The New Sudoku Players' Forum

[rjamil]

Hi tarek,

Whereas the ER would give something like this:

Code: Select all

+++++++++
| . X . |
| X . . |
| . . . |
+++++++++

I think the above Empty Rectangle pattern is called a Reduced Empty Rectangle, where ERI mini-row neighbor and ERI mini-col neighbor have only one cell that contain required value. If either mini-row or mini-col or both have two cells that contain required value then it is called an Empty-Rectangle.

Synonyms:
Reduced Empty Rectangle = Turbot Crane
Empty Rectangle = Grouped Turbot Crane

Code: Select all

                               X-Chains                   
                                   |                     
                             Turbot Fishes               
                                   |                     
           -----------------------------------------------   
          |                        |                      |
     [Skyscraper]           [2-String Kite]    [Reduced Empty Rectangle]

R. Jamil

[tarek]

Whereas the ER would give something like this:

Code: Select all

+++++++++
| . X . |
| X . . |
| . . . |
+++++++++

I think the above Empty Rectangle pattern is called a Reduced Empty Rectangle, where ERI mini-row neighbor and ERI mini-col neighbor have only one cell that contain required value. If either mini-row or mini-col or both have two cells that contain required value then it is called an Empty-Rectangle.

Thanks,

This reduced ER term IMO adds a bit of confusion.
These patterns rely on 2 strong links in location using a single value.
If the strong links are in parallel lines then it is a skyscraper
If the strong links are non-parallel lines the it is a 2-String kite
If one of the strong links is in a box then it is called Turbot Crane (2 boxes in theory can exist but simpler box-line reduction will have the same eliminations)

The empty rectangle would then describe the grouped strong link in a box … So you start from the Turbot and get to Empty rectangle & not start from the Empty rectangle to get to the Reduced Empty rectangle IMO.

My diagrams were just a visual interpretations of 2 different varieties of grouped strong links in a box. One of these varieties you would normally describe as Empty rectangle. The other I have no name for yet!

I am of the view that we need to move away from the Empty rectangle term because to me & some other players it is a type of grouped strong link in a box (I demonstrated a 3 strong link pattern composed only of Empty rectangles). Some other players use the term Empty rectangle to signify a 2 strong link pattern (or a grouped Turbot crane)

tarek

[SpAce]

Hi tarek and rjamil,

Regrading L1 & grouped strong link in box,

I hope that you agree with me this:

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+++++++++
| X X X |
| X X X |
| . . . |
+++++++++

Would constitute a grouped strong link in box different from that of an Empty rectangle.

Good point! I'm glad you brought this up because boxes can indeed host two kinds of grouped links! In this first example the strong link is between two mini-rows (could be two mini-cols) within the box. Since they're group nodes, it's a grouped link. If the same is a simple link (between just two candidates) then it can be seen as either a line-based (in this case column) or a box-based bilocation strong link, but the grouped link is obviously box-based only. (Btw, Franken Fishes have these kinds of box formations.)

An example:

Code: Select all

-1 -1 -1 | .  12 . | 2  2  2
 .  .  . | .  .  . | /  /  /
 .  12 . |-1 -1 -1 | 2  2  2
 --------+---------+--------


Grouped W-Wing: (1=2)r3c2 - r3c789 = r1c789 - (2=1)r1c5 => -1 r1c123,r3c456

On the other hand, in an ER the strong link is between a mini-row and a mini-col, so it's clearly different (like a hinge). It's still a grouped link because it's between group nodes as well (unless it has just two candidates). Thus I would call such a pattern "grouped" too. However, I have nothing against it if someone wants to distinguish between the two different types of box links otherwise, but "grouped" is equally applicable to both kinds anyway. "Grouped" simply means that a link is between more than two candidates, and I would attach that qualifier to any pattern that uses at least one such link. (Btw, Mutant Fishes have these hinge types of box formations.)

An example:

Code: Select all

 .  .  . | .  .  . | /  2    /
 .  12 . | .  .  . | 2  2-1  2
 .  .  . | .  .  . | /  2    /
 --------+---------+----------
 .  .  . | .  .  . | .  .    .
 . -1  . | .  .  . | .  12   .
 .  .  . | .  .  . | .  .    .

Grouped W-Wing: (1=2)r2c2 - r2c789 = r13c8 - (2=1)r5c8 => -1 r2c8,r5c2

If it's important to distinguish the two kinds of group links, I guess one could add another qualifier like "hinge" to the latter case, as in "Grouped (hinge) W-Wing". Personally I think the "Grouped" qualifier is more important and enough.

Whereas the ER would give something like this:

Code: Select all

+++++++++
| . X . |
| X . . |
| . . . |
+++++++++

Yes. Unlike the other simplified link, the "reduced" ER link is always box-only. I think StrmCkr's term for it is actually Minimal ER (unless it's changed lately). That link (or a pattern using it) would not be "grouped" because it only has two candidates. It's also a bit different from the grouped ER, because it can function in several different ways. The weak links of a grouped ER are fully defined as a single row and a single column, so it's easy to see as a well-defined strong hinge, but a minimal ER is more ambiguous as either candidate can link to both rows and columns. Its hinge gets fully defined only when used within a larger pattern. In fact, it doesn't have to be a hinge at all, as it can function like the first pattern too!

Reduced Empty Rectangle = Turbot Crane
Empty Rectangle = Grouped Turbot Crane

I agree with that, especially if Reduced -> Minimal (I think). Naturally I would strongly prefer the right-hand terms.

I am of the view that we need to move away from the Empty rectangle term because to me & some other players it is a type of grouped strong link in a box (I demonstrated a 3 strong link pattern composed only of Empty rectangles). Some other players use the term Empty rectangle to signify a 2 strong link pattern (or a grouped Turbot crane)

I wouldn't mind that. Like I've said before, Empty Rectangle is a horribly ambiguous term. However, it's also used very much so it might be difficult to change. But, it would be a good start if we could at least rename the simplest ER patterns as Turbot Crane and Grouped Turbot Crane. That would make me very happy!

Personally I would prefer "hinge" as a generic term to describe any orthogonal box-link (strong or weak). So, how about "strong hinge" and "weak hinge" to describe those box links? I think that would be very unambiguous!

The Turbot Fish family has examples of both: Turbot Crane has a strong hinge and 2-String Kite has a weak hinge, and that's true whether they're grouped or not. Because it's an invariant of those patterns, we can safely use the "Grouped" prefix with them because it tells exactly where the group nodes must be. A Grouped Crane has a "grouped strong hinge" while a Grouped Kite has a "grouped weak hinge" (and one or two grouped line-based strong links). A Grouped Skyscraper obviously has a single line-based grouped strong link. Thus there's zero ambiguity when using the "Grouped" prefix with Turbot Fishes. It just makes it easier to identify the pattern in the grid, knowing it's a bit more complex than the simple case.

[tarek]

I am mindful that this grouped weak link in a box may harbour an endofin at the intersection of the mini lines (I call it the heart cell to differentiate from the 4 rectangle cells which are empty and the 4 other cross cells). This endofin has to see eliminations for them to happen. At the moment I’m dropping any pattern that has an endofin from Sukaku explainer but it is a matter to be revisited later.

tarek

[StrmCkr]

think StrmCkr's term for it is actually Minimal ER (unless it's changed lately)

Yup, still refer it to as a minimal er.

[rjamil]

Hi tarek and SpAce,

I would like to add my last wordings for Empty Rectangle. From what I learned, Empty Rectangle acts upon three different ways/configurations within box as follows:

Code: Select all

1   ER        2 Reduced ER  3 Minimal ER
+---------+   +---------+   +---------+
| /  X  / |   | /  X  / |   | /  X  / |
| X  .  X |   | X  .  X |   | X  .  / |
| /  X  / |   | /  /  / |   | /  /  / |
+---------+   +---------+   +---------+
Where:
X means value must be included;
/ means value must not be included;
. means Empty Rectangle Intersection cell, value may or may not be included.

Please consider to rename above mentioned three different patterns of Empty Rectangle with whatever majority players' like/consent.

R. Jamil

[tarek]

Code: Select all

1   ER        2 Reduced ER  3 Minimal ER
+---------+   +---------+   +---------+
| /  X  / |   | /  X  / |   | /  X  / |
| X  .  X |   | X  .  X |   | X  .  / |
| /  X  / |   | /  /  / |   | /  /  / |
+---------+   +---------+   +---------+

To me the 1st 2 are the same and would call them Empty Rectangle, grouped Turbot or Grouped strong link. The last one is just a Turbot or strong link

[rjamil]

To me the 1st 2 are the same and would call them Empty Rectangle, grouped Turbot or Grouped strong link. The last one is just a Turbot or strong link

Well, is there any significant difference between 2nd and 3rd patterns in terms of number of exclusions? Please let me know in exemplars form.

R. Jamil

[tarek]

To me the 1st 2 are the same and would call them Empty Rectangle, grouped Turbot or Grouped strong link. The last one is just a Turbot or strong link

Well, is there any significant difference between 2nd and 3rd patterns in terms of number of exclusions? Please let me know in exemplars form.

I'm not differentiating them based on effect or eliminations, I'm differentiating them based on simplicity to spot. You descriptions of the ER are systemic and correct but I prefer to start from your minimal description and call it something else then describe the other patterns (reduced and full) compared to it

[SpAce]

I am mindful that this grouped weak link in a box may harbour an endofin at the intersection of the mini lines (I call it the heart cell to differentiate from the 4 rectangle cells which are empty and the 4 other cross cells). This endofin has to see eliminations for them to happen. At the moment I’m dropping any pattern that has an endofin from Sukaku explainer but it is a matter to be revisited later.

You're right that it should be considered an "endofin" if such patterns are used, but I don't think such complicated patterns have relevance in this context. In most cases patterns with endofins (i.e. base triplets in Allan Barker's terminology) can and should be avoided by using a work-around anyway.

A weak hinge simply can't exist if there's a candidate in the intersection, so I don't know why you'd even worry about it. That's the big difference between a strong hinge and a weak hinge, since the strong hinge can work with the intersection candidate (it just gets covered twice by the line links) but the weak hinge can't. In other words, such a strong hinge is actually a strong-only link. If there's no candidate in the intersection, then it can act as either a strong or a weak inference (but obviously only one or the other is used at a time).

Code: Select all

Strong-Weak Hinge:

+-------+
| / X X |  Grouped strong hinge : r1c23 = r23c1 (usage: r1cX - r1c23 = r23c1 - rYc1)
| X / / |  Grouped weak hinge   : r1c23 - r23c1 (usage: r1cX = r1c23 - r23c1 = rYc1)
| X / / |
+-------+

Strong-only Hinge:

+-------+
| X X X |  Grouped strong hinge : r1c123 = r23c1 (or r1c23 = r123c1)
| X / / |  Grouped weak hinge   : N/A; (unless ORed with the endofin: ((r1c23 - r23c1) | r1c1)
| X / / |
+-------+

Weak-only Hinge:

+-------+
| / X X |  Grouped strong hinge : N/A; (unless ORed with the exofins: ((r1c23 = r23c1) | b1p5678)
| X X X |  Grouped weak hinge   : r1c23 - r23c1
| X X X |
+-------+

---

Dual Strong-Weak Hinge:

+-------+
| / / X |  Strong hinge : r1c3 = r3c1 (usage: r1cX - r1c3 = r3c1 - rYc1, OR: r3cX - r3c1 = r1c3 - rYc3)
| / / / |  Weak hinge   : r1c3 - r3c1 (usage: r1cX = r1c3 - r3c1 = rYc1, OR: r3cX = r3c1 - r1c3 = rYc3)
| X / / |
+-------+


[tarek]

A weak hinge simply can't exist if there's a candidate in the intersection

Ah … good point

I'm almost there with the L1 Wing (3 strong links in location using 1 digit) and here is a beauty:

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+-------------+-------------+-------------+
| 9  *13  25  | 134 8   134 | 25  6    7  |
| 8   4  %35  | 2   6   7   |%359 %359 1  |
| 7  *123 6   | 13  5   9   | 8   4  ^23  |
+-------------+-------------+-------------+
| 1   9   7   | 56  3   8   | 4   25  26  |
|-36  5   4   | 7   9   2   | 1   8  ^36  |
| 236 *23 8   | 456 1   45  |-357-357 9   |
+-------------+-------------+-------------+
| 23  7   9   | 8   4   13  | 6   123 5   |
| 5   8   13  | 9   2   6   | 37  137 4   |
| 4   6   123 | 135 7   135 | 239 139 8   |
+-------------+-------------+-------------+
3 strong links 3c2r2c9 (grouped 3-String Kite) *%^ with r1c5, r6c78<>3

[SpAce]

I'm almost there with the L1 Wing (3 strong links in location using 1 digit) and here is a beauty:

Very nice! Btw, your grid seems to be a bit too tightly packed as it fails to paste correctly into Hodoku. I think the problem is that there's no space between r6c78. So, here's a working version (with appropriately marked eliminations too ):

Code: Select all

.-----------------.-------------.------------------.
| 9    *13    25  | 134  8  134 |  25    6      7  |
| 8     4    %35  | 2    6  7   | %359  %359    1  |
| 7    *123   6   | 13   5  9   |  8     4     ^23 |
:-----------------+-------------+------------------:
| 1     9     7   | 56   3  8   |  4     25     26 |
| 6-3   5     4   | 7    9  2   |  1     8     ^36 |
| 236  *23    8   | 456  1  45  |  57-3  57-3   9  |
:-----------------+-------------+------------------:
| 23    7     9   | 8    4  13  |  6     123    5  |
| 5     8     13  | 9    2  6   |  37    137    4  |
| 4     6     123 | 135  7  135 |  239   139    8  |
'-----------------'-------------'------------------'

Grouped L1-Wing / Grouped 3-String Kite:

(3)r6c2 = r13c2 - r2c3 = r2c78 - r3c9 = (3)r5c9 => -3 r5c1,r6c78

Btw, I like the name 3-String Kite! Did you invent it? However, I think we then need similar names for the 3-link versions of Skyscraper and Turbot Crane, too! (So far I've simply called them Extended Skyscraper and Extended Turbot Crane, but they're boring.)

Or:

Siamese Mutant 3x4-Fish:

(3)R2C29\b13[r5b4|r6b6] => -3 r5c1,r6c78

[SpAce]

Hi rjamil and tarek,

Code: Select all

1   ER        2 Reduced ER  3 Minimal ER
+---------+   +---------+   +---------+
| /  X  / |   | /  X  / |   | /  X  / |
| X  .  X |   | X  .  X |   | X  .  / |
| /  X  / |   | /  /  / |   | /  /  / |
+---------+   +---------+   +---------+

...
Well, is there any significant difference between 2nd and 3rd patterns in terms of number of exclusions?

As depicted, there's no difference in functionality between any of them. The first two are no different at all, and even the last one is only different if you replace the ERI cell with '/'. Only then it's the minimal pattern, not if there's possibly a candidate there.

Please let me know in exemplars form.

The burden of proof is on you. You should show an example of what you mean because I have no idea. You can't really talk about exclusions otherwise because those partial patterns have none by themselves, so there's nothing to compare. To do that you should show complete patterns that actually eliminate something.

For the most relevant functional differences, look at the examples I posted. They cover not only the strong hinges (fka ERs) but also the weak ones that never had names before. I showed the maximal and minimal patterns only, but anything between them works just like the maximal pattern. The only difference in the minimal pattern is that it can work in two different ways (in other words, the ERI cell can be in two different places).

To me the 1st 2 are the same

I agree so far.

and would call them Empty Rectangle, grouped Turbot or Grouped strong link. The last one is just a Turbot or strong link

Please don't bring up any Turbot reference here! It's only valid with complete patterns that fit the Turbot Fish definition of X-Chains of length four (or more in extended cases like your 3-String Kite). Using it here results in the same ambiguity we're trying to avoid with the new names! The examples above are just grouped strong links, specifically ERs or as I would prefer strong hinges. Not Turbots of any kind, though they naturally can be used as part of a Turbot Crane or a Grouped Turbot Crane. Also, the shorthand for those specific Turbot Fishes is "Crane". "Turbot" is the shorthand for the full family including Skyscrapers and Kites. (That's one reason why the "Loader Crane" might have actually been better.)

Let's be extra careful here, or otherwise we're back in the full ambiguity mode -- only with more names causing even more confusion!

[rjamil]

Hi SpAce,

The burden of proof is on you. You should show an example of what you mean because I have no idea. You can't really talk about exclusions otherwise because those partial patterns have none by themselves, so there's nothing to compare. To do that you should show complete patterns that actually eliminate something.

Allow me to show the difference between Full Empty Rectangle and Reduced/Minimal Empty Rectangles by means of number of eliminations by referring to look into my this post.

As depicted, there's no difference in functionality between any of them. The first two are no different at all, and even the last one is only different if you replace the ERI cell with '/'. Only then it's the minimal pattern, not if there's possibly a candidate there.

In my limited knowledge, if by replacing the last one's ERI cell with '/', the pattern will no longer be an ER and falls in to bilocal box wise strong link.

R. Jamil
----------
Hint: Interesting 6 vs 9, M vs W and P vs d perspectives instead of ambiguities.

[SpAce]

Allow me to show the difference between Full Empty Rectangle and Reduced/Minimal Empty Rectangles by means of number of eliminations by referring to look into my this post.

Thanks for those examples. Yes, when used that way you can get more simultaneous eliminations with the reduced ER. You might get even more with the minimal ER. The reason is simple: for the extra eliminations it's not used as a hinge but like the parallel box-wise strong link tarek demonstrated earlier. The full ER can only work as a hinge, i.e. its weak links only "shoot" orthogonally, but the reduced and minimal ERs can also shoot in parallel because at least one half of the strong link is a single candidate which sees both ways (rows and columns) and even into its own cell.

So, you're in fact using more than one type of linking there at the same time, but only one of them is truly ER-related (the one that shoots orthogonally). In other words, the actual ER functionality (hinging) is exactly the same in all cases, including the minimal ER. Any extra eliminations are due to the pattern being simultaneously something else, like a hybrid. The minimal ER is the most capable hybrid linker there is.

In my limited knowledge, if by replacing the last one's ERI cell with '/', the pattern will no longer be an ER and falls in to bilocal box wise strong link.

Well, it certainly is a "bilocal box-wise strong link" but it's also a minimal ER, as explained by StrmCkr here. Hodoku also counts it as an ER if you turn on the option "Allow ERs with two candidates" (and as far as I remember, that option is there because StrmCkr insisted). You don't have to trust me but I know you trust StrmCkr, so there should be no question about this. The minimal ER certainly functions like any other ER, except that it can act as two different ones. It's actually a double-hinge (*), in addition to being a hybrid linker, making it the most flexible ER of all.

(Btw, if you thought it couldn't be an ER without the ERI candidate, then you should have put an 'X' there instead of a dot, right?)

(*) I'm thinking, would it actually make more sense to call it "double-hinge" or "dual hinge" instead of "minimal hinge" like did in my diagrams? I know it definitely can't be called double-ER (or even less dual-ER) because they already mean something else.