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Vanhegan extreme - recycling "useless" wings

 
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nataraj



Joined: 03 Aug 2007
Posts: 1048
Location: near Vienna, Austria
PostPosted: Thu Aug 07, 2008 8:07 am    Post subject: Vanhegan extreme - recycling "useless" wings Reply with quote
Inspired by the recent discussion about using UR and/or finned wings as strong links in difficult puzzles, I tried to use the same approach with those otherwise "useless" xy- w- and m-wings.

I took today's Vanegan "Extreme"

Code:

+-----------+-----------+-----------+
|  .  8  .  |  9  .  4  |  3  1  .  |
|  .  .  .  |  2  .  .  |  .  .  8  |
|  5  .  9  |  8  .  .  |  7  .  4  |
+-----------+-----------+-----------+
|  .  .  8  |  .  4  9  |  .  .  .  |
|  9  .  .  |  7  .  5  |  .  .  6  |
|  .  6  .  |  1  3  .  |  8  .  .  |
+-----------+-----------+-----------+
|  6  .  1  |  .  .  8  |  2  .  7  |
|  8  .  .  |  .  .  1  |  .  .  .  |
|  .  9  4  |  5  .  7  |  .  8  .  |
+-----------+-----------+-----------+


play online

and, after some coloring (2 and 5),
xy-wing(s) 23-27-37,
w-wing (3,4) r5c8=r3c4 via strong link on 4 in col 7)
(maybe one other wing, not sure - sorry)

arrive at this position,
Code:

+--------------------------+--------------------------+--------------------------+
| 27      8       6        | 9       57      4        | 3       1       25       |
| 14      14      37       | 2       57      36       | 569     569     8        |
| 5       23      9        | 8       1       36       | 7       26      4        |
+--------------------------+--------------------------+--------------------------+
| 137     157     8        | 6       4       9        | 15      2357    1235     |
| 9       124     23       | 7       8       5        | 14#     34*     6        |
| 47      6       57       | 1       3       2        | 8       479     59       |
+--------------------------+--------------------------+--------------------------+
| 6       35      1        | 34      9       8        | 2       345     7        |
| 8       357     257      | 34*     26      1        | 4569#   4569    359      |
| 23      9       4        | 5       26      7        | 16      8       13       |
+--------------------------+--------------------------+--------------------------+




where it's either xy-chains or ...

maybe we can re-cycle one of those "useless" wings?

Remember the w-wing I mentioned (marked *-#=#-* in the grid above), the one that got rid of the 3 in r8c8?

It is now "useless" in the sense that it does not directly eliminate any candidates
But it still provides a nice strong link: (3)r5c8=r8c4

Luckily, I still have my little diagrams which picture strong links in single candidates. Let's pencil this new link in:


and, indeed, the additional strong link now provides one more link in the coloring chain and a new elimination is possible:

3 in r8c9 sees both r8c4 and r9c9 (connected via r9c1,r4c1,r5c3,r5c8)

All I had to do in order to find this elimination was to re-use an existing result (the w-wing) and draw a single line into an existing diagram. No new search, no re-thinking ... only one little step:

every time there is a "useless" wing, check to see if it provides a new strong link and, if yes, add it to the diagram.

Edit 1817 GMT+2: I included another "elimination" in the drawing, but that one is not valid:

3 in r7c2 "sees" both r9c1 and r7c4 (and those are connected via ...)

BUT THIS ELIMINATION IS NOT VALID! There is no ALTERNATING sequence of weak and strong links...

the link betweenr5c8 and r8c4 is a strong link that (as I see it) canNOT stand in for a weak link as usual!

In a row, if there are only two cells with a given candidate X, both statements are true at the same time:

a) at least one of the two cells must be X (definition of a strong link)
b) at most one of the two cells can contain X (weak link)

Whereas with the two pincers of an w-wing, for example, only (a) is guaranteed, not necessarily (b). In our grid, if r7c4<>3 then r8c4=3, but the w-wing will not prove that then r5c8<>3.

--------

BTW, there are tons of useless wings in this puzzle, but most of them duplicate strong links already found by coloring.

(edited to correct error: using strong link for a weak link is not valid in the general case.)
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Asellus



Joined: 05 Jun 2007
Posts: 865
Location: Sonoma County, CA, USA
PostPosted: Thu Aug 07, 2008 7:20 pm    Post subject: Reply with quote
nataraj wrote:
and, after some coloring (2 and 5),
xy-wing(s) 23-27-37,
w-wing (3,4) r5c8=r3c4 via strong link on 4 in col 7)
(maybe one other wing, not sure - sorry)

I believe you also used a couple of 23 bivalue M-Wings in the left tower: one exploiting the c1 conjugate <2>s that eliminates <3> from r45c2 and another exploiting the b4 conjugate <3>s that eliminates <2> from r8c2.

As for the invalid coloring elimination, it is indeed important to remember that including a strong only inference link requires multi-coloring techniques just as does including a weak only inference link.
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nataraj



Joined: 03 Aug 2007
Posts: 1048
Location: near Vienna, Austria
PostPosted: Thu Aug 07, 2008 7:52 pm    Post subject: Reply with quote
Asellus wrote:
I believe you also used a couple of 23 bivalue M-Wings in the left tower: one exploiting the c1 conjugate <2>s that eliminates <3> from r45c2 and another exploiting the b4 conjugate <3>s that eliminates <2> from r8c2.


I did. Spot on!
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