The New Sudoku Players' Forum

[SpAce]

I think, I missed an asterisks and tild signs interpretation!!

So I thought. The correct interpretation is not necessarily obvious just by looking at the grid.

If I am correct, then following are the exemplars of L3-Wings:

Yes. You're just missing the second elimination in the non-grouped cases (Z-Y), though. In the non-grouped dual cases it's technically Z-XY, but it's irrelevant because the Z becomes a placement anyway. For the same reason there's no real benefit to specifying a dual at all, because you get the same placement by using just one or the other (if the dual pattern exists). Perhaps such a pattern is a bit easier to spot, but that's all.

My question is, if cells containing +XY, +XZ and +YZ values have no more than two values each, then no need to check bivalues and all the above mentioned L3-Wing patterns become XY-Wing Hybrid move's pattern?

I think that would be a deadly pattern (i.e. one that is not possible in a valid puzzle).

[tarek]

Code: Select all

1. 3 strong cells, 3 total digits                                  : Y-Wing / XYZ-Wing  : VVV / V[V]V
2. 2 strong cells, 1 strength in location (house), 2 total digits  : W-Wing             : VLV
3. 2 strong cells, 1 strength in location, 3 total digits          : H3-Wing            : VVL
4. 1 strong cell, 2 strengths in location, 2 total digits          : M-Wing / S-Wing    : VLL / LVL
5. 1 strong cell, 2 strengths in location, 3 total digits          : H2-Wing            : VLL
6. 3 strengths in location, 1 digit                                : X-Chain (L1-Wing)  : LLL
7. 3 strengths in location, 2 digits                               : L2-Wing            : LLL
8. 3 strengths in location, 3 digits                               : L3-Wing            : LLL

(Note that I've replaced "candidates" with "digits" and added the corresponding wing-names and strong-link configurations.)

Just a note that Sukaku explainer supports both XY/XYZ wing and L1-wing (Colouring) under the name of "3 Strong links" ... Adding to the strength in location options is the Empty rectangle (which can be viewed as a strong link within a box).

I know that using so many terms (as SteveK mentions) is moot, but in a sudoku helper it may be helpful. I'm looking at hopefully programming these into Sukaku explainer in the near future as "Easier to spot" techniques.

tarek

[SpAce]

Hi tarek,

Just a note that Sukaku explainer supports both XY/XYZ wing and L1-wing (Colouring) under the name of "3 Strong links" ...

Does anyone actually use the name L1-Wing? I've never seen it, but I figured it would be a logical name for an X-Chain of length 6, so I've included it as a theoretical possibility to complete the family.

Adding to the strength in location options is the Empty rectangle (which can be viewed as a strong link within a box).

"Empty Rectangle" is actually the most annoying pattern name I know because it has so many levels of ambiguity (even more than "Unique Rectangle" which is never clear whether it means just the deadly pattern or a solving pattern with the included guardians). Sometimes "ER" is used like you just did, but the complete pattern has actually two strong links (like all Turbot Fishes): a box-based and a line-based one. At least some people (StrmCkr, rjamil) use "ERI" for the box-link only, which avoids some of the ambiguity. Furthermore, it's also less than clear whether ERs with just two candidates in the box-link are considered actual ERs or not, but I would prefer not so that ER would only mean a pattern with a grouped box-based strong link.

So what is the corresponding pattern with a non-grouped box-based strong link? That's another annoying one. The standard name would be "Turbot Fish" but that's obviously ambiguous as well, because it's also the family name. It also means that the more logical name for ER would be "Grouped Turbot Fish". Unfortunately "ER" was already used and named before Turbot Fishes, which I guess messed up the naming.

Here's more on my thoughts on the matter. Unfortunately it's not possible to fix the mess at this point, but I think we should do something at least. Based on that discussion I think the simplest fix would be to replace "Turbot Fish" with "Turbot Crane" to avoid the worst ambiguity (or else use something else for the family name, but I think that's more difficult). It's not ideal, but that would give us two logical hierarchies:

Code: Select all

                                 AICs
                                   |
                               X-Chains                   
                                   |                     
                             Turbot Fishes               
                                   |                     
           -----------------------------------------------   
          |                        |                      |
     [Skyscraper]           [2-String Kite]         [Turbot Crane]

Code: Select all

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

I would also love to replace "Empty Rectangle" with "Grouped Turbot Crane" to be similar to its grouped siblings, but obviously that's never going to happen.

I know that using so many terms (as SteveK mentions) is moot, but in a sudoku helper it may be helpful. I'm looking at hopefully programming these into Sukaku explainer in the near future as "Easier to spot" techniques.

That's good! I don't know any public solver that could name all of those wing types. Although I said the names aren't necessary, I also have no problem with simple patterns having names as long as they have a logical hierarchy and don't hide the fact that they're just specific instances of more generic fundamentals (like AICs).

For example, the one-letter wings are easy to remember once understood that they're just AICs with three strong links in different configurations. That forms a complete, logical family, instead of just random patterns to be memorized. The same with the Turbot Fishes (X-Chains with two strong links), though as previously mentioned, that hierarchy is a complete mess.

Anyway, I would also add the "grouped" versions of the one-letter wings (with "location" type links), just like the Turbots. On the other hand, for presentation I would skip all the complicated and arbitrary Type designators and only use logical and useful prefixes (like "Grouped") when necessary, as I just mentioned here.

I don't know what if any of these ideas you might want to consider for the Sukaku Explainer, but just some food for thought anyway Is it still Java-based, by the way? As a Mac user I would hope so...

[tarek]

It is a good idea to mention in the sukaku explainer that these "simpler to spot" techniques are part of generalised technique

From a programming point of view this would mean placing these techniques before the generalised one (which will catch them anyway) and the option to switch them off and keeping the generalised one is available.

I now can see the Turbot crane name. That can be changed in Sukaku explainer as it makes sense. I'm keeping the the name turbot (crane) when a strong link exists in box. It will be trumped by the name Kite when there are strong links in both column(s) and rows(s). Skyscraper is left for the special case of parallel strong links only.

I would like to keep the ER as a "grouped box-based strong link" because with 3 strong link patterns you can exclusively use ER to establish eliminations. Your Strong link in remote pairs could be an ER as well (and other examples I'm sure)

Code: Select all

3 Strong links (with Empty Rectangle) 444
+----------------+----------------+----------------+
| 7    1    8    | 24   24   3    | 5    6    9    |
| 5    34   34   | 6    9    1    | 2    78   78   |
| 9    6    2    | 8    5    7    | 13   134  34   |
+----------------+----------------+----------------+
| 38   9    5    | 7   #136  2    | 4   *138  68   |
| 348  2    7    |*349 *1346 69   |*13   5   *368  |
| 1    34   6    | 5   #34   8    | 7    9    2    |
+----------------+----------------+----------------+
| 23   8    9    | 1    237  4    | 6   *37   5    |
| 2346 57   34   | 23   8    56   | 9   *347  1    |
| 346  57   1    | 39   67-3 569  | 8    2   #347  |
+----------------+----------------+----------------+
3 empty rectangles forming 3 strong links r9c5<>3
which is simmilar to a Finned Mutant Swordfish b569\r5c8b8
fins (r9c9, r46c5) r9c5<>3

I'll probably get to naming these things properly but for the time being the ER term is staying (sorry ). I can improve the wording in the Hint explanation to include "Strong link in location using the same digit" or similar terms

[StrmCkr]

not a reference source but some information on pattern formations:

local-wing-t34685.html
inverted-w-wing-t34338.html
hybrid-wings-work-in-progress-t34212.html?hilit=%20hybrid%20wing
l2-wings-examples-and-exemplar-2-digit-3-strong-links-t32239.html?hilit=%20hybrid%20wing
split-wing-exemplars-and-examples-t30016.html?hilit=%20split%20wing
two-definitions-of-w-wing-terminology-question-t6452.html
m-wings-m-rings-exemplars-examples-t30030.html

Does anyone actually use the name L1-Wing? I've never seen it, but I figured it would be a logical name for an X-Chain of length 6, so I've included it as a theoretical possibility to complete the family.

the local wing aka local 1-wing was the generic name for L-wings on daily Sudoku form and they still used 2-3 digits but limited to size and very stringent formation .. {similar to my first link}

[tarek]

Thanks StrmCkr,

I hope that nobody will get angry when I start using the term (empty line) when I start using the grouped strong link in a line then

[StrmCkr]

Err, my links borked

Ill have yongo fix them...

[SpAce]

Does anyone actually use the name L1-Wing? I've never seen it, but I figured it would be a logical name for an X-Chain of length 6, so I've included it as a theoretical possibility to complete the family.

the local wing aka local 1-wing was the generic name for L-wings on daily Sudoku form and they still used 2-3 digits but limited to size and very stringent formation .. {similar to my first link}

That doesn't really answer my question. The only type numbers I've seen associated with L-Wings are L2-Wing and L3-Wing, which have a clear logic behind them (unlike the messy H-Wings whose type numbering has zero logic) -- L2 has two digits and L3 has three digits.

However, the same pattern with three (bi)local strong links can exist with just one digit, which could logically be called L1-Wing. I've just never seen that name used because that pattern is obviously an X-Chain and not commonly seen as part of the wing family. Logically it could be included, however, and I wouldn't mind if it were. That would complete the one-letter-wing family, making it include all reasonable configurations of three native strong links.

So, I repeat my question: has the name L1-Wing ever been used that way (for an X-Chain of length 6)? It's the only logical L1-Wing there is. I really hope nothing else has been (and more importantly ever will be) called L1-Wing, because it wouldn't make any sense.

--

PS. If I could choose, I would use the L-Wing numbering logic for the other wings too (well, only the M-Wing really). It means that I would move the illogically placed H2-Wing into the M-Wing family and call it M3-Wing (VLL with three digits), while the normal M-Wing would be M2-Wing (VLL with two digits). That move would simplify the horribly complicated and arbitrarily numbered H-Wing family too -- it currently has six (!) numbered top-level types but it should only have one (current H3).

H3-Wing is really the only thing that should be there, which means it could be named just H-Wing without any number. H1-Wing should be dropped altogether because it's really an L3-Wing, H2 should be renamed M3, H5 and H6 don't deserve their own types at all, and H4 is a different beast altogether because of the overlapping ALS (clearly more complex than any of the basic wings). That leaves just the H3 as the only top-level H-Wing -- which means it doesn't even need any type number. Too bad none of this is going to happen at this point, but one can always dream things were named simply and logically:

Code: Select all

VVV (3 digits) : Y-Wing                      : (a=b) - (b=c) - (c=a) => -a (common peers)
VVL (3 digits) : H-Wing  (currently H3-Wing) : (a=b) - (b=c) - c = c => -a (last cell)
VLL (2 digits) : M2-Wing (currently M-Wing)  : (a=b) - b = (b-a) = a => -a (common peers)
VLL (3 digits) : M3-Wing (currently H2-Wing) : (a=b) - b = (b-c) = c => -a (last cell)
VLV (2 digits) : W-Wing                      : (a=b) - b = b - (b=a) => -a (common peers)
LVL (2 digits) : S-Wing                      : a = a - (a=b) - b = b => -a (last), -b (first)
LLL (1 digit)  : L1-Wing (currently X-Chain) : a = a - a = a - a = a => -a (common peers)
LLL (2 digits) : L2-Wing                     : a = (a-b) = b - b = b => -a (last), -b (first)
LLL (3 digits) : L3-Wing                     : a = (a-b) = (b-c) = c => -a (last), -c (first)

Note 1: Only S-Wing and M2-Wing can loop, i.e. become Rings -- and the end result is identical.

Note 2: At least these unnamed configurations are also technically possible:

Code: Select all

VLL: (a=b) - b = b - b = b => -a (last), -b (first)
LLL: a = (a-b) = (b-a) = a => -a (common peers)

However, they're both degenerate because they contain a turbot fish with the same end result.

Note 3: Any of the basic biValue links can be replaced with ALSs, and the biLocation links with group links or even AHSs. Those complications should be noted with the appropriate prefix, as in "Grouped W-Wing" or "ALS-XY-Wing". Group links can always (?) be used without any major change in the wing logic, so the naming works nicely, but using an ALS in a V-link (or AHS in an L-link) can change the elimination logic compared to the basic type, making it more questionable when and if such extensions should be counted as wings and under what names. For example, (using the current names) the H3 and H4 have the same VVL link configuration and even the same number of digits, but they work quite differently:

Code: Select all

H3-Wing:  (a=b) - (b=c) - c = c => -a (last cell)
H4-Wing: (ab=c) - (c=a) - a = a => -a (common peers)

The H4 logic is not even possible without the ALS complication, so it's clearly logically different despite the same link configuration. On the other hand, the H3 logic can be used with an ALS, so this would be a correctly named extension:

Code: Select all

ALS-H3-Wing: (ab=c) - (c=d) - d = d => -a (last cell)

I'm not really sure where I'd put the H4 in my dreamed naming system, if anywhere, because it doesn't really belong (not having a simple counterpart). Besides, it's unnecessarily restricted in its current form anyway. I think such interesting ALS patterns might deserve their own category outside of the one-letter-wing families.

[SpAce]

It is a good idea to mention in the sukaku explainer that these "simpler to spot" techniques are part of generalised technique

From a programming point of view this would mean placing these techniques before the generalised one (which will catch them anyway) and the option to switch them off and keeping the generalised one is available.

Exactly! That's how it should work. In Hodoku it mostly works, but it has a problem with some patterns -- specifically Turbot Fishes, which is what got me thinking about the naming. Unlike Skyscrapers, Kites, and (grouped) ERs, there's no way to look for the Turbot Crane variant specifically because its official name (Turbot Fish) is the same as the generalized type! Those patterns do get found and listed under Turbot Fishes (but it shows them all) and ERs if the 2-candidate option is turned on (but it lists the grouped ones too), as well as generic X-Chains, AICs, and complex fishes (but they obviously have even wider scopes). Hodoku doesn't know Grouped Skyscrapers and Grouped Kites either, so they would be a nice addition.

I now can see the Turbot crane name. That can be changed in Sukaku explainer as it makes sense.

Excellent! If you browsed the link I gave earlier, you might have noticed that we considered other names too. StrmCkr suggested "Loader Crane" and I initially liked it a lot (and still do, actually), and even suggested using the same idea for the grouped version ("Tower Crane"). However, I don't think those names have much of a chance to be adopted at this point in time because they have no (obvious) connection to the Turbot family. "Turbot Crane" isn't necessarily that much of a stretch.

I'm keeping the the name turbot (crane) when a strong link exists in box. It will be trumped by the name Kite when there are strong links in both column(s) and rows(s). Skyscraper is left for the special case of parallel strong links only.

Yes. All three have exactly two strong inferences:

Code: Select all

Skyscraper: strong links in two parallel lines   (weakly linked by an orthogonal line)
Kite:       strong links in two orthogonal lines (weakly linked by a box)
Crane:      strong links in one line and one box (weakly linked by an orthogonal line)

There can be more strong links but they're used for weak inferences. In other words, it doesn't matter if a Kite's box link is strong or weak, because it's always used as a weak link. I guess that's what you meant anyway.

I would like to keep the ER as a "grouped box-based strong link" because with 3 strong link patterns you can exclusively use ER to establish eliminations. Your Strong link in remote pairs could be an ER as well (and other examples I'm sure)

I'm fine with that. In fact, I think "ER" makes much more sense as such a generic term instead of being specifically one of the Turbot Fish types. Liker I said earlier:

I would also love to replace "Empty Rectangle" with "Grouped Turbot Crane" to be similar to its grouped siblings, but obviously that's never going to happen.

What do you think, would it be possible to go all the way? Then we'd have a fully logical naming hierarchy for the Turbot family, and "ER" could be freely used for the box-based grouped strong link only without ambiguity. I'd love that. Then the Grouped Turbot Crane (and other ER patterns) would include an ER but it wouldn't be an ER. That's how it should have been in the first place.

Code: Select all

3 Strong links (with Empty Rectangle) 444
+----------------+----------------+----------------+
| 7    1    8    | 24   24   3    | 5    6    9    |
| 5    34   34   | 6    9    1    | 2    78   78   |
| 9    6    2    | 8    5    7    | 13   134  34   |
+----------------+----------------+----------------+
| 38   9    5    | 7   #136  2    | 4   *138  68   |
| 348  2    7    |*349 *1346 69   |*13   5   *368  |
| 1    34   6    | 5   #34   8    | 7    9    2    |
+----------------+----------------+----------------+
| 23   8    9    | 1    237  4    | 6   *37   5    |
| 2346 57   34   | 23   8    56   | 9   *347  1    |
| 346  57   1    | 39   67-3 569  | 8    2   #347  |
+----------------+----------------+----------------+
3 empty rectangles forming 3 strong links r9c5<>3
which is simmilar to a Finned Mutant Swordfish b569\r5c8b8
fins (r9c9, r46c5) r9c5<>3

I guess that could also be called Grouped L1-Wing

(3)r46c5 = r5c45 - r5c79 = r4c8 - r78c8 = (3)r9c9 => -3 r9c5

Too bad there's no similar term (to "ER") for when boxes are used for weak linking. Personally I think it's even cooler (way less obvious), which is why the Grouped 2-String Kite is my favorite Turbot Fish variant. I can't find a real world example for a three-box extension, but it should surely work:

Code: Select all

.------------------------------------.---------------------------------.------------------------------------.
|  123456789   12346789-5  123456789 | 123456789  123456789  123456789 |  123456789  *123456789   123456789 |
| *123456789   12346789   *123456789 | 12346789   12346789   12346789  | *123456789   12346789   *123456789 |
|  123456789   12346789-5  123456789 | 123456789  123456789  123456789 |  123456789  *123456789   123456789 |
:------------------------------------+---------------------------------+------------------------------------:
|  123456789   123456789   123456789 | 123456789  123456789  123456789 |  123456789   12346789    123456789 |
|  123456789   123456789   123456789 | 123456789  123456789  123456789 |  123456789   12346789    123456789 |
|  123456789   123456789   123456789 | 123456789  123456789  123456789 |  123456789   12346789    123456789 |
:------------------------------------+---------------------------------+------------------------------------:
|  123456789   123456789   123456789 | 123456789  123456789  123456789 |  123456789  *123456789   123456789 |
|  12346789   *123456789   12346789  | 12346789   12346789   12346789  | *123456789   12346789   *123456789 |
|  123456789   123456789   123456789 | 123456789  123456789  123456789 |  123456789  *123456789   123456789 |
'------------------------------------'---------------------------------'------------------------------------'

Grouped L1-Wing: (5)r2c13 = r2c79 - r13c8 = r79c8 - r8c79 = (5)r8c2 => -5 r13c2

Or:

Mutant 3x4-Fish: (5)R28C8\c2b139 => -5 r13c2

[tarek]

In Sukaku explainer I’m introducing the “suffix” which follows the technique name. This suffix is a series of numbers that are particular to that technique. For the Fishes here they would describe the strong links (base sectors)

For your 3-string kite it would be something like “121” at the moment the grouped turbot has a “4” but when I’ve got all the algorithm complete I’ll show here some examples before I finalise the presentation and suffix.

The “turbot crane” is very specific to the pattern With 1 strong link in box and 1 strong link in line. In short it implies the presence of 2 links at least so similar to 2-string kite and Skyscraper. While the “ER” to me is closer to “grouped strong link in a box” which is what I would carry on doing until I finalise the appropriate terminology

I have a redundant example of 2 strong links (in boxes) joined by a weak link in a line ( I would still call it a Turbot fish) ... redundant because there is a simpler box-line interaction b\line which is also equivalent to a Franken X-Wing (also redundant)

Tarek

[tarek]

the Grouped 2-String Kite is my favorite Turbot Fish variant.

Here is a taster with 2 strong links:

Code: Select all

+-------------------+-------------------+-------------------+
| 36789 34678 3689  | 589   59    1     | 5679  2     45789 |
| 1     678   2689  | 3     4     2578  | 5679  689   5789  |
| 2789  478   5     | 289   6     278   | 3     1489  14789 |
+-------------------+-------------------+-------------------+
| 4     3568  3689  | 589-1  2     58   |*169   7     1389  |
| 89    2     7     | 6     19    3     | 4     5     189   |
| 35689 1     3689  | 4589  7     458   | 69    3689  2     |
+-------------------+-------------------+-------------------+
| 2567  567   4     |%125   3     256   | 8    %19   %1579  |
| 2357  357   123   | 1245  8     9     |*1257  134   6     |
| 23568 9     12368 | 7     15    2456  |*125   134   1345  |
+-------------------+-------------------+-------------------+
2-String Kite r7c7 r4c4<>1

[SpAce]

Here is a taster with 2 strong links:

Code: Select all

2-String Kite r7c7 r4c4<>1

Nice! I'd still prefer to call it "Grouped 2-String Kite" to let the reader know it's more complicated (and fun) than the normal variant. In manual spotting the difference is significant. Simple bilocation strong links are easier to see than grouped ones.

Btw, today's puzzle had a fun almost-triple-ER formation.

[StrmCkr]

That doesn't really answer my question. The only type numbers I've seen associated with L-Wings are L2-Wing and L3-Wing, which have a clear logic behind them (unlike the messy H-Wings whose type numbering has zero logic) -- L2 has two digits and L3 has three digits.

to be fair: Local wings where generic and very stringent specific formations.
L-WING x=(x-z)=(z-y)=y y may be eliminated from the beginning cell and x can be eliminated from the ending cell. { which is an l3 wing by today's standards}

(1)L used extremely rarely was the generalized form of the listed above and could have 2,3 digits and could expand the link count when it was +- digits. the term one wasn't used to represent digits but a generalized form...

number counts where added later 2,3 to represent digit size and thus linking application pretty much making the use of (1) not really sound, Local wing was then listed representing stringent and generic version of the move set and the generalized version.
some adopted the 2,3 designation and it went from there.

does that help.. not really.

its like what happened to M-wing and a (g)M-wing where the g represents the Generalized version which could use grouped links between nodes. where the original M wing could not.
{bit of an annoying part of evolution of the techniques when different forms collaborate but some one says no to including it into their "name" so something is added to note the difference}

id stick to the naming system you purposed earlier it is well thought out and concise, i agree with it.

{yes i know the H wing system was basically type cased numbers based on the new stuff I/we was exploring, not a formation concentric analysis and grouping found in other "wings" }

has the name L1-Wing ever been used that way (for an X-Chain of length 6)

not to my knowledge asides from you in a few discussions with me.
does the 1 make sense representing digit counts to follow suit to the rest , absolute.

[SpAce]

Hi StrmCkr,

Thanks for the history lesson, once again! I love them.

its like what happened to M-wing and a (g)M-wing where the g represents the Generalized version which could use grouped links between nodes. where the original M wing could not.

I've thought the (g)M wasn't about group links but the weird fact that the original M-Wing required two identical bivalue cells for no reason:

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M-Wing: (a=b) - b = (b-a) = a => -a

Apparently the original inventor required that the cell with the weak link (a-b) was also bivalue like the first one with the strong link, but that makes no sense, of course. Such a pattern may be easier to spot (like a W-Wing) but it has the huge price of missing many valid cases.

{bit of an annoying part of evolution of the techniques when different forms collaborate but some one says no to including it into their "name" so something is added to note the difference}

Yes. Backwards compatibility is important but a pain in the butt, too. Fortunately we no longer have the problem with the M-Wing, as far as I know, since the g-version has become the norm.

id stick to the naming system you purposed earlier it is well thought out and concise, i agree with it.

Thanks! Problem is, I have a hard time seeing it adopted at this point. I'm pretty sure some people would get really confused and annoyed, if we started talking about M2- and M3-Wings. Then again, who knows. Things that make sense are easier to learn and to remember.

{yes i know the H wing system was basically type cased numbers based on the new stuff I/we was exploring, not a formation concentric analysis and grouping found in other "wings" }

I understand that. When exploring new stuff it's useful to list all the encountered cases, and it's not necessarily easy to see the big picture until the cases have been collected and analyzed. Your (and others') work in that area is very much appreciated. Now that the big picture is clearer because of those huge earlier efforts, the proper hierarchy is much easier to see. That's why I understand that some of my criticism may sound a bit unfair. That's not my intention. I'd just like to make things simpler for new learners.

has the name L1-Wing ever been used that way (for an X-Chain of length 6)

not to my knowledge asides from you in a few discussions with me.
does the 1 make sense representing digit counts to follow suit to the rest , absolute.

Thanks for the confirmation!

[tarek]

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.

The above example would give a strong link equivalent of this:

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

Whereas the ER would give something like this:

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

in the 2 strong link in location with 1 digit you wouldn't encounter it but I'm sure it is possible to have a 3 strong link pattern with it!

Any thoughts?