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[Marty R.]

This is an illustration of an X-Chain from Sudopedia. At first glance it looks like a simple coloring chain until one notices the third 3 in column 5. Two questions:

1) How can the chain proceed with the third 3 in column 5?

2) People have tried explaining this before, but it hasn't penetrated my thick skull: what makes the link in column 8 a weak one but the links in boxes 5, 6 and 9 strong ones?

[Asellus]

Let me attempt the answer:

In inference logic, a conventional strong "conjugate" or "either-or" link can be thought of as a "wild card" link since it fulfills both strong and weak inference logic.

So... yes, the link in C8 is "strong" in the conjugate sense, but it can serve as a weak link for inference purposes.

In Alternate Inference Chaining (AIC, which is what this is), one can also exploit what can be called a "strong inference link" or "inferential strong link," which is different from the conventional strong conjugate link. The readers of this forum are already familiar with such links. For example, the pincers digits of an XY-Wing/Chain or of a W-Wing, etc., generally have a strong inferential link rather than a conjugate link because it is possible that both are true, but impossible that both are false.

Once you let a weak link or a strong inferential link into your chain, you must keep strict track of the alternating link types: strong-weak-strong-weak-strong, etc. Weak links are only allowed at the weak positions. Strong inferential links are only allowed at the strong positions. Strong conjugate links are allowed anywhere. The two pincer ends of the chain must be connected to the chain by strong link positions. (Eureka notation is very helpful for keeping track of this. And, there is no particular reason to limit such chains to a single digit. But, those are whole other discussions.)

By the way, if your chain meets up with itself and forms a complete alterating implication loop, then all of the links become conjugate links and you can eliminate all candidates that can see both pairs of each successive "node". (An XY Loop is an example of this. But it works the same with any AIC Loop.)


For those so interested...

About that inference logic:

Weak Inference exists whenever we can say: "If candidate A is true, candidate B must be false." This exists whenever it is impossible for both candidates to be true. Both weak links and conjugate (strong) links meet this requirement.

Strong Inference exists whenever we can say: "If candidate A is false, candidate B must be true." This exists whenever it is impossible for both candidates to be false. Both conjugate (strong) links and inferential strong links meet this requirement.

I hope this helps to clear up this perennially confusing issue.

[nataraj]

Marty, I assume by "third 3 in col 5" you refer to third "3" in row 8 which happens to be in col 5. I'll try and answer your 2 questions.

Q: "How can the cain proceed"
A: The complete loop would include two more (weak) links:
one in col 4 one in row 8. That r8c4 has TWO weak links is what's called the discontinuity (because all the rest of the loop has alternating weak and strong links). Any cell that has two weak links in an otherwise strictly alternating loop cannot contain the candidate (that's why 3 is removed from r8c4)

Q:"what makes the link in col 8 a weak one ..."
A:Because we WANT it to be weak.
Any strong link can stand in for a weak link (not vice versa).
To view this link as "weak" is only a matter of choice but it allows us to see the whole loop as alternating between strong and weak links (except for the discontinuity).

The diagram shows that in order to use the logic of AIC (alternating inference chains) or "nice loops", you can take consecutive sequences of 1,3,5,... strong links (must be an even number. with 3 strong links just view the middle one as weak)

Hope this helps.

[Marty R.]

Thank you both for again trying to explain this.

[keith]

Marty,

Let me try.

Strong link =:

Two candidates in a house. One must be true.


Weak link -:

Three or more candidates in a house. One must be false.


(House = row, column, or box.)


Here is a skyscraper:

x=x-x=x

[Asellus]

[Marty R.]

Can I be obnoxious and ask one more time: despite various explanations, I am no closer to understanding the part of the chain in row 5 with three occurrences of 3.

Please, tell me the example is in error or explain it as if talking to your eight-year-old granddaughter.

Thank you for your patience.

[re'born]

[Marty R.]

Thanks re'born, I finally see it, but not because I can understand the notation. What I'm seeing is what I call an ordinary XY-Chain. If r4c4 =3, then the 3 in r8c4 is zapped. If r4c4 is not = 3, then the chain proceeds to r8c7 for the pincer effect. I've been staring at the chain thinking that it was supposed to work with either value in r4c4.

Thanks again.