The New Sudoku Players' Forum

[rjamil]

Hi SpAce,

I derived "Grouped Empty Rectangle" terminology from this post.

BTW, old is gold.

R. Jamil

[SpAce]

I derived "Grouped Empty Rectangle" terminology from this post.

In that case it seems that you probably misinterpreted something in that discussion.

you can link two type E's similar to that of
grouped empty rectangles.

That only proves that StrmCkr used the term "grouped empty rectangles" with an example that looks like yours. Did he call that whole pattern a Grouped ER, though? That remains somewhat unclear, but I don't think so. Did you look at the context of the discussion? It was about group links in general. I would presume that StrmCkr used the term "grouped" similarly with his ER example, meaning that the ERs contained group links (as opposed to minimal ERs which have simple links). Only he knows what he meant, but that would be the only explanation that makes sense in that context. It appears that you interpreted it differently, and probably incorrectly.

In short, there are two kinds of ERs: grouped (the default) and minimal. A pattern with multiple connected ERs is not a Grouped ER (I don't know what it is, but it's not that -- that I'm sure of). I'd be really surprised if StrmCkr disagreed with that.

[rjamil]

Hi SpAce,

As per my observations, StrmCkr pointed out thrice here, here and here that all those posts and single thread where gone.
In that case, let's wait for an expert/senior opinion. Meanwhile, I will continue using the same terminology for the time being.

Note: maybe, using the Weak Link wordings as per point 5.5 (instead of Strong Link) is acceptable? But, what about "Grouped" wording?

R. Jamil

[StrmCkr]

http://forum.enjoysudoku.com/post51042.html#p51042
http://forum.enjoysudoku.com/post261726.html#p261726

to highlight the technique from past posts

naming is muddled
roughly ---- from the hos-posh of non existent hard copy notes i had as THE thread for multiple ERI's connected was lost and most of those that contributed to it are defacto.

the name list that raj is using and referring was pm'ed to him directly from my self

posted below hopefully with some insight/clarity and amendment.

ERI - is the center cell of a row col interaction within A box.

Dual - references to the number of active links coming off each of the ERI

Linked - references that the ERI used are looped back to each other.

Grouped: refers to a collection of 3+ ERI's ----> each ERI acting as a weak link

Code: Select all

    Empty Rectangle (1 exemplar, 1 exclusion)   Dual Empty Rectangle (1 exemplar, up to 2 exclusions)
      --------------+-------------+-----------  --------------+-------------+-----------
    01) /  +Z    /  |  .   .   .  |  .  .   . 02) /  +Z    /  |  .   .   .  |  .  /   .
       +Z  +Z   +Z  |  .   .   .  |  . -Z   .    +Z  +Z   +Z  |  .   .   .  |  . -Z   .
        /  +Z    /  |  .   .   .  |  .  .   .     /  +Z    /  |  .   .   .  |  .  /   .
      --------------+-------------+-----------  --------------+-------------+-----------
        .   .    .  |  .   .   .  |  .  .   .     .   .    .  |  .   .   .  |  .  /   .
        .   .    .  |  .   .   .  |  .  .   .     .   .    .  |  .   .   .  |  .  /   .
        .   .    .  |  .   .   .  |  .  .   .     .   .    .  |  .   .   .  |  .  /   .
      --------------+-------------+-----------  --------------+-------------+-----------
        .   .    .  |  .   .   .  |  .  .   .     .   .    .  |  .   .   .  |  .  /   .
        /  +Z    /  |  /   /   /  |  / +Z   /     /  -Z    /  |  /   /   /  |  / +Z   /
        .   .    .  |  .   .   .  |  .  .   .     .   .    .  |  .   .   .  |  .  /   .
      --------------+-------------+-----------  --------------+-------------+-----------

Code: Select all

    Dual Linked Empty Rectangle (1 exemplar, up to 15 exclusions)
      --------------+-------------+-----------
    01) /  +Z    /  |  .   .   .  |  / +Z   /
       +Z  -Z   +Z  | -Z  -Z  -Z  | +Z -Z  +Z
        /  +Z    /  |  .   .   .  |  / +Z   /
      --------------+-------------+-----------
        .  -Z    .  |  .   .   .  |  . -Z   .
        .  -Z    .  |  .   .   .  |  . -Z   .
        .  -Z    .  |  .   .   .  |  . -Z   .
      --------------+-------------+-----------
        .  -Z    .  |  .   .   .  |  . -Z   .
        /  +Z    /  |  /   /   /  |  / +Z   /
        .  -Z    .  |  .   .   .  |  . -Z   .
      --------------+-------------+-----------

Code: Select all

    *Grouped Empty Rectangle (1 exemplar, 1 exclusion)
      --------------+-------------+-----------
    01) /  +Z    /  |  .   .   .  |  .  .   .
       +Z  +Z   +Z  |  .   .   .  |  . -Z   .
        /  +Z    /  |  .   .   .  |  .  .   .
      --------------+-------------+-----------
        .   .    .  |  .   .   .  |  .  .   .
        .   .    .  |  .   .   .  |  .  .   .
        .   .    .  |  .   .   .  |  .  .   .
      --------------+-------------+-----------
        /  +Z    /  |  .   .   .  |  / +Z   /
       +Z  +Z   +Z  |  .   .   .  | +Z +Z  +Z
        /  +Z    /  |  .   .   .  |  / +Z   /
      --------------+-------------+-----------

Code: Select all

    *Dual Linked Grouped Empty Rectangle (1 exemplar, up to 16 exclusions)
      --------------+-------------+-----------
    01) /  +Z    /  |  .   .   .  |  / +Z   /
       +Z  -Z   +Z  | -Z  -Z  -Z  | +Z -Z  +Z
        /  +Z    /  |  .   .   .  |  / +Z   /
      --------------+-------------+-----------
        .  -Z    .  |  .   .   .  |  . -Z   .
        .  -Z    .  |  .   .   .  |  . -Z   .
        .  -Z    .  |  .   .   .  |  . -Z   .
      --------------+-------------+-----------
        /  +Z    /  |  .   .   .  |  / +Z   /
       +Z  -Z   +Z  | -Z  -Z  -Z  | +Z -Z  +Z
        /  +Z    /  |  .   .   .  |  / +Z   /
      --------------+-------------+-----------

the last example "daul" can be dropped from the name as its redundant and adds more confusion it did make sense at the time as each of the eri's has 2 links.

and amended as the following.

Code: Select all

    *Grouped Linked Empty Rectangle (1 exemplar, up to 16 exclusions)
      --------------+-------------+-----------
    01) /  +Z    /  |  .   .   .  |  / +Z   /
       +Z  -Z   +Z  | -Z  -Z  -Z  | +Z -Z  +Z
        /  +Z    /  |  .   .   .  |  / +Z   /
      --------------+-------------+-----------
        .  -Z    .  |  .   .   .  |  . -Z   .
        .  -Z    .  |  .   .   .  |  . -Z   .
        .  -Z    .  |  .   .   .  |  . -Z   .
      --------------+-------------+-----------
        /  +Z    /  |  .   .   .  |  / +Z   /
       +Z  -Z   +Z  | -Z  -Z  -Z  | +Z -Z  +Z
        /  +Z    /  |  .   .   .  |  / +Z   /
      --------------+-------------+-----------

[SpAce]

Hi StrmCkr! Thanks for chiming in!

http://forum.enjoysudoku.com/post51042.html#p51042

Thanks for that link! I found this and this especially interesting parts, as they highlight the central piece of confusion I've mentioned about ERs before. It's never clear whether one talks about the box pattern only or includes the elimination producing strong link in the classic pattern. It gets even more confusing when larger ER combinations are being used.

Anyway, I guess I must absolve rjamil, at least partly, because it seems that some of his ER nomenclature did indeed come from you, StrmCkr. That, however, doesn't change my opinion about some of those names

Dual - references to the number of active links coming off each of the ERI

I don't really understand what that means. What do you mean by "active links"? To me the defining feature of a classic Dual Empty Rectangle is that it has strong (bilocation) links on both sides of a single ERI, which means that it's actually a combination of two classic ER patterns having two different eliminations (just like a Siamese fish). Like Siamese fish, it could always be seen as two separate moves, so it's really just a convenience feature. I also don't really see how that scales to larger ER patterns, so unless otherwise convinced, I'd reserve "Dual" for this single special case. In the other examples I've seen it seems mostly redundant and confusing, because they're all looped but the classic Dual ER is not.

Linked - references that the ERI used are looped back to each other.

Thus a better descriptor would be "Looped" or "Loop". "Linked" doesn't imply looping behavior, or much of anything at all. I would find "Doubly Linked" somewhat acceptable, as it's also used for looping ALS XZs, though I don't really like it much there either. Just "Linked" or "Dual Linked" doesn't work for me at all. Simple "Loop" as a postfix seems clearly best to me.

Grouped: refers to a collection of 3+ ERI's

This is terrible. You really shouldn't overload a term that has a well-known meaning in the very same context. How about "Chained", for example? That would be descriptive and not at all confusing. Just like we have chained ALSs we can have chained ERs -- or just ALS Chains and ER Chains. What about simply adding Double, Triple or Quad as a prefix (with or without "Chain" as a postfix)? I think I'd like that best, because it tells exactly how many ERs are present in the ER chain.

----> each ERI acting as a weak link

So, can we agree that rjamil's "Strong Link" part was wrong?

Code: Select all

Empty Rectangle (1 exemplar, 1 exclusion)   Dual Empty Rectangle (1 exemplar, up to 2 exclusions)

These are standard; no problem with them.

Code: Select all

    Dual Linked Empty Rectangle (1 exemplar, up to 15 exclusions)
      --------------+-------------+-----------
    01) /  +Z    /  |  .   .   .  |  / +Z   /
       +Z  -Z   +Z  | -Z  -Z  -Z  | +Z -Z  +Z
        /  +Z    /  |  .   .   .  |  / +Z   /
      --------------+-------------+-----------
        .  -Z    .  |  .   .   .  |  . -Z   .
        .  -Z    .  |  .   .   .  |  . -Z   .
        .  -Z    .  |  .   .   .  |  . -Z   .
      --------------+-------------+-----------
        .  -Z    .  |  .   .   .  |  . -Z   .
        /  +Z    /  |  /   /   /  |  / +Z   /
        .  -Z    .  |  .   .   .  |  . -Z   .
      --------------+-------------+-----------

That's a poor name in many ways. First, the pattern has two Empty Rectangles and not one, so it shouldn't be called any type of singular ER. That applies to all of these multi-ERs. Then you have the keyword "Dual" which means two of something, which one could logically interpret to mean those two ERs in this case -- which is apparently not its purpose at all (though I don't really understand what its meaning is here anyway). Thus I'd drop "Dual" (and its mystery purpose) and replace it with "Double" to actually mean those two ERs. Also, since "Linked" really means looped, it should rather be called "Looped" or just "Loop".

I'd simply call it something like: Double-ER Loop (or just generic ER Loop).

Code: Select all

    *Grouped Empty Rectangle (1 exemplar, 1 exclusion)
      --------------+-------------+-----------
    01) /  +Z    /  |  .   .   .  |  .  .   .
       +Z  +Z   +Z  |  .   .   .  |  . -Z   .
        /  +Z    /  |  .   .   .  |  .  .   .
      --------------+-------------+-----------
        .   .    .  |  .   .   .  |  .  .   .
        .   .    .  |  .   .   .  |  .  .   .
        .   .    .  |  .   .   .  |  .  .   .
      --------------+-------------+-----------
        /  +Z    /  |  .   .   .  |  / +Z   /
       +Z  +Z   +Z  |  .   .   .  | +Z +Z  +Z
        /  +Z    /  |  .   .   .  |  / +Z   /
      --------------+-------------+-----------

This I would call: Triple-ER (or Triple-ER Chain, or just generic ER Chain).

I really hope everyone eventually understands why "Grouped" should be totally out of the question.

Code: Select all

    *Dual Linked Grouped Empty Rectangle (1 exemplar, up to 16 exclusions)
      --------------+-------------+-----------
    01) /  +Z    /  |  .   .   .  |  / +Z   /
       +Z  -Z   +Z  | -Z  -Z  -Z  | +Z -Z  +Z
        /  +Z    /  |  .   .   .  |  / +Z   /
      --------------+-------------+-----------
        .  -Z    .  |  .   .   .  |  . -Z   .
        .  -Z    .  |  .   .   .  |  . -Z   .
        .  -Z    .  |  .   .   .  |  . -Z   .
      --------------+-------------+-----------
        /  +Z    /  |  .   .   .  |  / +Z   /
       +Z  -Z   +Z  | -Z  -Z  -Z  | +Z -Z  +Z
        /  +Z    /  |  .   .   .  |  / +Z   /
      --------------+-------------+-----------

This I would call: Quad-ER Loop (or just generic ER Loop).

They're all pretty simple chain/loop patterns, really, so why complicate them with complex and confusing names?

I would probably call rjamil's solution simply Triple-ER or more generically ER Chain. Short, descriptive, and not easily confused with anything else.

[rjamil]

Hi SpAce,

So, can we agree that rjamil's "Strong Link" part was wrong?

Well, if each ERI is acting as a weak link then maybe two ERIs are acting as strong link?

R. Jamil

[StrmCkr]

That's a poor name in many ways. First, the pattern has two Empty Rectangles and not one, so it shouldn't be called any type of singular ER. That applies to all of these multi-ERs. Then you have the keyword "Dual" which means two of something, which one could logically interpret to mean those two ERs in this case -- which is apparently not its purpose at all (though I don't really understand what its meaning is here anyway). Thus I'd drop "Dual" (and its mystery purpose) and replace it with "Double" to actually mean those two ERs. Also, since "Linked" really means looped, it should rather be called "Looped" or just "Loop".

as defined dual indicates the number of links from the ERI cell
yes there is 2 eris technically 1 of them acts as the 2nd link so they both have "dual" links.

so its clearly different then the Siamese version below.

Code: Select all

 +---------------+-----------+---------------+
| .    (1)  .   | 1  .    1 | .    (1)  .   |
| (1)  (1)  (1) | 1  (1)  1 | (1)  (1)  (1) |
| .    (1)  .   | 1  .    1 | .    (1)  .   |
+---------------+-----------+---------------+
| 1    1    1   | 1  .    1 | 1    1    1   |
| 1    -1   1   | 1  (1)  1 | 1    -1   1   |
| 1    1    1   | 1  .    1 | 1    1    1   |
+---------------+-----------+---------------+
| 1    1    1   | 1  .    1 | 1    1    1   |
| 1    1    1   | 1  .    1 | 1    1    1   |
| 1    1    1   | 1  .    1 | 1    1    1   |
+---------------+-----------+---------------+


the name grouped isn't overloaded as its a collection of ERI cells not the pattern in whole.
it has the exact same purpose of grouped cells in the simplified version row/col.

instead:
ERI's use [ Row * col ] * box - > space
so yes you can "group together eri cells" and perform eliminations as a cover set problem. but you tend to see everything as "chains"

anyways, like i said before old thread that's gone besides it had little usages and muddled.
names for these can change i'm okay with that.

ill let you guys hash that out as long as ERI definition remains untouched.

the only useful part was the ERI definition as it gave a way to node link ERI in chains easily with other techniques

[StrmCkr]

Hi SpAce,

SpAce wrote:
So, can we agree that rjamil's "Strong Link" part was wrong?


Well, if each ERI is acting as a weak link then maybe two ERIs are acting as strong link?

R. Jamil

in chain form a weak link in 2 directions changes to a "Strong link", which is what happens to the middle eri, when looking at the application of both side boxes in chain form.
so technically yes its strong link in a sense, but as its own identity its a weak link to a & b

[SpAce]

Regarding the strong and weak links, I have no idea what either of you are talking about. So, let's recap what those terms actually mean.

Strong Link: logical OR between two entities; i.e. at least one is true
Weak Link: logical NAND between two entities; i.e. at most one is true

Based on those simple definitions, we could safely say that our three ERIs are not weakly linked because they're all true (they clearly exist), and we know it just by looking at the grid. In theory, that fact makes them actually strongly linked -- because at least one of them is true (since they're all true). However, that makes the whole concept of strong and weak links totally meaningless in this context, because we're talking about static patterns that happen to be true in our grid.

We use strong and weak links to prove something useful when we have only partial knowledge, like when we know that one or the other of some group must be true (or false) but we don't know which one. Here we already know that all of our ERIs are true so there's nothing to prove about them; thus they have no useful links between each other. They're connected, yes, but not strongly or weakly linked. Only the candidates within them have useful links. Let's look at those:

Code: Select all

 +------------+-------------------+-----------------+
 | 3   1    6 | 2(4)8  [4]8  9    | (5)   2[4]  7   |
 | 5   24   8 | 2[4]   3     7    | 1     9     6   |
 | 24  9    7 | 6      5     1    | 2[4]  3     8   |
 +------------+-------------------+-----------------+
 | 89  5    1 | (3)    7     [4]8 | 2-4   6     249 |
 | 6   7    4 | 9      2    5     | 8     1     3   |
 | 89  3    2 | [4]8   1    6     | 7     5     49  |
 +------------+-------------------+-----------------+
 | 1   6    5 | 7      9    248   | 3     248   24  |
 | 7   48   3 | 5      6    248   | 9     248   1   |
 | 24  248  9 | 1      48   3     | 6     7     5   |
 +------------+-------------------+-----------------+

We could write rjamil's logic as two slightly different chains (because the grouped ERI in box 2 can be used in two ways):

(4): b3[r3c7=r1c8] - b2[r1c45=r2c4] - b5[r6c4=r4c6] => -4 r4c7
(4): b3[r3c7=r1c8] - b2[r1c5=r12c4] - b5[r6c4=r4c6] => -4 r4c7

From those chains we can clearly see what the strong and weak links are. The native strong links are inside the boxes (between two halves of an ERI), and the native weak links are between halves of different ERIs (and between the end points and the target, which is not explicitly depicted).

Native Strong Links:

(4)r3c7 = (4)r1c8
(4)r1c45 = (4)r2c4
(4)r1c5 = (4)r12c4
(4)r6c4 = (4)r4c6

Derived Strong Links:

(4)r3c7 == (4)r2c4
(4)r3c7 == (4)r12c4
(4)r1c45 == (4)r4c6
(4)r1c5 == (4)r4c6
(4)r3c7 == (4)r4c6 * (this is what proves our elimination; see the * weak links below)

Native Weak Links:

(4)r1c8 - (4)r1c45
(4)r1c8 - (4)r1c5
(4)r2c4 - (4)r6c4
(4)r12c4 - (4)r6c4

External Native Weak Links:

(4)r3c7 - (4)r4c7 *
(4)r4c6 - (4)r4c7 *

(both of the strongly linked end points have a weak link to the target, so it can be eliminated)

Derived Weak Links:

(4)r1c8 -- (4)r6c4
(4)r1c8 -- (4)r4c7
(4)r1c45 -- (4)r4c7
(4)r1c5 -- (4)r4c7
(4)r2c4 -- (4)r4c7
(4)r12c4 -- (4)r4c7
(4)r6c4 -- (4)r4c7
(4)r4c7 -- (4)r4c7 ** (self-contradiction; also proves the elimination)

So, one last time: the full ERIs themselves aren't linked with each other or anything else at all -- they're simply true, because they happen to exist in our grid. However, the two halves of each ERI are internally strongly-linked and externally weakly-linked (in this case to halves of other ERIs). That makes the pattern work, as it creates a derived strong link between the two end nodes (r3c7 and r4c6, which are halves of b3 and b5 ERIs) guaranteeing that at least one of them is true -- and since both have a weak link to the target (r4c7) it must be false and can be eliminated.

In an ER Chain or Loop all linking action happens between ERI halves (strongly internally, weakly externally). It's meaningless to talk about strong links or weak links between ERIs themselves. Can we now drop this?

[StrmCkr]

I have no idea what either of you are talking about

you should for some one that looks at chains a lot.

Strong Link: logical OR between two entities; i.e. at least one is true
Weak Link: logical NAND between two entities; i.e. at most one is true

Code: Select all

+---------------+---------+---------------+
| .    (1)  .   | .  .  . | .    (1)  .   |
| (1)  (1)  (1) | .  .  . | (1)  .    (1) |
| .    (1)  .   | .  .  . | .    (1)  .   |
+---------------+---------+---------------+
| .    .    .   | .  .  . | .    .    .   |
| .    .    .   | .  .  . | .    .    .   |
| .    .    .   | .  .  . | .    .    .   |
+---------------+---------+---------------+
| .    (1)  .   | .  .  . | .    .    .   |
| (1)  .    (1) | .  .  . | .    -1   .   |
| .    (1)  .   | .  .  . | .    .    .   |
+---------------+---------+---------------+

States of all eri :
Row OR Col is true & ERI cell is false
Or
Row & Col is true via ERI is true}

now, that gives us the following relation ships. between 3 ERI's

box 3 {row or COl ) is true -> IF Row is true [ box 1 Col is true ] -> [box 3 Row is true]
-> IF Col is true [ box 2 and 3 are all states above]

box 7 {row or COl ) is true -> IF Col is true [ box 1 Row is true ] -> [box 3 Col is true]
-> IF Row is true [ box 1 and 3 are all states above]


Box 1 {row or COl or ERI {True/false} ) -> IF Row is true [ box 3 Col is true ] -> [box 7 is in all states above]
-> IF Col is true [ box 7 Row is true ] -> [box 3 is in all states above]
-> IF ERI is true [ box 3 Col is true ] -> [box 7 Row is true ]
-> IF ERI is false [ box 3,7 is in all states above] & [ box 1 row or Col is true ]

this shows that box 1 has weak links to both box 3 & 7 and a strong link to box 3,7 via ERI cell

box 1 has a strong link in the sense that either: Row or Col turns on the exact opposite in either of the other eri boxes.

[SpAce]

I have no idea what either of you are talking about

you should for some one that looks at chains a lot.

You're right, I should. I guess there's a couple of strongly linked reasons if I don't, but who knows which one of them is true. Btw, relating to that you said earlier:

so yes you can "group together eri cells" and perform eliminations as a cover set problem. but you tend to see everything as "chains"

Are you subtly implying that I'm so narrow-mindedly chain-oriented that I can't see or understand other possibilities? Well, obviously I feel most at home with chains, but I'm slowly getting more comfortable with the base\cover POV as well. Didn't you see that I produced a fish conversion of rjamil's solution (and all the other presented X-Chains for this puzzle) to exercise just that? These days I can pretty well understand and even produce base\cover solutions (not just single-digits either), but I don't think they're very human-friendly compared to chains. So, I'm not really sure what your point is.

this shows that box 1 has weak links to both box 3 & 7 and a strong link to box 3,7 via ERI cell

box 1 has a strong link in the sense that either: Row or Col turns on the exact opposite in either of the other eri boxes.

What you call "Row" and "Col" I called "halves". Thus, I don't really see what in your demonstration was supposed to disprove my point -- which you apparently missed because you kept using inaccurate terminology: boxes certainly don't have strong or weak links with anything, and neither do ERIs (as full patterns). ERI halves, ERI Rows, ERI Cols, and ERI Cells can have strong and weak links (because they can be either true or false), but not full ERIs (because they're simply true if they exist in the grid) or much less boxes (which are always true, of course). I guess that's self-evident only to those of us who "see everything as chains".

Now, it's very much possible that I didn't quite grasp your point either, but part of it is because you communicate inaccurately, which makes it quite frustrating to try to follow your thinking. For example, when you say "ERI" it's not clear whether you mean the full pattern (I do) or just the ERI Cell (which you sometimes, but not always, mention specifically). That's part of the larger problem with the whole ER/ERI terminology, which makes me dislike it. Everything in it is very ambiguous, and I don't like ambiguity. When ER patterns are written as chains or fishes there's no ambiguity or unnecessary complexity at all, which is why I prefer them.

Edit: Btw, I've now reread your last post several times and I still have no idea what you were trying to say.

[rjamil]

Hi,

I have limited knowledge about fishes and no knowledge about chains.

However, let compare ER patterns with fish:

Code: Select all

1) X-Wing
---------+----------+----------
 . -Z  . |  .  .  . |  . -Z  .
 / +Z  / |  /  /  / |  / +Z  /
 . -Z  . |  .  .  . |  . -Z  .
---------+----------+----------
 . -Z  . |  .  .  . |  . -Z  .
 . -Z  . |  .  .  . |  . -Z  .
 . -Z  . |  .  .  . |  . -Z  .
---------+----------+----------
 . -Z  . |  .  .  . |  . -Z  .
 / +Z  / |  /  /  / |  / +Z  /
 . -Z  . |  .  .  . |  . -Z  .
---------+----------+----------
2) Dual Linked Empty Rectangle
---------+----------+----------
 / +z  / |  .  .  . |  / +z  /
+z -Z +z | -Z -Z -Z | +z -Z +z
 / +z  / |  .  .  . |  / +z  /
---------+----------+----------
 . -Z  . |  .  .  . |  . -Z  .
 . -Z  . |  .  .  . |  . -Z  .
 . -Z  . |  .  .  . |  . -Z  .
---------+----------+----------
 . -Z  . |  .  .  . |  . -Z  .
 / +Z  / |  /  /  / |  / +Z  /
 . -Z  . |  .  .  . |  . -Z  .
---------+----------+----------
3) Dual Linked Grouped Empty Rectangle
---------+----------+----------
 / +z  / |  .  .  . |  / +z  /
+z -Z +z | -Z -Z -Z | +z -Z +z
 / +z  / |  .  .  . |  / +z  /
---------+----------+----------
 . -Z  . |  .  .  . |  . -Z  .
 . -Z  . |  .  .  . |  . -Z  .
 . -Z  . |  .  .  . |  . -Z  .
---------+----------+----------
 / +z  / |  .  .  . |  / +z  /
+z -Z +z | -Z -Z -Z | +z -Z +z
 / +z  / |  .  .  . |  / +z  /
---------+----------+----------

1st one is an X-Wing that contains two strong links (bilocation cells) within two rows; and whose both end points are also aligned within columns.

2nd one is Dual Linked Empty Rectangle that contains one strong link of bilocation cells within one row and another strong link end points are replaced with ERI cells.

3rd one is Dual Linked Grouped Empty Rectangle that contains both strong links end points replaced by ERI cells.

Code: Select all

4) Empty Rectangle
---------+----------+----------
 / +z  / |  .  .  . |  .  .  .
+z  . +z |  .  .  . |  . -Z  .
 / +z  / |  .  .  . |  .  .  .
---------+----------+----------
 .  .  . |  .  .  . |  .  .  .
 .  .  . |  .  .  . |  .  .  .
 .  .  . |  .  .  . |  .  .  .
---------+----------+----------
 .  .  . |  .  .  . |  .  .  .
 / +Z  / |  /  /  / |  / +Z  /
 .  .  . |  .  .  . |  .  .  .
---------+----------+----------
5) Grouped Empty Rectangle
---------+----------+----------
 / +z  / |  .  .  . |  .  .  .
+z  . +z |  .  .  . |  . -Z  .
 / +z  / |  .  .  . |  .  .  .
---------+----------+----------
 .  .  . |  .  .  . |  .  .  .
 .  .  . |  .  .  . |  .  .  .
 .  .  . |  .  .  . |  .  .  .
---------+----------+----------
 / +z  / |  .  .  . |  / +z  /
+z  . +z |  .  .  . | +z  . +z
 / +z  / |  .  .  . |  / +z  /
---------+----------+----------

4th one is an ER that is strongly linked with bilocation cells.

and, 5th one is an ER that is linked with two ERI cells.

It is now clearly shows that when one bilocation cells end point are replaced with ERI cells then it act like a strong link (or sometimes more than that).

Edit: ER/ERI ambiguity (negligible).

R. Jamil

[SpAce]

And that is my cue. It's now clear that nothing I said made a dent, and I'm done wasting my time. Have fun with your lovely Strongly Linked Siamese Dual Linked Grouped Empty Rectangles

[StrmCkr]

No I don't imply anything

I just figured since your chain based you'd see that box row implies box col as the weak link interface
where an eri directly implies a box row and box col in the others.

Eri is the intersection of row * col * box
Which can be true or false. For the cell, col, or row

Its a refrence to the row*box*col denoting and marking the center cell of an Er formation regardless. If it exsits or dosent.
ERI represents the mini row. And min col of the box.

The center cell is crtical as it's exisitance means this. Cell. Can be be either true or. False, where fake. Still means row/col is true or falee

Er is a name of a technique , where eri is a link type.

Just like a. Grouped link represents a box *col or box * row ie 3 cells.

I get your point you see box row on turns off other box row which implies box col as on

To me the boxs can imply something as they are the third dimension interaction and are a summation of three spaces.

Instead of 2 d grouped nodes like mini rows (3 cells)
eris use 3d space
so row. On turns on opposite field in line of sight boxes. So if I'm clear enough box -. Box has row/col implications as a directional. Change . Thus a strong and weak link composition and where eris are active both are plausible.

FOR CLAIRTY
ERI : MEANS ROW/COL. OR CENTER CELL IS TRUE for an ER formation in a box

Since its a link type I'm only using link relation ships from the 3 cells, and not all 15 cells from the diagram.

If the eri cell. Is missing In box one. Then picture the for formation as an xy wing with row col as the interaction instead of abc

If that makes any sense to you.

[SpAce]

StrmCkr, one thing I don't understand in your explanations is why you keep talking about the ERI Cell as if it had any significance in ER patterns whether it's occupied or not. The grouped strong link between the ERI Row and ERI Col in that box is valid either way, and it's all that matters for any inferences: the strong link guarantees that at least one of the ERI Row or ERI Col is true (or both) and that's enough.

If the ERI Cell is occupied and true, then yes, both the ERI Row and ERI Col are true and fire along their weak links, but it makes no difference in any examples I've seen. The case of both being true is covered by the strong link (OR) and, as far as I see, we don't need to consider it separately. However, if the ERI Cell is occupied, then there's no weak link between the ERI Row and ERI Col, so the link is strong-only, but that doesn't matter for ER patterns -- they only use the strong link anyway.

If the occupancy of the intersection cell does matter, then we have something else, such as a (Grouped) 2-String Kite. Then the box houses a (grouped) weak link instead of a strong link, and the weak link only exists if the intersection cell is not occupied. The (grouped) strong links are then on the row and column, so it's the exact opposite of the ER situation.

PS. If you're talking about something much more complex, like Allan Barker's triplets and varying rank regions, then you should probably show an example where such concepts actually matter. I don't think they do in normal ER situations.

(Added) PPS. Incidentally, if you look at my solutions for today's puzzle, you'll see that I used a grouped strong link on 9s (box 9) in the first solution (similarly to an ER situation) and also an almost-weak-link on 4s (box 1) in the third solution (Almost Kite). Only in the latter case it matters that the intersection cell is occupied, because it denies the weak link and thus obstructs the Kite pattern. That's why I needed to prove the same elimination with the intersection candidate being both false and true. That is not needed for the strong-link case.