The New Sudoku Players' Forum

[yzfwsf]

ERI Pair or other?

ERI.png (32.76 KiB) Viewed 1422 times

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-ab  .   .   | .   .   .   | .   .   . 
-ab  .   .   | .   .   .   | .   .   . 
-ab  .   .   | .   .   .   | .   .   . 
----------- -+-------------+-------------
-ab  *   *   | -ab ab -ab  |-ab -ab -ab 
*     /   /  | .   .   .   | .   .   . 
*    /    /  | .   .   .   | .   .   . 
-------------+-------------+--------------
 ab  .   .   | .   .   .   | .   .   . 
-ab  .   .   | .   .   .   | .   .   . 
-ab  .   .   | .   .   .   | .   .   . 

[Leren]

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*-----------------------------------------------------*
| 4      6789  16789 | 89 2   3     | 1679 1569  159  |
| 3      79    1579  | 4  6   59    | 179  8     2    |
| 89-26  2689  25689 | 1  7   589   | 69   3     4    |
|--------------------+--------------+-----------------|
|b26789 a2689 a2689  |a26 3   4     | 5    19-26 1789 |
|b2689   5     3     | 7  189 12689 | 689  4     89   |
|b26789  1     4     | 5  89  2689  | 3    269   789  |
|--------------------+--------------+-----------------|
| 5      3     789   | 89 4   189   | 2    179   6    |
|b26     4     26    | 3  159 7     | 189  159   1589 |
| 1      789   789   | 26 589 26    | 4    579   3    |
*-----------------------------------------------------*

ALS XZ Rule (Loop 1) : ALS 1 r4c234; ALS 2 r4568c1; Z = 8 & 9 => - 26 r3c1, r4c8; stte

Leren

[Cenoman]

OK with Leren.
I use also the name "Doubly linked ALS-XZ rule"
Note that you can place cell r4c1 in ALS 1: ALS 1 r4c1234; ALS 2 r568c1; Z = 8 & 9, as well
Using AIC presentation, you can write it as a loop:
(8=269)r4c234 - (9=2678)r4568c1@ or (9=268)r4c234 - (8=2679)r4568c1@

Considering AALS r456c1, ALS 1 r8c1 (Z = 2 & 6), ALS 2 r4c234 (Z = 8 & 9), you can also name it Sue de Coq.

Anyhow, it is a rank-0 logic, so why not name it a MSLS (as proposed by SpAce) ?
MSLS
7 truths: cells r4c1, r4c234, r568c1
7 links: 2r4, 6r4, 2c1, 6c1, 7c1, 8b4, 9b4
6 eliminations -26 r3c1, r4c8, r4c1
I isolate cell r4c1: you can assign 2r4c1 to the link 2r4 or 2c1, similarly 6r4c1 to 6r4 or 6c1, even mixing 2r4 & 6c1 or 2c1 & 6r4 (leading to four symmetric pigeonhole matrices, all demonstrating the same eliminations, including cannibalistic -26r4c1)

[yzfwsf]

Anyhow, it is a rank-0 logic, so why not name it a MSLS (as proposed by SpAce) ?
MSLS

My code only check ERI+2 Cells with 2 same candidates，Although it is MSLS/ALS-XZ, it is only a special case of MSLS/ALS-XZ. Just like XY-Wing is also ALS-XZ, but we didn't name it ALS-XZ.

[Mauriès Robert]

HI all,
Another point of view is that of the TDP, which does not try to find out to which model such a configuration belongs, but reasons with anti-tracks:
P'(26r4c23) : -26r4c23->26r456c1->-26r8c1 contradiction => 26r4c234 => -26r4c1c8
P'(26r56c1) : -26r56c1->26r4c123->-26r4c4 contradiction => 26r568c1 => -26r1c34
Robert

[champagne]

Anyhow, it is a rank-0 logic, so why not name it a MSLS (as proposed by SpAce) ?
MSLS
7 truths: cells r4c1, r4c234, r568c1
7 links: 2r4, 6r4, 2c1, 6c1, tc1, 8b4, 9b4
4 eliminations -26 r3c1, r4c8
I isolate cell r4c1: you can assign 2r4c1 to the link 2r4 or 2c1, similarly 6r4c1 to 6r4 or 6c1, even mixing 2r4 & 6c1 or 2c1 & 6r4 (leading to four symmetric pigeonhole matrices, all demonstrating the same eliminations)

Hi Cenoman,

Interesting, and surely a general rule. If combining ALSs you come to a point where you have a locked situation, you certainly have a corresponding rank0 logic using the cells as truths (I don't use the MSLS name just because I am not sure of the precise definition).

[Leren]

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-ab      .       .      |  .   .   .   | .   .   . 
-ab      .       .      |  .   .   .   | .   .   . 
-ab      .       .      |  .   .   .   | .   .   . 
------------------------+--------------+-------------
-abcd   *abcd   *abcd   | -ab *ab -ab  |-ab -ab -ab 
#abcd-e  /       /      |  .   .   .   | .   .   . 
#abcd-e  /       /      |  .   .   .   | .   .   . 
------------------------+--------------+-------------
#ab      .       .      |  .  -ab  .   | .   .   . 
-ab      .       .      |  .   .   .   | .   .   . 
-ab      .       .      |  .   .   .   | .   .   . 

I'm going to take the pattern as it appears in the puzzle and see if I can take take it a bit further. Referring to the exemplar above, we have an ALS (abcd) in r4c235 (* cells), and an AALS (abcde) in r567c1 (# cells).

Now it's clear that one of r4c23 must be c or d, otherwise there is an immediate contradiction ab in 3 cells r4c235.

Now suppose both cd were in r4c23. Then r567c1 would be a locked triple abe, abe, ab. So r4c1 <> abcd and r56c1 must be ae or be in some order. It can't be ab ab otherwise we have another contradiction ab in 3 cells r567c1.

But if its ae, then, since r7c1 = b then b can't be in Box 3 Column 1 so it can only be in r4c23. Similar reasoning for the case be.

The net result of all this is that one of ab must be in r4c23. ie exactly one of ab and one of cd must be in r4c23 and also in r56c1 and as well one of r4c5 and r7c1 is a and the other is b and you get the eliminations shown in the exemplar.

There is actually only one e = 7 in the puzzle, but nevertheless, there are 9 eliminations, r3c1 <> 26, r4c8 <> 26, r4c1 <> 2689, r6c1 <> 7.

Does this reasoning look OK ? Leren

[Cenoman]

Hi Leren and yzfwsf,

Does this reasoning look OK ?

I felt uneasy when reading your demo, because it was questioning all what I knew about ALS, AALS chains.
As you do not use r4c1 cell content in your rationale, you seem to have succeeded to demonstrate that an AALS doubly linked to an ALS is a rank-0 logic (as if it were a Sue de Coq or doubly linked ALs's).

it's clear that one of r4c23 must be c or d

Agreed (c OR d = True results from the als r4c234)

Now, you demonstrate that (cd)r4c23 = False.
That's true. I'd use another way:
r4c1 can't be a nor b. Thanks to the Empty Rectangle in box 4, there are two W-wings, grouped and cannibalistic
(a=b)r4c5 - r4c123 = r56c1 - (b=a)r7c1 => r4c1<>a
(b=a)r4c5 - r4c123 = r56c1 - (a=b)r7c1 => r4c1<>b

If cd were in r4c23, Then r56c1 would be a locked pair ab, in contradiction with (a|b)r7c1.

Then you get configurations of this type (where a and b can be shifted, also c and d can be shifted, also assignment of a|b, c|d to r4c23 can be shifted):

Code: Select all

-ab      .       .      |  .   .   .   | .   .   .
-ab      .       .      |  .   .   .   | .   .   .
-ab      .       .      |  .   .   .   | .   .   .
------------------------+--------------+-------------
-d+     *c      *a      | -ab *b  -ab  |-ab -ab -ab
#bde     /       /      |  .   .   .   | .   .   .
#bde     /       /      |  .   .   .   | .   .   .
------------------------+--------------+-------------
#a       .       .      |  .  -ab  .   | .   .   .
-ab      .       .      |  .   .   .   | .   .   .
-ab      .       .      |  .   .   .   | .   .   .

Now, further conclusions about eliminations of d and e in column 1 depend on the candidates present in r4c1. Some more assumptions on cells marked (/) in box 4, should also be stated. At least, box 4 is an Empty Rectangle for digits a and b.
Your conclusions, Leren, are true, if you add in the pattern assumptions that r4c1 has no more candidates than abcde.

My code only check ERI+2 Cells with 2 same candidates

BTW, I use Empty Rectangle, shortcut ER. yzfwsf uses ERI acronym. But what does ERI stands for ? (To me: Empty Rectangle Intersection... but I may be wrong)

I guess that yzfwsf intention was rather a pattern like this (An ER + 2 bivalue cells ab, aligned with the ER row and ER column):

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-ab       .       .      |  .   .   .   | .   .   .
-ab       .       .      |  .   .   .   | .   .   .
-ab       .       .      |  .   .   .   | .   .   .
-------------------------+--------------+-------------
cde+,-ab *abcd   *abcd   | -ab *ab -ab  |-ab -ab -ab
#abcde    /       /      |  .   .   .   | .   .   .     / cells without a or b
#abcde    /       /      |  .   .   .   | .   .   .
-------------------------+--------------+-------------
#ab       .       .      |  .  -ab  .   | .   .   .
-ab       .       .      |  .   .   .   | .   .   .
-ab       .       .      |  .   .   .   | .   .   .

My own proposals of connecting it to Doubly linked ALS's or Sue de Coq or MSLS are beside the point, sice they use the limited assignment of abcde to r4c1 (as seen in the exemplar).

For oddagons with two digits in commons cells, we have retained "Bivalue Oddagon". For the present pattern, I would similarly propose: "Bivalue Empty Rectangle".

[yzfwsf]

Hi Cenoman & Leren
I mean the following structure, Only two bivalue cells with the same two candidates + ER of these two candidates are considered.

ERI_Pairs.png (60.65 KiB) Viewed 1320 times

[Leren]

Cenoman wrote : Your conclusions, Leren, are true, if you add in the pattern assumptions that r4c1 has no more candidates than abcde.

I now can't see the extra eliminations regarding c & d being valid in the general case. The reason is that c or d could be in r4c1.

So, you could have something like r4c1 = c, r4c2 = d, r4c3 = a & r4c4 = b. In Column 1 you could have r4c1 = c as before, r5c1 = b, r6c1 = e & r7c1 = a

I can't see this resulting in a contradiction in the general case, even thought it did not work out that way in the sample puzzle.

However, even in the case of the ER in just 2 digits ab, it's now clear why you can remove them from r4c1. It also follows that the 2 digits can't be in site of just one bi-value cell, one must be in site of one cell and one the other.

Thus, the bi-value cells have opposite parity and the elimination of - ab from r7c5 remains valid in the most general case, and should be added to the first exemplar.

Leren

[Cenoman]

I mean the following structure, Only two bivalue cells with the same two candidates + ER of these two candidates are considered.

Sorry. My diagram was still dealing with Leren's reasoning. But I had in mind exactly what is shown in your XSudo output. Just be aware that Leren's rationale on the opposite parity of the bivalue cells is right. It leads to the additional elimination: r7c5<>ab in the diagram below (no longer referring to c, d, e candidates). Such an elimination is not seen in the XSudo diagram, since r8c4 is a given.

Code: Select all

-ab       .       .      |  .   .   .   | .   .   .
-ab       .       .      |  .   .   .   | .   .   .
-ab       .       .      |  .   .   .   | .   .   .
-------------------------+--------------+-------------
+,-ab    *ab+    *ab+    | -ab *ab -ab  |-ab -ab -ab
#ab+      /       /      |  .   .   .   | .   .   .     / cells without a or b
#ab+      /       /      |  .   .   .   | .   .   .     + extra candidates <> a OR b
-------------------------+--------------+-------------
#ab       .       .      |  .  -ab  .   | .   .   .
-ab       .       .      |  .   .   .   | .   .   .
-ab       .       .      |  .   .   .   | .   .   .

[yzfwsf]

Hi Cenoman/Leren
I found a perfect sample as below.

ERI_Pairs.png (28.01 KiB) Viewed 1276 times

[eleven]

Nice pattern (new for me).
I think ERI pair is an appropriate name for it.

[edit: double kite was not a good alternative]

[tarek]

From several discussions elsewhere …
ER should describe the grouping in a box
The Crane description was coined to describe that single digit pattern with 2 strong links, One of those links is an ER.

Here you are using a bivalue ER with each arm doubly linked to a bivalue cell achieving a loop … Very nice!

I see it as a special form of ALS XY with the center ALS Doubly linked to each of the other 2 ALSs

tarek

[eleven]

I see it as a special form of ALS XY with the center ALS Doubly linked to each of the other 2 ALSs

Which center ALS ? Wouldn't you see the strong links first ?

Also if you can identify it as a special case of a more general doubly linked construct, it is an own pattern for me. For a manual solver it can be spotted almost as easy as a single ERI or kite (grouped or not), but probably the additional eliminations would not be found.