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[keith]

I decided to look for another of these, which ravel pointed out a week or two ago.

[Victor]

Nice! This is really an AIC I suppose? (Kind of thing I search for these days, mostly without success.)
The chain is 3R4C5 = 4R4C5 - ..... = 3R6C6, killing any 3s seen by both ends.

[Marty R.]

[keith]

[Marty R.]

[Victor]

Marty, not sure that it's quite an XY-chain. I thought they were constructed purely from bivalue cells. You could solve this puzzle with an XY-chain: look at R4C359 & R5C4 - cells 29, 39, 34, 24 in that order. If either end is NOT a 2, then the other end must be - works like an XY-wing, in this case removing 2 from R4C4. (But Keith's method is nicer!)

Here are some general thoughts about AICs and things. I've come to a dead end sometimes when doing puzzles - fairly certain that there's no more colouring / skyscrapers / etc. to be found. So I went looking for other techniques and took a liking to ALS - elegant idea, easy to understand if not always easy to find. They've turned out to be quite powerful in the sense that they've cracked some puzzles I couldn't otherwise do, and they seem to turn up quite often in Menneske puzzles. (You could solve this puzzle with ALS.) BUT there are 2 cons:
(a) I can't be sure I've spotted them even if they're there, in the way that be with say XY-wings.
(b) I'm fairly sure that there are Menneskes - and others - that they won't crack.

So what else? Well, you've gone for Medusa colouring, and I can see how powerful that is, but I'm sort of mentally prejudiced against this technique - no sound reason! I've watched people crack puzzles with what are essentially AICs I think, like Steve doing the puzzle you posted in the Solving Techniques .. section. And Nataraj, Ravel & Asellus at work on this one: http://www.dailysudoku.co.uk/sudoku/forums/viewtopic.php?t=2290&start=0&postdays=0&postorder=asc&highlight=

[Victor]

Sorry, accidentally clicked Submit before I'd finished.
The point I'm making is that I need something beyond ALS to try to solve difficult puzzles, but I'm not sure how to go about it without doing Medusa.
(Not sure if Medusa solves all puzzles that can be solved by AICs?)
There are of course rare techniques that might solve a puzzle here or there, like Ravel's recent smart sort of repeating W-wing, or Sue de Coq. Nice to meet things like that, but they're icing on the cake, and won't help with many puzzles.

[Marty R.]

[Asellus]

Victor,

There are two ways to see this AIC. The simplest (conforming to Keith's description) is:

(3)r5c6-(3=4)r4c5-(4)r4c7=(4-3)r6c7=(3)r6c6-(3)r5c6; r5c6<>3

Only slightly different (and conforming more, I suspect, to Marty's description) is:

(3)r5c6-(3=4)r4c5-(4={37})r45c7-(3)r6c7=(3)r6c6-(3)r5c6; r5c6<>3

The is almost an ALS Chain, which might be what Marty means by calling it an "XY Chain." Instead of the strongly linked <4>s in c7 exploited in the first example, it exploits the 2-cell {347} ALS in r45c7.

An XY Chain is just the simplest form of ALS Chain... one limited entirely to bivalue cells (the simplest form of ALS). But, the same concept applies if you use larger ALSs along the way, such as that 2-cell 3-digit ALS used here. In all cases, an ALS collapses to a locked set along the inference chain sequence and thus propogates the chain. (A naked single is the "locked set" when a bivalue collapses.)

The second example deviates from a true ALS Chain by exploiting the strongly linked <3>s in r6. So, we probably have to say that the more generic term "AIC" is the most accurate description.

By the way, the key to understanding the Eureka notation for an ALS chain node is to realize that in an ALS each digit is strongly linked to the alternate locked set. So here, the <4> in the <347> ALS is strongly linked to the {37} locked set: (4={37})r45c7. Had we needed, we could also have exploited (3={47}) or (7={34}).

And for "extra credit": The "xz" ALS technique is simply a 2-node ALS Chain:

(x)victims - ({ax}=z)ALS1 - (z={bx})ALS2 - (x)victims; victims<>x

where a and b are any other digits that form the ALSs. The "shared exclusive" z is weakly linked (cannot both be true) between the two ALSs.

[Victor]

Thanks Marty - no I didn't know that I could edit a post - thanks for telling me.

And thanks Asellus. You're turning me into a proper sudoku-er! I do appreciate it.