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[Marty R.]

This one is really bugging me. I've made seven moves involving five different techniques, solved lots of cells with little remaining, yet I've hit a brick wall except for a possible BUG+4.

[Steve R]

Marty

I don’t know which puzzle this is but it is definitely serious. The classification (“hard,” “fiendish,” etc) is a general guide. The numerical ratings Gaby provides offer a more precise indication of difficulty. Here I suspect you need a chain or some almost locked sets. If so, the rating might include a level 4 technique.

The computer tells me that one way forward is to eliminate 4 from r8c8 using the XY-chain
-4- r8c5 -9- r2c5 -8- r2c3 -4- r5c3 -7- r3c3 -2- r1c1 -7- r1c8 -4-

Steve

[Marty R.]

Thanks Steve. I thought I had exhausted the XY-Chain possibilities, but obviously not.

I'll have to look at the site a little more closely, as I didn't realize that the puzzles had ratings more precise than the Hard, Fiendish, etc.

By the way, the first Fiendish I tried was too easy, as it solved with only a Type 1 UR. The one here was the second, so I'm hoping there's a happy medium somewhere.

[Asellus]

Marty,

Since you posted elsewhere about appreciating the technique of coloring pincers, you might find the following interesting.

Note the two XY Wings with <4> pincers at R1C14/R5C1 and R2C35/R8C5. Note also the <4> X Wing at R25C13.

This means that IF the X Wing corners R2C3 and R5C1 are false ("green"), R1C4 and R8C5 must both be true ("red"). But simple coloring produces a "Wrap": R8C8 would also be true. I'm pretty sure it is not valid to eliminate all the "red" values in this case. (The pincer ends are "strongly inferential," not conjugate pairs.) Instead, the X-Wing assumption is false: R2C3 and R5C1 must be <4>.

[Marty R.]

As always Asellus, I appreciate the help, but it's too advanced for me. I noticed the wings, but linking them and talking about wraps and inferences is beyond my skill level, sorry to say.

[Asellus]

While it may or may not be very useful generally, I find this particular situation to be rather elegant and worthy of note.

Maybe a diagram will help, for Marty or for others:

[Marty R.]

[keith]

[Asellus]

Marty,

Provided that you only use a single color chain (starting from one of the pincers), then you have nothing to worry about.

It is when coloring is extended from both pincers that things get tricky. Let's say the pincer cells are "R" and "g". There would then be an "RG" chain and a "gr" chain. Eliminations are only valid between cells that match the pincers, i.e. between "R" and "g". Eliminations cannot be made between "r" and "G". (The pincer cells serve as a "bridge" between the two color chains. "R" and "g" are indeed the odd-numbered cells along each chain.)

By the way, "strongly inferential" means that "at least one of the two cells must be true" (they can't both be false). That is the case with pincer cells. We know that at least one of them is true, but both of them might be true. It is not a conventional "strong" link (a "conjugate" or "either/or" link).

A strongly inferential link is the logical opposite of a "weak" link, which we have when "at least one of the two cells must be false" (they can't both be true, but can both be false).

In conventional multi-coloring with a weak link "bridge" that is colored "R" and "g", eliminations occur between colors "r" and "G" (the even-numbered cells along each chain). For multi-coloring with a strongly inferential "bridge", the situation is the opposite, as noted above.

[Marty R.]

My pincer coloring has been limited to a single chain.

[Asellus]

The pincers of an XY-Wing are strongly inferential. It is possible for both to be true. We only know that if one is false, the other must be true. But, if one is true, we don't know anything about the other.

The same is true for pincers of W Wings and of XY Chains provided that the pincers of the XY Chain do not share a House. And, strongly inferential links can arise in other ways as well.

The conventional "strong" link is the same as a "conjugate" link or "either/or" link. They are synonymous. And, such strong links can exist between cells that do not "see" each other (i.e. do not share a House). For instance, that is how basic coloring works: a "red" and "green" pair in remote cells eliminate candidates from common "buddies". The remote red-green pair have a strong link.

Weak and strong links can occur between any two cells. Strongly inferential links can only occur between remote cells since two cells in a House can't both be true.

Hope this makes it clearer.

[Marty R.]