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[Victor]

. . . or skyscrapers or ERs, etc. in sight in this one, M3620506 (72). Post basics:

[Asellus]

Victor,

Your grid isn't quite simplified: the <7> in r5c6 can be removed due to locked candidates. Since the <7>s in b6 are strongly linked, a bit of coloring not only removes that r9c9 <7>, it makes some determinations:

[Myth Jellies]

Here is a Medusa/3D Coloring implementation for finding AICs.

Three of the conjugate strings (aA, bB, dD) are highlighted below

[Asellus]

The preceeding is a good instructive example of the use of Medusa multi-coloring and how it relates to inference logic in general.

However, it also exemplifies for me why it is that I prefer the "extended" form of single cluster Medusa that I've been using lately. First, at least in this case, both approches produced the same overall result (the identical BUG+1). Second, the extended single-cluster Medusa does not require any need to make explicit note of weak link connections between color clusters/strings/chains. Nor does it require one to determine which pair/pairs of various colors are strongly linked and which are not. This makes the "extended" approach much more "user-friendly."

Perhaps the Medusa multi-coloring technique can reveal more underlying implications than can the "extended" technique. But then again, maybe not. So far, I suspect that it is the latter.

[Myth Jellies]

The problem with the "extended method" is that it marks up the page based on assuming a particular color is true (or false). It can do basically anything that T&E can do (because assuming a color is true is the same as assuming a set of candidates is true) and its results are dependent on the color/candidate that you start with. The real multi-color Medusa method makes no such assumptions and has no start point dependence. It just marks up each readily apparent individual conjugate string and then you can use a simple who-sees-who to create AIC color molecules. It is not as powerful as the extended method, and still requires a lot of searching to note the deductions; but to my way of thinking it is much more satisfying. It is a way of coming up with AICs without using a single if-then which allows you to put AICs in the same toolbox as singles, naked pairs, and x-wings.

[Asellus]

I felt much the same when I was first exploring the method. However, I now believe that T&E has nothing (or something vanishingly close to nothing) to do with it. For one thing, the extensions work for both colors simultaneously: it isn't just picking one color to assume as true. The truth assumption is a shorthand for the extended marking rules. One could, alternately, construct "extended" marking rules that didn't explicitly mention any truth assumption. It is truly a bifurcation method and has exactly the same issues as any such method (i.e. assuming that one set of candidates or another is true).

On the other hand, the multi-coloring involves selecting weak links to exploit. To my mind, there is more T&E involved in that process. (And I say this as someone who has exploited involved AICs to a considerable extent in solutions and enjoyed doing so. Finding them has a substantial amount of trial and error feel... though, admittedly, an "educated" sort of T&E that improves the odds of success.) The typical advanced grid has a very large number of weak links present. How does one choose? It isn't so easy to see where any particular weak link "bridge" of two clusters will lead. Even if one says, "Well, first color multiple clusters then examine the weak link connection possibilities," this only adds another T&E dimension: In exchange for fewer weak links to consider between the established clusters, one still must choose which clusters to color.

With extended Medusa, I don't have to choose any weak links to exploit or more than one cluster to color in hopes of linking them effectively. Yet, the results seem to be the same! Once I make my initial choice of where to start the Medusa coloring, everything else follows strictly mechanically. Either it will produce eliminations or it won't. Perhaps a different starting choice would produce better results. But that's true of Medusa in general, of whatever sort.

So for me, I'm much more comfortable with extended Medusa, aesthetically, than I am with Medusa multi-coloring... precisely the opposite assessment, and even though the logical underpinnings of the multi-coloring is interesting to someone like me.

As for that question of "assuming something is true," don't ALL sudoku solving methods involve precisely that? And, aren't all such methods (other than forcing) bifurcative? An AIC typically assumes that a victim is true, then shows that this leads to the conclusion that the victim is false. So, we accept the alternate. Even DP solutions assume that either the puzzle is valid or it isn't. Everywhere we look we have alternate assumptions of truth, as we should have (in my humble opinion). In fact, that isn't the interesting question at all.

The interesting question is: which is easier to use?

In answer to that question, I believe it is no contest: "extended" Medusa wins by a mile!

[Myth Jellies]

[Asellus]

Well, we certainly see this world differently. I can't say I'm pursuaded.

When I said "ALL sudoku methods," I should have been clear I was speaking of all advanced methods. I did not mean to include basic methods such as naked locked sets.

[Victor]

Thanks guys, found that very interesting and helpful. I'm not qualified to comment on whether or not T & E is involved in extended Medusa. I'll just remark, as a solver, that different styles of thought / approach suit different people. I happen to be comfortable thinking in chains, tho' I acknowledge of course that you have to find them first - guess I'll have to try Medusa.
I did find the first elimination, of the 8 in R2C2, using a certain amount of T & E (albeit of an 'educated' kind). But I did then see that R5C5 <> 2 without T & E, by looking at how conjugate links interact, and that felt good.
Anyway, thanks