The New Sudoku Players' Forum

[daj95376]

Here are the individual JExocet eliminations (as I see them):

Code: Select all

 +--------------------------------------------------------------------------------+
 |  9       8       234     |  7       36      1234    |  1236    125     12356   |
 |  1237    127     6       |  138     5       1238    |  4       12789   1239    |
 |  12347   5       2347    |  13468   368     9       |  12367   1278    1236    |
 |--------------------------+--------------------------+--------------------------|
 |  8       1279    2479    |  14569   679     147     |  1269    3       12569   |
 |  1246    3       249     |  145689  689     148     |  1269    12459   7       |
 |  1467    179     5       |  13469   2       1347    |  8       149     169     |
 |--------------------------+--------------------------+--------------------------|
 |  5       6       23789   |  389    Q1       378     | R39-27  q79-2    4       |
 |  237     4       23789   | r39-8   Q79-38   6       | R5       1279    1239    |
 | B37     B79      1       |  2       4       5       |  379     6       8       |
 +--------------------------------------------------------------------------------+
 

Hi Danny,

In my view, this is more than just direct eliminations from a JE, but you are using the word Qexocet.

In fact, looking for all exocets in band for that puzzle, my solver finds

r7c4 r7c6 r8c1 r8c3
r7c4 r7c6 r8c3 r9c7
r9c1 r9c2 r7c7 r7c8
r9c1 r9c2 r7c7 r8c5
r9c1 r9c2 r8c4 r8c5

and we have the properties you apply. (r7c8==r8c5 r7c7==r8c4)
Eliminations made are very close to what can be done using the double exocet.

With the JE4, we have also r8c1 == r9c7

Most of the posts in this thread have gone beyond DPB's original definition of a JExocet. That said, I think "secondary equivalences" are trivial and qualify as ...

JExocet Corollary #1

Code: Select all

When the JExocet pattern has:

*) a diagonal distribution of solutions -- Q & R -- and

*) a diagonal distribution of non-base candidates -- (/) -- and

*) non-base candidates in one cell (~) of a mini-unit containing Q and/or R and parallel to BB

Then secondary equivalences are forced for Q and/or R, and additional eliminations may exist.


Code: Select all

 R==r follows from: R eliminated from [r1] and [r3] and only possible remaining cell is r in [r2]
 |---------------+---------------+---------------|
 |   B   B   .   |   .   .   .   |   .   .   .   |
 |   .   .   .   |   Q   ~   r   |   .   .   /   |
 |   .   .   .   |   /   .   .   |   .   .   R   |
 |---------------+---------------+---------------|

 Q==q follows from: Q eliminated from [r1] and [r2] and only possible remaining cell is q in [r3]
 |---------------+---------------+---------------|
 |   B   B   .   |   .   .   .   |   .   .   .   |
 |   .   .   .   |   Q   .   .   |   .   .   /   |
 |   .   .   .   |   /   .   .   |   q   ~   R   |
 |---------------+---------------+---------------|

 both conditions apply
 |---------------+---------------+---------------|
 |   B   B   .   |   .   .   .   |   .   .   .   |
 |   .   .   .   |   Q   ~   r   |   .   .   /   |
 |   .   .   .   |   /   .   .   |   q   ~   R   |
 |---------------+---------------+---------------|


When this corollary exists, you can expect me to treat any resulting eliminations as part of the original JExocet.

In fact, it's way past time for DPB to add this to his original definition.

[champagne]

Thank's dany,

your point is quite clear

[blue]

Just a quick note: I made a couple of lengthy additions to my previous two posts, and posed an intereting (?) question. In case they might me missed, I've added this post.

For Danny: were the eliminations that David made in his "solution path", part of what you had in mind ... r7c7 <> 7 and r8c5 <> 3 ?
Do you have a "simple" way of viewing them as following on account of "secondary equivalences" ?
I ask, because when I use "XSudo" and add the truths and links that support the secondary equivalences (r7c8 == r8c5) and (r8c4 == r7c7) ... then XSudo reports eliminations for 7r7c7 and 3r8c5 -- even before I add "cell links" for r7c8 and r8c5. Given the mystery behind some of XSudo's eliminations, I don't know if they have simple explanations or not.

For the "XSudo" enabled, the expansion(s) adding the truths/links supporting the secondary equivalences, can be taken as either of these:

Code: Select all

17 Truths = {379R7 379R8 3C357 7C357 9C357 9N12}
20 Links = {3r139 7r349 9r459 8n5 7n7 3b789 7b789 9b789}
6 Eliminations --> r4c26<>7, r7c7<>27, r8c5<>38,

or

17 Truths = {3C357 7C357 9C357 9N12 379B8 379B9}
20 Links = {3r13789 7r34789 9r45789 8n5 7n7 379b7}
13 Eliminations --> r4c26<>7, r7c37<>7, r8c59<>3, r8c89<>9, r13c7<>3, r7c7<>2, r8c8<>7, r8c5<>8,

Note: They don't produce the same set of eliminations ... hence the comment about the "mystery" factor.
However, "r7c7 <> 7, r8c5 <> 3" are common to both, in addition to the usual exocet eliminations.

[

With the cell links for r7c7 and r8c5, they're:

Code: Select all

17 Truths = {379R7 379R8 3C357 7C357 9C357 9N12}
22 Links = {3r139 7r349 9r459 7n78 8n45 3b789 7b789 9b789}
8 Eliminations --> r4c26<>7, r7c78<>2, r8c45<>8, r7c7<>7, r8c5<>3,

and

17 Truths = {3C357 7C357 9C357 9N12 379B8 379B9}
22 Links = {3r13789 7r34789 9r45789 7n78 8n45 379b7}
17 Eliminations --> r7c3<>379, r4c26<>7, r7c78<>2, r8c59<>3, r8c45<>8, r8c89<>9, r13c7<>3,  r7c7<>7, r8c8<>7,

]

Also: the adding just the usual truths/links for the JExocet:

Code: Select all

11 Truths = {3C357 7C357 9C357 9N12}
14 Links = {3r139 7r349 9r459 8n5 7n7 379b7}
4 Eliminations --> r4c26<>7, r7c7<>2, r8c5<>8

... we get the eliminations for digit 7 in r4c26 (but not for digit 7 in r3c18 !). I haven't looked into how those might be explained, but since it doesn't include candidates in both rows, I think they must be another example of "XSudo mystery factor" eliminations. (They're (also) common to both of the truth/link extensions above).

Note2: the truths & links are applied to the original PM grid for the puzzle:

Code: Select all

+--------------------+--------------------+---------------------+
| 9      8     234   | 7       36    1234 | 1236   125    12356 |
| 1237   127   6     | 138     5     1238 | 4      12789  1239  |
| 12347  5     2347  | 13468   368   9    | 12367  1278   1236  |
+--------------------+--------------------+---------------------+
| 8      1279  2479  | 14569   679   147  | 1269   3      12569 |
| 1246   3     249   | 145689  689   148  | 1269   12459  7     |
| 1467   179   5     | 13469   2     1347 | 8      149    169   |
+--------------------+--------------------+---------------------+
| 5      6     23789 | 389     1     378  | 2379   279    4     |
| 237    4     23789 | 389     3789  6    | 5      1279   1239  |
| 37     79    1     | 2       4     5    | 379    6      8     |
+--------------------+--------------------+---------------------+

Regards,
Blue.

[David P Bird]

Hi Blue

Regarding your points about the inferences between target cells, base cells etc in a JE pattern, you may remember that I put the case to Denis that I regarded JEs not only as elimination patterns but also inference patterns. Clearly, as such, these inferences would have to be provable and stated as mini-theorems for example as in < this post >. In the course of a formal solution we would then cite the pattern being used and pull out the relevant inference.

This is why I wanted to define the JED and JEC pattern classes as their theorems are different, but from the reception that idea got it's clear it was another lead balloon. (DAJ refers to them as corollaries).

Included in those inferences is the weak inference between two instances of a digit in the target cells that becomes conjugate when the digit is known to occupy in the base cells that you mentioned.

I take your points about the errors in my notation.

Regarding your peace making efforts, I thank you kindly, but as Einstein said "Insanity is doing the same thing over and over again and expecting different results" or as my mother put it "A leopard doesn't change its spots". So I'm sorry but I'm right out of slack.

Any member of the forum can choose to post here, and I can choose to respond to them or not. I won't guarantee to respond to anyone who is disrespectful or insulting to me, is unwilling to admit it when they are wrong, or only responds sometimes. I also take a poor view of those who ask questions and can't be bothered to say thanks when they get answers.

I apologise to those that feel they are the ham in the middle of a sandwich as a result of this, as I have done in other cases.

As you agree so strongly with ronk, I accept the term "fins" is to be scrubbed. On May 9th When I wrote the piece about "Partial Fish" I asked for suggestions for convenient and memorable terms then, but got no response.

I've been trying to steer a middle line between the different sects in our community – those that accept nets but not uniqueness and vice versa, and those that want to solve puzzles as a form of mental exercise and those who want to write solving programs. Wanting a human "recognisable pattern" was what started this JE malarkey off in the first place, but I'm the only one here that doesn't use a computer solver.

Now it seems Champagne is riled with me because we can't understand one another (sorry Champagne).

A little while back you announced that you had had enough of JEs and were turning off. Now it's my turn. While I'm not trying to respond to various side issues on this thread, I may be able to pull together the drafts I've been making of the various JE scenarios that are possible.

Regards

David

[daj95376]

For Danny: were the eliminations that David made in his "solution path", part of what you had in mind ... r7c7 <> 7 and r8c5 <> 3 ?
Do you have a "simple" way of viewing them as following on account of "secondary equivalences" ?

Although DPB references the same eliminations, the timing is completely different between our solutions. Here's my breakdown on the generation of eliminations:

Code: Select all

 +--------------------------------------------------------------------------------+
 |  9       8       234     |  7       36      1234    |  1236    125     12356   |
 |  1237    127     6       |  138     5       1238    |  4       12789   1239    |
 |  12347   5       2347    |  13468   368     9       |  12367   1278    1236    |
 |--------------------------+--------------------------+--------------------------|
 |  8       1279    2479    |  14569   679     147     |  1269    3       12569   |
 |  1246    3       249     |  145689  689     148     |  1269    12459   7       |
 |  1467    179     5       |  13469   2       1347    |  8       149     169     |
 |--------------------------+--------------------------+--------------------------|
 |  5       6       23789   |  389    Q1       378     | R39-27  q79-2    4       |
 |  237     4       23789   | r39-8   Q79-38   6       | R5       1279    1239    |
 | B37     B79      1       |  2       4       5       |  379     6       8       |
 +--------------------------------------------------------------------------------+
 # 147 eliminations remain

 ### -379- QExocet   Base = r9c12   Target = r8c5==r7c8,r7c7==r8c4


Code: Select all

*) Perform JExocet eliminations ala DPB's original definition.

*)  =>  r7c7<>2, r8c5<>8


*) Determine if secondary equivalences exist and eliminate additional camdidates.

*) r7c7==r8c4  =>  39 common base candidates  =>  r7c7<>7, r8c4<>8
*) r8c5==r7c8  =>  79 common base candidates  =>  r7c8<>2, r8c5<>3


After these eliminations, there is a Hidden Single, r7c3=2, that leads to the subsequent cracking of the puzzle through basics.

I don't know anything about XSUDO, so I have no idea as to the cause of your problems there.

Regards, Danny

[blue]

Hi David,

Regarding your points about the inferences between target cells, base cells etc in a JE pattern, you may remember that I put the case to Denis that I regarded JEs not only as elimination patterns but also inference patterns. Clearly, as such, these inferences would have to be provable and stated as mini-theorems for example as in < this post >. In the course of a formal solution we would then cite the pattern being used and pull out the relevant inference.

Included in those inferences is the weak inference between two instances of a digit in the target cells that becomes conjugate when the digit is known to occupy in the base cells that you mentioned.

On May 9th When I wrote the piece about "Partial Fish" I asked for suggestions for convenient and memorable terms then, but got no response.

I apologize for not recalling these posts. Some posts I read, some I don't.
Any answer to the question: have people used the weak inferences to build chain-based inferences ?

Regarding your peace making efforts, I thank you kindly, but as Einstein said "Insanity is doing the same thing over and over again and expecting different results" or as my mother put it "A leopard doesn't change its spots". So I'm sorry but I'm right out of slack.

Any member of the forum can choose to post here, and I can choose to respond to them or not. I won't guarantee to respond to anyone who is disrespectful or insulting to me, is unwilling to admit it when they are wrong, or only responds sometimes. I also take a poor view of those who ask questions and can't be bothered to say thanks when they get answers.

I apologise to those that feel they are the ham in the middle of a sandwich as a result of this, as I have done in other cases.

I must say (no disrespect intended -- quite the contrary, in fact), that I'm surprised at your (apparent) attitude. After exchanging several PM's with you (over the course of time), the best I expected was a humble apology to ronk, and the worst, was an acknowledgement that he had asked a reasonable question in a respectful manner.
With that said, I'll step back.

A little while back you announced that you had had enough of JEs and were turning off.

I haven't lost interest completely. I've been looking into exocets (not necessarily JExocets) with AHS targets. Some of the things that I said about them (about most of them being "worthless"), were untrue. I've been continuing to look into them, and waiting for an opportune moment to comment further. (Not ready to comment yet, though). Some of Leren's recent posts, have touched on a large percentage of the cases from champagne's "potential hardest" list -- cases where one of Danny's "secondary equivalences" has the "2nd cell" replaced by a single digit AHS (usually a single digit). Another "large percentage" of the cases (the largest, and first for me to filter out), are cases where applying the eliminations for a standard exocet, leaves a hidden set for non-base digit candidates. After that, seem to be a lot of worthless "large AHS" cases, a lot of "twin" (J)exocets, and a lot of "double twins"/"twin twins" ... don't know what to call them, and a lot of cases that are only justified via "obscure logic", unrelated to JExocets. I haven't seen any cases except for "double twins" targets for the same AHS digit ... where a row AHS is (or can be) a target for JExocet with S cells in columns. I do remember that you asked about them, after a comment that I made.

Best Regards,
Blue.

[blue]

Hi Danny,

Code: Select all

 +--------------------------------------------------------------------------------+
 |  9       8       234     |  7       36      1234    |  1236    125     12356   |
 |  1237    127     6       |  138     5       1238    |  4       12789   1239    |
 |  12347   5       2347    |  13468   368     9       |  12367   1278    1236    |
 |--------------------------+--------------------------+--------------------------|
 |  8       1279    2479    |  14569   679     147     |  1269    3       12569   |
 |  1246    3       249     |  145689  689     148     |  1269    12459   7       |
 |  1467    179     5       |  13469   2       1347    |  8       149     169     |
 |--------------------------+--------------------------+--------------------------|
 |  5       6       23789   |  389    Q1       378     | R39-27  q79-2    4       |
 |  237     4       23789   | r39-8   Q79-38   6       | R5       1279    1239    |
 | B37     B79      1       |  2       4       5       |  379     6       8       |
 +--------------------------------------------------------------------------------+
 # 147 eliminations remain

 ### -379- QExocet   Base = r9c12   Target = r8c5==r7c8,r7c7==r8c4


Code: Select all

*) Perform JExocet eliminations ala DPB's original definition.

*)  =>  r7c7<>2, r8c5<>8


*) Determine if secondary equivalences exist and eliminate additional camdidates.

*) r7c7==r8c4  =>  39 common base candidates  =>  r7c7<>7, r8c4<>8
*) r8c5==r7c8  =>  79 common base candidates  =>  r7c8<>2, r8c5<>3


I understand the eliminations for 2r7c8 and 8r8c4, but not the ones for 7r7c7 and 3r8c5.
No doubt, I'm missing something obvoius. Can you explain in excruciating detail ?

Ahh, wait -- I see it ... "39 common base candidates", etc.
Thank you !

Blue.

[daj95376]

[Edit: withdrawn]

[champagne]

I haven't lost interest completely. I've been looking into exocets (not necessarily JExocets) with AHS targets. Some of the things that I said about them (about most of them being "worthless"), were untrue. I've been continuing to look into them, and waiting for an opportune moment to comment further. (Not ready to comment yet, though). Some of Leren's recent posts, have touched on a large percentage of the cases from champagne's "potential hardest" list -- cases where one of Danny's "secondary equivalences" has the "2nd cell" replaced by a single digit AHS (usually a single digit). Another "large percentage" of the cases (the largest, and first for me to filter out), are cases where applying the eliminations for a standard exocet, leaves a hidden set for non-base digit candidates. After that, seem to be a lot of worthless "large AHS" cases, a lot of "twin" (J)exocets, and a lot of "double twins"/"twin twins" ... don't know what to call them, and a lot of cases that are only justified via "obscure logic", unrelated to JExocets. I haven't seen any cases except for "double twins" targets for the same AHS digit ... where a row AHS is (or can be) a target for JExocet with S cells in columns. I do remember that you asked about them, after a comment that I made.

Best Regards,
Blue.

Hi Blue,

before restarting the run in my grey and green files (whatever can be the cut off) I tried to integrate the last findings to be sure to catch all puzzles of interest (but to discard as many as possible all puzzles having no attractive value).

I went through your posts related to box extension, but if I see the elimination logic, I had some difficulty to design an identification process. As a result, I postponed that topic.

If you have now a synthetic view, easy to use for such a design, it would be grate.

[Leren]

BTW do you have an example of a JE with an orthogonal lines digit that is opposite a target cell, I'm keen to try out that move but haven't found one yet.

Sorry I'm afraid not.

David

David, I found an example in this puzzle Blue posted in this thread on p 15 of this thread on 04/06/2013 (in a list of 129 grey area puzzles).

......7....2..6.54.5.....12...67.....2..8...76.........1....4...8..5.....753..2.8

Code: Select all

                         X
 *--------------------------------------------------------------*
 | 139   6     1349   | 259   2349  2359   | 7     8     39     |
X| 78    39    2      | 78    1     6      | 39    5     4      |X
 | 3789  5     3489   | 789   349   3789   | 6     1     2      |
 |--------------------+--------------------+--------------------|
 | 13589 349   1389   | 6     7     239    |T359   24    139    |
 |B1359  2    B139    | 149   8     349    | 359   6     7      |
 | 6     349   7      |T15-9  239   2359   | 8     24    139    |
 |--------------------+--------------------+--------------------|
 | 239   1     6      | 2789  29    2789   | 4     37    5      |
 | 239   8     39     | 2479  5     2479   | 1     37    6      |
 | 4     7     5      | 3     6     1      | 2     9     8      |
 *--------------------------------------------------------------*
                         X


The Exocet is r5c1;r5c3;r4c7;r6c4;1359

The PM shows the position following -2 r4c3. Base digit 9 can be covered in the S cells by two orthogonal lines (Row 2 and Column 4).
This arrangement of 9's in the S cells => if 9 occupies a Base cell it can't occupy Target cell r4c3. Thus r4c3 <> 4.

This extra elimination is enough to crack the puzzle - a summary of the solution being:

Exocet r5c1;r5c3;r4c7;r6c4;1359

r4c3 <> 2 - non Base digit in Target
r4c3 <> 9 - described above
Secondary equivalence r4c3==r6c9, r6c7 => r4c3 <> 5, r6c9 <> 3, 9
Singles + XWing (2) c14 r78
Singles + Skyscraper (9) r26 c79; stte

Leren

[daj95376]

Since I couldn't follow Leren's logic in his latest example, I decided to look for eliminations on my own.

Code: Select all

 +-----------------------------------------------------------------------+
 |  139    6      1349   |  259    2349   2359   |  7      8      39     |
 |  78     39     2      |  78     1      6      |  39     5      4      |
 |  3789   5      3489   |  789    349    3789   |  6      1      2      |
 |-----------------------+-----------------------+-----------------------|
 |  3589-1 349    389-1  | Q6      7      239    | R359    24     139    |
 | B1359   2     B139    |  49-1   8      349    |  359    6      7      |
 |  6      349    7      | Q1259   239    2359   | R8      24     139    |
 |-----------------------+-----------------------+-----------------------|
 |  239    1      6      |  2789   29     2789   |  4      37     5      |
 |  239    8      39     |  2479   5      2479   |  1      37     6      |
 |  4      7      5      |  3      6      1      |  2      9      8      |
 +-----------------------------------------------------------------------+
 # 91 eliminations remain

 (1)r5c13                 -  r5 c4 = (1)r6c4
 (3)r5c13* - r46c2 = r2c2 - *r52c7 = (3)r4c7
 (5)r5c13                 -  r5 c7 = (5)r4c7
 (9)r5c13* - r46c2 = r2c2 - *r52c7 = (9)r4c7


Since any pairing combination of <359> in BB forces a contradiction in r4c7, then we must have one of r5c13=1 true and r4c13,r5c4<>1.

[Leren]

I was being a bit terse with my wording of that last move because David P Bird and I had discussed it before we had
found an example.

So more formally: Suppose either of r5c13 = 9 then JE => exactly one of r4c7, r6c4 = 9 and one of them <> 9.
Suppose r4c7 <> 9. Then we have (9) r45c7 = r2c7 - r2c2 = r46c2 which contradicts r5c13 = 9.
Thus r5c13 = 9 => r4c7 = 9 and r6c4 <> 9. But also r5c13 <> 9 = JE => both r4c7, r6c4 <> 9. Common outcome r4c7 <> 9.

Code: Select all

Single JExocet

B B . | . x . | . . .   B = base cells 
. . . | . \ . | T . .   T = target cell
. . . | . # . | \ . .   # = fin target cell
------+-------+------     
. . \ | . O . | \ . .   
x x O | x x x | O x x <   
. . \ | . O . | \ . .   O = occupied (maximum)
------+-------+------   \ = excluded
. . \ | . O . | \ . .   x = fin cells
. . \ | . O . | \ . .     
. . \ | . O . | \ . .

David had shown me the above diagram that details eliminations (x's and #) that can be made for a JE digit
(that can be covered by 2 orthogonal lines in the S cells) when it has been established that it must occupy one of the JE Base cells.

The # elimination can be made whether or not the digit occupies a Base cell.

Leren

[Leren]

Here is an even more interesting case of the same pattern a few puzzles further on in the same collection.

.5..3.1...31..5.72...7..........46...23.91.4...........123....8.97.......6....219; JExocet r1c6;r3c6;r5c4;r8c5;268

Code: Select all

                                                           X
 *-----------------------------------------------------------------------*
 | 46     2      3456    | 135    7      15      | 8      3456   9       |
 | 689    7      35689   | 23589  4      2589    | 56     356    1       |
 | 1      489    34589   | 6      58     589     | 7      345    2       |
 |-----------------------+-----------------------+-----------------------|
 |B268    5     B68      | 4      1      3       | 9      268    7       |
 | 3      468    1       | 2578   9      2567    | 456   T28-6   458     |
 | 24689  4689   7       | 258   T68     2568    | 3      1      458     |
 |-----------------------+-----------------------+-----------------------|
 | 4689   1      4689    | 589    3      45689   | 2      7      458     |
 | 5      3      48      | 78     2      478     | 1      9      6       |
X| 7      4689   2       | 189    568    14589-6 | 45     58     3       |X
 *-----------------------------------------------------------------------*
                                                          X

The digit of interest is 6 and the orthogonal cover lines are Row 9 and Column 8.

A similar pattern is established as in the previous example leading to Target cell r5c8 <> 6

An added feature of this case is that r4c5 <> 6 (this is r1c7 in David''s exemplar diagram in my previous post) ie Target Cell r5c6 has the only 6 in its mini-column.

When this is the case, the digit of interest can be eliminated from the x positions (in Row 5 of the exemplar diagram) ie in Row 9 in the above PM - leading to r9c6 <> 6.

The reason for this is that if Target Cell r6c5 <> 6 then r9c5 = 6. If r6c5 = 6 then the JE => r4c13 = 6 => r56c2 <> 6 => r9c2 = 6. Thus all 6's in Row 9 except for
Columns 2 and 5 can be eliminated (in the exemplar diagram the digit of interest can be eliminated from all positions in Row 5 except for the 2 O positions).

I'll certainly be doing more testing to see how often this and the previous situation occurs. Congratulations David for thinking up this move !

Leren

PS Oops, on re-reading this post I realise that I'd rotated the original puzzle 90 deg anticlockwise so that the JExocet base cells were aligned in a row
and the pattern would read in the same way as David's exemplar diagram.

The actual puzzle that corresponds to the PM is: .2....8.9.7..4...11..6....2.5.41....3...9......7...3...1..3.27.53..2.196.........; JExocet r4c1; r4c3; r6c5; r5c8; 268

Leren

[champagne]

just to feed the system, another group of puzzles with different properties

98.7..6....5.4..3......9...8..6....2.7...2.....1.8..5.14.....8.....3...1......5.3
98.7..6....5.9........8..4.83....7....2..7.6....3....272...83....6.........2....1
98.7..6....5.9..4......3..98.42.....7.2...1...1...6....2.1..7......7...5.....4.3.
98.7..6....5.9..4......6...53.....9.....3..........2.41..8......9...1..7..3.4..5.

[champagne]

and still different

98.7..6..7..5..4...56.3....5..8..9...6...2.....8..7.1..7.6...94....857...........
98.7..6..5..9..4....7.3....4..5..8...7.....2...8..1....6..597.....6...48........5
98.7..6..7..........5.64...3.....9....86...4......2..5..94..56...75...8.....1...9
98.7..6..7..5..4....5.3....4..9..8...5.....2...8..1....6..975.....6...84........7
98.7..6..7..8..59...4.....38..3..9...2.....31....1....3...86.5..9.5..........38..
98.7..6..7..8..9...56.4....3..5..2...6...9.....5.3..1..7.6...29....785...........
98.7..6..7..8..9...56.4....8..5..3...6...9.....5.3..2..7.6...19....785...........
98.7..6..7.5....4..6......85.6.8..3....5........3....94.76...8..2..71.......4.1..
98.7..6..7.5....4..6......95.6.9..3....5........3....24.76...9..1..72.......4.3..
98.7..6..7.6.5..9...5.4....63.5....9.....4.6.......58.37.6...5...2..........1...7
98.7..6..5..9..4....7....3.6..5..8...7......2..8.1.....4..597.....4...86.......5.
98.7..6..5..9..7....4....3.6..5..4...48.2.........4..1.9.4...76....589...........
98.7..6..7...5..8...4....7.36.2...9...7...2.......13..23.8.......6..7.......6.9..
98.7..6..7..5..94.....3...88...75.9..649..............6..8..5...2.....84.....1...
98.76.5..7...4.9....3..8...6..97.4.....5.3..........2.1..4...56.7....1......1...4
98.7..6..7...5......4..8.3.4....975..7....2....5.8...1.4...5.8....1....6....2.3..
98.7..6..5.4.6..3......8...62...57....8....1.........317..5.9...4.9........8.7..2