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keith · Last edited by keith on Mon Oct 27, 2008 9:16 pm; edited 1 time in total

When Marty asked the question about a definition of grouped coloring I looked, and was surprised to find very little. Here is my effort to write something down. This is a first draft. As always, comments and further contributions are welcome. Keith

If a candidate occurs only twice in a row, column, or box, the two cells in which it occurs are said to be a "strong link" in that candidate. Strong links are a fundamental building block of many Sudoku techniques. An excellent guide has been written by Havard: Strong Links for Beginners

keith · Posted: Sun Oct 26, 2008 8:10 pm Post subject: Some Examples

This post is out of chronological sequence: It is reserved to be edited, to provide examples at later dates.

March 1, 2009:
http://www.dailysudoku.com/sudoku/forums/viewtopic.php?t=3303
I think this is a great example.

Keith

Marty R. · Posted: Sun Oct 26, 2008 9:36 pm Post subject:

Thanks Keith. I appreciate the time it must've taken to put this together.

Asellus · Posted: Mon Oct 27, 2008 7:51 am Post subject:

Marty,

Diagram 4 does not require multi-coloring for the # elimination; it is accomplished by "basic" (single cluster) coloring, which is what I believe Keith meant by simple coloring. All of the links connecting the "pincers" are conjugate. Diagram 5 requires ("non-simple") multi-coloring due to the weak (only) link in Box 1.

Asellus · Posted: Mon Oct 27, 2008 9:09 am Post subject:

Regarding the ER, I will attempt to provide additional explanation according to my understanding of such things:

Yes, the ER, too, uses groups. To understand how requires a bit of preliminary clarification of definitions. The strong links discussed by Keith are conjugate links, meaning that one side is true and the other false. It is true that conjugate links are strong links. However, conjugate links are also weak links. Conjugate links satisfy the logical requirements of both strong and weak links (which is why they are so easy to use). [See below if you need more explanation of strong and weak inferences.]

There also exist strong links that are only strong links. The ER shown above in Diagram 6 is an example of one. But first, let's look at the ER structure in Box 1 of Diagram 5. It can be expressed as a grouped conjugate link between r1c23 on the one side and r23c1 on the other. As such, it is both a grouped strong link and a grouped weak link. (When it functions as an ER, it is functioning as a grouped strong link. For the elimination at @, it is functioning as a grouped weak link, not as an ER.)

[To be precise, the grouped link is between the grouped candidate digit in cells r1c23 and the grouped candidate digit in cells r23c1. Saying that the link is between the grouped cells is a shorthand for this.]

Since Keith was considering only weak links and conjugate strong links, his statement that the groups on either side of a link cannot share a candidate is true. To understand the grouped link of the ER in Diagram 6, we must consider the case where this restriction does not apply. Here, the grouped link is between r1c123 and r123c1. Notice that both sides of the link contain the candidate in r1c1. Thus, if r1c1 is true, then both sides of the link are true. Still, since this link includes all instances of the candidate in Box 1, it is impossible for both sides to be false. This is a "strong only" link, sometimes called a "strong inference" link. But, really "strong link" is a perfectly fine description provided it is understood that strong link is not a synonym for conjugate link.
__________

True and False as applied to groups:

It might help to explain what it means for a group to be true or false. A group is true if at least one member of the group is true. A group is false if all members of the group are false.
__________

Strong and Weak Inferences:

For those not familiar with strong and weak inference concepts, this might be a good place to provide a quickie explanation. A strong inference exists between two things if when one of those things is assumed to be false the other must be true. The minimum requirement for this occurs when it is impossible for both things to be false. A conjugate link, such as exists between two candidates in a House which are the only two instances of that candidate in that House, meets this requirement: if they were both false, then the House would lack that candidate, which is not possible. A grouped link that encompases all of the instances of a candidate in a House does the same. Note that a link in which both sides can be true also satisfies this requirement as long as it is impossible for both sides to be false.

A weak inference exists between two things if when one of those things is assumed to be true the other must be false. The minimum requirement for this occurs when it is impossible for both things to be true. A conjugate link satisfies this requirement and is thus also a weak link. Any two candidates within a House that contains more than two of those candidates is also a weak link; both sides might be false, but both sides cannot be true. And, a grouped link within a House that does not share a candidate on both sides of the link is also a weak link, whether or not it includes all instances of the candidate digit within that House.

The examples given for strong and weak links do not exhaust the possibilities. But, more some other time!

Marty R. · Posted: Mon Oct 27, 2008 4:27 pm Post subject:

daj95376 · Joined: 23 Aug 2008 Posts: 3854

I deliberately avoided using the full Empty Rectangle pattern in another post on grouped strong links because I knew that it would be difficult to explain. However, since Pandora's Box (pun intended) has been opened here, I would like to add my opinion.

An ER pattern contains two usable grouped strong links.

Asellus · Posted: Mon Oct 27, 2008 8:49 pm Post subject:

keith · Posted: Mon Oct 27, 2008 9:31 pm Post subject:

Marty, you may already have your answer, but here is my spin: