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Another very hard one for P&P solvers
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ravel



Joined: 21 Apr 2006
Posts: 536

PostPosted: Thu Jul 19, 2007 6:47 pm    Post subject: Another very hard one for P&P solvers Reply with quote

TTHsieh posted this puzzle on the other forum.
Code:
 +-----------------------+
 | 1 . . | . . . | . . 9 |
 | . . . | 3 . 8 | . . . |
 | . . 4 | . . . | 1 . . |
 | - - - + - - - + - - - |
 | . 8 . | . 2 . | . 3 . |
 | . . . | 6 . 1 | . . . |
 | . 3 . | . 9 . | . 7 . |
 | - - - + - - - + - - - |
 | . . 8 | . . . | 6 . . |
 | . . . | 7 . 5 | . . . |
 | 9 . . | . . . | . . 4 |
 +-----------------------+

Take your time. It can be solved with x-, xy-, W-wing, UR and strong links (also 3 or Color wing).
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Earl



Joined: 30 May 2007
Posts: 677
Location: Victoria, KS

PostPosted: Fri Jul 20, 2007 12:51 am    Post subject: VH for P & P Reply with quote

I got this far, but I do not see the next step.

Any help out there?

Earl

Code:

+------------------+------------+------------------+
| 1    2567  3     | 4   57 26  | 2578  2568  9    |
| 2567 25679 25679 | 3   1  8   | 2457  2456  257  |
| 8    2567  4     | 29  57 269 | 1     256   3    |
+------------------+------------+------------------+
| 46   8     169   | 5   2  7   | 49    3     16   |
| 2457 24579 2579  | 6   3  1   | 24589 24589 258  |
| 256  3     1256  | 8   9  4   | 25    7     1256 |
+------------------+------------+------------------+
| 2357 1257  8     | 129 4  239 | 6     125   257  |
| 34   14    26    | 7   68 5   | 39    19    28   |
| 9    12567 2567  | 12  68 23  | 23578 1258  4    |
+------------------+------------+------------------+

Play this puzzle online at the Daily Sudoku site
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Earl



Joined: 30 May 2007
Posts: 677
Location: Victoria, KS

PostPosted: Fri Jul 20, 2007 2:17 am    Post subject: Tough PuzzleP & P Solvers Reply with quote

I have made some progress.
But still need help.

Earl

Code:

+---------------+------------+---------------+
| 1    257   3  | 4   57 26  | 2578 2568 9   |
| 257  9     6  | 3   1  8   | 257  2456 25  |
| 8    257   4  | 29  57 269 | 1    256  3   |
+---------------+------------+---------------+
| 6    8     9  | 5   2  7   | 4    3    1   |
| 2457 2457  57 | 6   3  1   | 2589 2589 258 |
| 25   3     1  | 8   9  4   | 25   7    6   |
+---------------+------------+---------------+
| 235  125   8  | 129 4  239 | 6    125  7   |
| 34   14    2  | 7   68 5   | 39   19   28  |
| 9    12567 57 | 12  68 23  | 2358 1258 4   |
+---------------+------------+---------------+

Play this puzzle online at the Daily Sudoku site
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Marty R.



Joined: 12 Feb 2006
Posts: 5770
Location: Rochester, NY, USA

PostPosted: Fri Jul 20, 2007 4:03 am    Post subject: Reply with quote

Ravel, this was pretty interesting, very similar to the one you posted two weeks ago, in that a lot of techniques were required and each gave up only a very little ground. I used seven different techniques, one of them twice and another three times, for a total of 10 moves needed in order to bust the puzzle.
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jLo



Joined: 30 Apr 2007
Posts: 55

PostPosted: Fri Jul 20, 2007 4:17 am    Post subject: Reply with quote

Quote:

Take your time. It can be solved with x-, xy-, W-wing, UR and strong links (also 3 or Color wing).


One xy-wing and one UR did it for me.

From Earl's position -

Code:

+---------------+------------+---------------+
| 1 257 3 | 4 57 26 | 2578 2568 9 |
| 257 9 6 | 3 1 8 | 257 2456 25 |
| 8 257 4 | 29 57 269 | 1 256 3 |
+---------------+------------+---------------+
| 6 8 9 | 5 2 7 | 4 3 1 |
| 2457 2457 57 | 6 3 1 | 2589 2589 258 |
| 25 3 1 | 8 9 4 | 25 7 6 |
+---------------+------------+---------------+
| 235 125 8 | 129 4 239 | 6 125 7 |
| 34 14 2 | 7 68 5 | 39 19 28 |
| 9 12567 57 | 12 68 23 | 2358 1258 4 |
+---------------+------------+---------------+


I see a single elimination at R8C9 that should really open
things up. The UR that I used has already been applied.
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jLo



Joined: 30 Apr 2007
Posts: 55

PostPosted: Fri Jul 20, 2007 4:45 am    Post subject: Reply with quote

Quote:


One xy-wing and one UR did it for me.




Oops, my UR was not quite valid.

I still think it is nothing but singles from Earl's position though.
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ravel



Joined: 21 Apr 2006
Posts: 536

PostPosted: Fri Jul 20, 2007 10:28 am    Post subject: Reply with quote

This is, what i had:
Code:
 *----------------------------------------------------------*
 | 1     2567   3      | 4    57  26   | 2578   2568   9    |
 | 2567  25679  25679  | 3    1   8    | 2457   2456   257  |
 | 8     2567   4      | 29   57  269  | 1      256    3    |
 |---------------------+---------------+--------------------|
 |-46    8      169    | 5    2   7    |#49     3      16   |
 | 2457  24579  2579   | 6    3   1    | 24589  24589  258  |
 | 256   3      1256   | 8    9   4    | 25     7      16   |
 |---------------------+---------------+--------------------|
 | 2357  1257   8      | 129  4   239  | 6      125    257  |
 |#34    14     26     | 7    68  5    |#39     19     28   |
 | 9     12567  2567   | 12   68  23   | 23578  1258   4    |
 *----------------------------------------------------------*
xy-wing
 *----------------------------------------------------*
 | 1    U257   3   | 4   U57  26   | 2578  2568  9    |
 |-257   9     6   | 3    1   8    | 257   4     257  |
 | 8    U257   4   | 29  U57  269  | 1     256   3    |
 |-----------------+---------------+------------------|
 | 6     8     9   | 5    2   7    | 4     3     1    |
 | 2457 -2457 #57  | 6    3   1    | 89    89    25   |
 |#25    3     1   | 8    9   4    |#25    7     6    |
 |-----------------+---------------+------------------|
 | 357   157   8   | 129  4   239  | 6     125   257  |
 | 34    14    2   | 7    6   5    | 39    19    8    |
 | 9     6    #57  | 12   8   23   | 23-57 125   4    |
 *----------------------------------------------------*
UR (U), kite (#)
 *-------------------------------------------------*
 | 1     2-57 3   | 4   @57  26   | 578  568  9    |
 |@57    9    6   | 3    1   8    | 257  4    257  |
 | 8    @257  4   | 29  @57  269  | 1    56   3    |
 |----------------+---------------+----------------|
 | 6     8    9   | 5    2   7    | 4    3    1    |
 | 2457  457 #57  | 6    3   1    | 89   89   2-5  |
 | 25    3    1   | 8    9   4    | 25   7    6    |
 |----------------+---------------+----------------|
 | 357   157  8   | 129  4   239  | 6    125 #57   |
 | 34    14   2   | 7    6   5    | 39   19   8    |
 | 9     6   #57  | 12   8   23   |#37   125  4    |
 *-------------------------------------------------*
W-wings
 *----------------------------------------------*
 | 1    27   3   | 4    57  26   | 78  568  9   |
 |#57   9    6   | 3    1   8    | 2   4   #57  |
 | 8   #257  4   | 29   57  269  | 1   56   3   |
 |---------------+---------------+--------------|
 | 6    8    9   | 5    2   7    | 4   3    1   |
 | 457  457  57  | 6    3   1    | 89  89   2   |
 | 2    3    1   | 8    9   4    | 5   7    6   |
 |---------------+---------------+--------------|
 | 357  1-57 8   | 129  4   239  | 6   125 #57  |
 | 34   14   2   | 7    6   5    | 39  19   8   |
 | 9    6    57  | 12   8   23   | 37  125  4   |
 *----------------------------------------------*
kite
 *----------------------------------------------*
 | 1    27   3   | 4    57  26   | 78  568  9   |
 |#57   9    6   | 3    1   8    | 2   4   #57  |
 | 8    257  4   | 29   57  269  | 1   56   3   |
 |---------------+---------------+--------------|
 | 6    8    9   | 5    2   7    | 4   3    1   |
 | 45-7 457 #57  | 6    3   1    | 89  89   2   |
 | 2    3    1   | 8    9   4    | 5   7    6   |
 |---------------+---------------+--------------|
 | 357  17   8   | 129  4   239  | 6   125 #57  |
 | 34   14   2   | 7    6   5    | 39  19   8   |
 | 9    6   #57  | 12   8   23   |#37  125  4   |
 *----------------------------------------------*
Color wing (3 strong links)
 *----------------------------------------------*
 | 1    27   3   | 4    57  26   | 78  568  9   |
 |#57   9    6   | 3    1   8    | 2   4   #57  |
 | 8    257  4   | 29   57  269  | 1   56   3   |
 |---------------+---------------+--------------|
 | 6    8    9   | 5    2   7    | 4   3    1   |
 | 45   457  57  | 6    3   1    | 89  89   2   |
 | 2    3    1   | 8    9   4    | 5   7    6   |
 |---------------+---------------+--------------|
 |#357  1-7  8   | 129  4   239  | 6   125 #57  |
 | 34   14   2   | 7    6   5    | 39  19   8   |
 | 9    6    57  | 12   8   23   | 37  125  4   |
 *----------------------------------------------*
x-wing
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Mogulmeister



Joined: 03 May 2007
Posts: 1151

PostPosted: Fri Jul 20, 2007 10:41 am    Post subject: Reply with quote

Very nice puzzle.
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nataraj



Joined: 03 Aug 2007
Posts: 1048
Location: near Vienna, Austria

PostPosted: Tue Aug 14, 2007 6:39 pm    Post subject: Reply with quote

ravel,

could you please explain how your second W-wing works? I get the one (@) all right, but spent a long time to make something out of the 57s in the lower right corner, without success. I am referring to your diagram:
Code:
*-------------------------------------------------*
 | 1     2-57 3   | 4   @57  26   | 578  568  9    |
 |@57    9    6   | 3    1   8    | 257  4    257  |
 | 8    @257  4   | 29  @57  269  | 1    56   3    |
 |----------------+---------------+----------------|
 | 6     8    9   | 5    2   7    | 4    3    1    |
 | 2457  457 #57  | 6    3   1    | 89   89   2-5  |
 | 25    3    1   | 8    9   4    | 25   7    6    |
 |----------------+---------------+----------------|
 | 357   157  8   | 129  4   239  | 6    125 #57   |
 | 34    14   2   | 7    6   5    | 39   19   8    |
 | 9     6   #57  | 12   8   23   |#37   125  4    |
 *-------------------------------------------------*
W-wings

As I understand W-wing, you are trying to eliminate 5 from r5c9. In order to do this, there has to be a chain starting at one of the pincers with '7' and ending with '5' at the other pincer.

Like the other one: if r1c5 is 5 => r1c2 cannot be 5
If r1c5=7, then r3c5=5 then r3c2=7 then r2c1=5 => r1c2 cannot be 5.

I just don't see it. After the W-wing top left, I had to resort to three rather long x-cycles, an xy-wing and xy-chains to come to the solution.

very tough puzzle indeed.
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nataraj



Joined: 03 Aug 2007
Posts: 1048
Location: near Vienna, Austria

PostPosted: Tue Aug 14, 2007 6:44 pm    Post subject: Reply with quote

on second thought, I think I see how it works:

it is sufficient to show that r5c3 and r7c9 cannot BOTH be 7.

so lets assume both are 7.
Then r9c3=5, r9c7=3 => r9c6=2 => r9c4=1 and no possible value for r9c8.
Therefore at least one of the two pincers must be 5 and we can remove 5 from r5c9.

Neat.
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Asellus



Joined: 05 Jun 2007
Posts: 865
Location: Sonoma County, CA, USA

PostPosted: Wed Aug 15, 2007 12:06 am    Post subject: Reply with quote

Many different routes. Mine:

1. XY Wing (same as Ravel's)
2. Sashimi Swordfish: -7 in R7C2
3. Color Wing (or ER or XY Chain) in C3,R5,B6: -5 in R9C7
4. Finned X-Wing: -5 in R1C7
5. 29 UR in R37C46: -2 in R7C6 (involves 9 X-Wing)
6. Finned X-Wing: -2 in R1C7 ("copy" of step 4!)
7. XY Chain from R1C7 to R9C3 (via C7,R8,B7): -7 in R9C7

I didn't determine whether or not every step was necessary.

Note: I use "Color Wing" as defined by Ruud. What Ravel calls a "Color Wing" above, Ruud calls a "Color Trap." More naming confusion!
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TKiel



Joined: 22 Feb 2006
Posts: 292
Location: Kalamazoo, MI

PostPosted: Wed Aug 15, 2007 2:27 pm    Post subject: Reply with quote

I reached this point, at which Simple Sudoku had no more hints, which means there are no locked candidates, naked/hidden subsets, fish of any size, XY-wings or simple/multi coloring chains. (It verifies that the puzzle has only one solution, so it must use techniques other than what I've listed, but it doesn't give them as hints.)

Code:
 
 *-----------------------------------------------------------*
 | 1     257   3     | 4     57    26    | 78    2568  9     |
 | 257   9     6     | 3     1     8     | 257   4     257   |
 | 8     257   4     | 29    57    269   | 1     56    3     |
 |-------------------+-------------------+-------------------|
 | 6     8     9     | 5     2     7     | 4     3     1     |
 | 2457  2457  57    | 6     3     1     | 89    89    25    |
 | 25    3     1     | 8     9     4     | 25    7     6     |
 |-------------------+-------------------+-------------------|
 | 357   157   8     | 129   4     39    | 6     125   257   |
 | 34    14    2     | 7     6     5     | 39    19    8     |
 | 9     6     57    | 12    8     23    | 237   125   4     |
 *-----------------------------------------------------------*


In focusing on the 5's, I noticed this possibility, because I've been trying to understand (and more importantly, use) 'grouped' single-digit (simple) coloring and this seemed like an opportunity. I guess I'm wondering if excluding the 5 at # is logical and if it would be considered grouped coloring.

Code:

 *-----------------------------------------------------*
 |       5         |       5         |       5B-       |
 | 5#              |                 | 5A-         5+  |
 |       5         |       5         |       5B-       |
 |-----------------+-----------------+-----------------|
 |                 |                 |                 |
 | 5     5     5B+ |                 |             5   |
 | 5A-             |                 | 5A+             |
 |-----------------+-----------------+-----------------|
 | 5     5         |                 |       5     5   |
 |                 |                 |                 |
 |             5B- |                 |       5B+       |
 *-----------------------------------------------------*
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TKiel



Joined: 22 Feb 2006
Posts: 292
Location: Kalamazoo, MI

PostPosted: Wed Aug 15, 2007 2:41 pm    Post subject: Reply with quote

After posting the above and looking at it some more, it seems that the letters I used to distinguish the different chains can be junked and the whole thing considered one grouped coloring chain, which basically makes this what Asellus would call a 'Color Wrap', in that C1 has two (-) cells. Therefore, no cells marked with (-) contain 5.

Code:

 *-----------------------------------------------------*
 |       5         |       5         |       5-        |
 | 5-              |                 | 5-          5+  |
 |       5         |       5         |       5-        |
 |-----------------+-----------------+-----------------|
 |                 |                 |                 |
 | 5     5     5+  |                 |             5   |
 | 5-              |                 | 5+              |
 |-----------------+-----------------+-----------------|
 | 5     5         |                 |       5     5   |
 |                 |                 |                 |
 |             5-  |                 |       5+        |
 *-----------------------------------------------------*
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Asellus



Joined: 05 Jun 2007
Posts: 865
Location: Sonoma County, CA, USA

PostPosted: Wed Aug 15, 2007 6:33 pm    Post subject: Reply with quote

Tracy,

As I see it, there are only two color clusters on 5, which, following your approach, I've labelled A and B:
Code:
*-----------------------------------------------------*
|       5         |       5         |       5         |
| 5#              |                 | 5AR         5   |
|       5         |       5         |       5         |
|-----------------+-----------------+-----------------|
|                 |                 |                 |
| 5     5     5Br |                 |             5AR |
| 5AR             |                 | 5AG             |
|-----------------+-----------------+-----------------|
| 5     5         |                 |       5     5   |
|                 |                 |                 |
|             5Bg |                 |       5Br       |
*-----------------------------------------------------*

There are only two weak links (or "Bridges") between these two clusters: one in Box 4 and one in R5. I'll focus only on the one in Box 4. Assigning "red" (r and R) to the weak link cells produces the results shown. Eliminations can only occur in cells that can "see" g and G, and there are no such cells containing <5> candidates.

One must be careful of weak links in coloring. It is always safest to consider only a single weak link serving as a "Bridge" between two "Clusters."

(I always assign matching polarities to the Bridge cells even if I have to reverse the coloring on one cluster. For me, it's too easy to make a mistake otherwise. However, it's not a necessity. Your "A- and B+" weak link in Box 4 means eliminations occur between A+ and B-. For me, that's a little trickier to see.)

There should be no weak links within a cluster. That means that your two-cell B conjugate in Box 3 is not valid due to the <5> at R7C8. (The link to R9C8 is weak.)

Considered as a single implication chain, it violates the requirement for alternate strong links:
[5]R2C1-[5=5]R26C7-[5]R6C1=[5]R5C3...
It is apparent already at the Box 4 link that there is trouble, the link is weak, not strong as required for the implication logic.

For a "Color Wrap," it must be a single color cluster containing only strong links. So, no Color Wrap here.

Note: If there were no <5> candidate in R7C8, then your two-cell B- conjugate in Box 3 would be okay. The result, however, would be elimination of <5> at R2C7, since it can "see" this B- conjugate and the A+ cell at R6C7 (the A+ and B- elimination requirement I mentioned above).

I hope this is helpful. Things can get tricky when one attempts to take coloring to creative levels.
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Steve R



Joined: 24 Oct 2005
Posts: 289
Location: Birmingham, England

PostPosted: Wed Aug 15, 2007 6:33 pm    Post subject: Reply with quote

Tracy

I’m afraid I couldn’t follow the eliminations you made in colouring terms. They are certainly right enough. Perhaps someone with a keener eye will do better.

In the meantime it may be helpful to mention one elimination derived from (group) colouring on 5. What I have done is start at r7c9, marked “A.” The conjugate group in box 9 is marked “b” and so on: the alphabetical sequence enumerates the chain and polarity of each group is shown by whether it is upper or lower case.

Code:
 *----------------------------------------------------------------*
 |  1     f257   3     | 4     57    26    |  78    C2568   9     |
 | E257    9     6     | 3     1     8      |d257    4     d257   |
 |  8     f257   4     | 29    57    269   |  1     C56     3     |
 |---------------------+-------------------+----------------------|
 |  6      8     9     | 5     2     7     |  4      3      1     |
 |  2457   2457  57    | 6     3     1     |  89     89     25    |
 |  25     3     1     | 8     9     4     |  25     7      6     |
 |-------------------+---------------------+----------------------|
 |  357    157   8     | 129   4     39    |  6     b125   A257   |
 |  34     14    2     | 7     6     5     |  39     19     8     |
 |  9      6     57    | 12    8     23    |  237   b125    4     |
 *----------------------------------------------------------------*

As r7c2 is at the same time an associate of all the members of an upper case group (A) and all the members of a lower case group (f), the 5 may be eliminated.

Steve
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Asellus



Joined: 05 Jun 2007
Posts: 865
Location: Sonoma County, CA, USA

PostPosted: Wed Aug 15, 2007 6:37 pm    Post subject: Reply with quote

Steve,

Take a look at my "ER Loop" posting on the Techiques board. [Edit: It's actually on the "Other Puzzles" board.] It may look familiar.
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TKiel



Joined: 22 Feb 2006
Posts: 292
Location: Kalamazoo, MI

PostPosted: Wed Aug 15, 2007 9:47 pm    Post subject: Reply with quote

I think I recognize the flaw in my reasoning. I assumed that since r5c3 & r6c1 were peers, they would end up having opposite polarity, which is true if one of them were +. But both of them could also be -, which throws off the polarity of my chain and negates the exclusion.

Steve,

Thanks. Something like that is exactly what I was looking for.



Asellus wrote:
For a "Color Wrap," it must be a single color cluster containing only strong links. So, no Color Wrap here.


So you mean to tell me there is a separate term for when a coloring chain that includes weak links wraps around on itself?
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Asellus



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PostPosted: Wed Aug 15, 2007 10:13 pm    Post subject: Reply with quote

Tracy wrote:
I think I recognize the flaw in my reasoning. I assumed that since r5c3 & r6c1 were peers, they would end up having opposite polarity

Yes, due to the weak link, they can both be false.
Tracy wrote:
So you mean to tell me there is a separate term for when a coloring chain that includes weak links wraps around on itself?

Well...

If we're talking about two separate strongly-linked color clusters connected with one weak link "Bridge," then I've never seen it happen and am not sure if it's logically possible. If it is, then I suppose it would be a "Color Wing Wrap" (!) in the parlance I use. (The same-house polarity contradiction could only occur between cells of the opposing clusters and of polarities opposite those involved in the Bridge.)

If we're throwing more than two weak links into the coloring array, then I hesitate to make any assertions because the logic rapidly becomes treacherous.

I suspect that it is probably preferrable to work within the implication chain framework, rather than using coloring notions, if one wants to exploit multiple weak links. The rules of implication chains are better suited to such efforts.
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keith



Joined: 19 Sep 2005
Posts: 3355
Location: near Detroit, Michigan, USA

PostPosted: Wed Aug 15, 2007 10:32 pm    Post subject: Reply with quote

Quote:
So you mean to tell me there is a separate term for when a coloring chain that includes weak links wraps around on itself?


Tracy,

Take a look at Sudopedia

http://www.sudopedia.org/wiki/Main_Page

As I understand it, a color wrap is a single coloring chain, let's say

a-A-a-A-a-A-a

all strong links. Then, say you find that the ends of the chain are weakly linked, they are in the same line or box. a must be false, A is true.

A fancy name for one type of conclusion from simple coloring.

Keith
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Asellus



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PostPosted: Wed Aug 15, 2007 10:39 pm    Post subject: Reply with quote

After giving it a bit of thought, I'm pretty sure a "Color Wing Wrap" is not logically possible. Here's the reasoning:

We have two color clusters, A and B, with polarities within each denoted as Aa and Bb. We then posit a weak link somewhere between two cells with polarity "A" and "B" that forms a "bridge" between the clusters.

Assuming all eliminations/contradictions due to either individual cluster alone have already been exploited, the only additional eliminations/contradictions possible are between cells of polarity "a" and "b".

Now, assume that somewhere in the grid an "a" cell and "b" cell share a house. They thus have a weak link (since the clusters are separate there is no strong link). Because they cause eliminations, all other candidates for the subject digit within that house are eliminated. As a result, they now acquire a strong link! There is no "Wrap" contradiction here. Instead, the two separate clusters combine into a single cluster and all "b" cells become "A" cells and all "B" cells become "a" cells.

However, if, as a result of this merger into a single cluster, we discover two "A" cells (or two "a" cells) within a single house, then there is a resulting "Color Wrap." However, we are back to a single strongly-linked cluster.

I don't know if that will make the situation clear; however, I'm fairly confident that the reasoning is sound.
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