really worth the effort, though. Again I will say that if one is familiar with using AIC's with ALS's (and now weak ALS's might be occasionally helpful, too), then you've got many of the solution techniques covered, even if you don't know them by name. If we try to give a name to every class of position according to the particular arrangement of sets and links within it, we're going to generate a lot of terms, and since the same technique is often independently discovered by different individuals, there will be aliases for the same technique or concept. That's my problem when I try to study other forums.
I will comment that if the best possible use of the ALC technique is to be made in the general case, then the ALC conclusions must make mention of other digits besides u and v.
I will also add that I have convinced myself that any position which satisfies the above constraints for ALC eliminations also contains "2 ALS's with 2 restricted common digits" (as I discussed in my long-winded spiel in this thread). I will leave the proof of that as an exercise for the reader. (Hint: consider the set of unsolved cells complementary to D in house CD.) If any readers are wondering why I chose C, D, E and u,v as names for the key sets and digits here, I was just trying to avoid any possible confusion between the two posts.
So if you understand the ideas I presented in that spiel, you don't absolutely have to use the ideas behind ALC eliminations (and conversely). However, sometimes the alternate approach will involve smaller sets and will be easier to spot.