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Knotty Puzzle.

 
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wapati



Joined: 10 Jun 2008
Posts: 472
Location: Brampton, Ontario, Canada.

PostPosted: Fri Dec 10, 2010 4:20 pm    Post subject: Knotty Puzzle. Reply with quote

This one has a lot of stuff going on.
I know I have a program that will check for backdoors but I can't find it/remember which. Anyways, Carcul convinced me that most all puzzles may have at least one so good luck all!

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



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

PostPosted: Fri Dec 10, 2010 5:23 pm    Post subject: Re: Knotty Puzzle. Reply with quote

Many steps, but none harder than UR, m-wing or w-wing. Krampus Wink type ...

wapati wrote:
... backdoors ... Carcul convinced me that most all puzzles may have at least one so good luck all!



Hm. Maybe what appears to be a "backdoor" is just one more method that happens to solve the particular puzzle and - in that instance - turns out to be a short and elegant one.

One of our visiting professors at my university once told me (Derflinger's conjecture): "if a problem has a very simple solution, evenually someone will find a very simple, elegant proof". He might have added "even if it takes a couple of centuries" (and the development of a sufficiently powerful theory).

Since all sodokus have a very simple solution (81 numbers in 81 cells), eventually we will develop the theories that'll make the solution path seem elegant... and all "backdoors" will become part of the common arsenal of what we will then call "VH steps" Smile

Be that as it may, in the last couple of years we've come a long way ...
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peterj



Joined: 26 Mar 2010
Posts: 974
Location: London, UK

PostPosted: Fri Dec 10, 2010 5:51 pm    Post subject: Re: Knotty Puzzle. Reply with quote

wapati wrote:
Anyways, Carcul convinced me that most all puzzles may have at least one

In this case, this a backdoor elimination which leads to singles... Exclamation
Quote:
r3c6<>1 (Courtesy of Hoduku)

Unfortunately I have no logical basis for this elimination Sad - yet! Wink
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keith



Joined: 19 Sep 2005
Posts: 3341
Location: near Detroit, Michigan, USA

PostPosted: Fri Dec 10, 2010 9:34 pm    Post subject: Reply with quote

A definition from SudoCue:

Quote:
Backdoor: A candidate which, when placed, leads to the solution without the need for any advanced solving techniques. Every sudoku, no matter how difficult, has a few backdoors. They are the targets for guessing. The best backdoors are those that allow the puzzle to be completed with singles only.


I am not sure I buy this assertion: "Every sudoku ... has a few backdoors."

I seem to recall that Danny went on a mission to generate puzzles without backdoors at one time.

Keith
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daj95376



Joined: 23 Aug 2008
Posts: 3855

PostPosted: Fri Dec 10, 2010 11:43 pm    Post subject: Reply with quote

Puzzles without a backdoor exist but are not typically generated by a puzzle generator. This puzzle has a fist full of them. As for a solution, I was able to reduce it to:

Code:
 r6  b5  Locked Candidate 1              <> 3    r6c1

   c368  Swordfish (222)                 <> 4    r2c9,r9c14

   c2b7  Locked Pair                     <> 28   r7c1,r15c2,r9c3

 c24\r14 Sashimi X-Wing   (Skyscraper)   <> 9    r4c3

 +-----------------------------------------------------------------------+
 |  6      39     2389   | d489    389    7      |  24589  1      245    |
 |  89     5      1      |  2      6     c48     |  7     b489    3      |
 |  7      4      2389   |  5      1389   138    |  289    689    26     |
 |-----------------------+-----------------------+-----------------------|
 |  2      6      48     | e189    1589   158    |  3     a48-9   7      |
 |  1      39     389    |  7      4      2      |  589    689    56     |
 |  489    7      5      |  6      389    38     |  489    2      1      |
 |-----------------------+-----------------------+-----------------------|
 |  45     28     7      |  148    158    6      |  24     3      9      |
 |  3      28     6      |  48     7      9      |  1      5      24     |
 |  59     1      49     |  3      2      45     |  6      7      8      |
 +-----------------------------------------------------------------------+
 # 69 eliminations remain

 L2-Wing:  (4)r4c8 = r2c8 - r2c6 = (4-9)r1c4 = (9)r4c4  =>  r4c8<>9

 r4      Naked  Pair                     <> 48   r4c456

 r4  b5  Naked  Triple                   <> 159  r6c5

 r6  b5  Naked  Pair                     <> 38   r6c17
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peterj



Joined: 26 Mar 2010
Posts: 974
Location: London, UK

PostPosted: Sat Dec 11, 2010 12:08 am    Post subject: Reply with quote

I missed the swordfish Crying or Very sad

But there were a lot of "hub cells" for strong links on 4 and 9 which proved fruitful.
Code:
l-wing(49) ; (4)r6c1=r4c2 - r4c8=(4-9)r2c8=r2c1 ; r6c1<>9

Which opened up another AIC on 4 and 9 which is a sort of conjoined double m-wing!
Code:
*-----------------------------------------------------------------------*
 | 6      2389   2389   | 3(4)8(9) 389    7      | 24589   1      2345   |
 | 389    5      1      | 2        6      3(4)8  | 7       89-4   34     |
 | 7      4      2389   | 5        1389   138    | 289     689    236    |
 |----------------------+------------------------+-----------------------|
 | 2      6      489    | 18(9)    1589   158    | 3       (4)89  7      |
 | 1      389    389    | 7        4      2      | 589     689    56     |
 | 48     7      5      | 6        38(9)  38     | (4)8(9) 2      1      |
 |----------------------+------------------------+-----------------------|
 | 458    28     7      | 148      1258   6      | 24      3      9      |
 | 348    238    6      | 348      7      9      | 1       5      24     |
 | 3459   1      2349   | 34       235    345    | 6       7      8      |
 *-----------------------------------------------------------------------*
AIC ; (4)r4c8=(4-9)r6c7=r6c5 - r4c4=(9-4)r1c4=r2c6 ; r2c8<>4
Then
Code:
skyscraper(9) ; r1c4<>9, r6c5<>9

I also looked for a logical move for the "backdoor" elimination above - but couldn't get close!
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Marty R.



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

PostPosted: Sat Dec 11, 2010 2:28 am    Post subject: Reply with quote

I managed to solve it with extreme inelegance. Three Multi-colorings, M-Wing, ER and two XY-Wings. A number of the moves included extensions/transports.
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tlanglet



Joined: 17 Oct 2007
Posts: 2468
Location: Northern California Foothills

PostPosted: Mon Dec 13, 2010 7:14 pm    Post subject: Reply with quote

My initial move involved an AW-wing(34) in r2c9 & r8c1 with fin (4)r8c1. I playing around with potential deletions provided by the fin, I found the following contradiction but do not know how to write it appropriately.

Code:
 *--------------------------------------------------------------------*
 | 6      2389   2389   | 3489   389    7      | 24589  1      2345   |
 | 389    5      1      | 2      6     b348    | 7     w489  av34     |
 | 7      4      2389   | 5      1389   138    | 289    689    236    |
 |----------------------+----------------------+----------------------|
 | 2      6     y489    | 189    1589   158    | 3     x489    7      |
 | 1      389    389    | 7      4      2      | 589    689    56     |
 | 489    7      5      | 6      389    38     | 489    2      1      |
 |----------------------+----------------------+----------------------|
 | 458    28     7      | 148    1258   6      | 24     3      9      |
 | 348    238    6      | 348    7      9      | 1      5      24     |
 | 3459   1     z2349   | 34     235   c345    | 6      7      8      |
 *--------------------------------------------------------------------*

If the fin is true, then two implications are possible as follows
Path 1 marked abc: (4)r2c9-r2c6=(4)r9c6
Path 2 marked vwxyz: (4)r2c9-r2c8=r4c8-r4c3=r9c3
So this resulting conflict means that r2c9<>4, but I am unable to easily describe this explicit result.

Ted
P.S. It was only when I read other posts that I realized the existence of the swordfish.
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peterj



Joined: 26 Mar 2010
Posts: 974
Location: London, UK

PostPosted: Mon Dec 13, 2010 7:58 pm    Post subject: Reply with quote

Ted, your cells bcwxyz are the swordfish. I think what you describe is a contradiction that would exist if an elimination due to the fish was assumed true. It would result in one one of the rows/columns having two true values.

Other than just being a swordfish you could write it as a loop causing a contradiction to the assumption...

(4)r2c9 - r2c6=r9c6 - r9c3=r4c3 - r4c8=r2c8 - contradiction ; r2c9<>4
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daj95376



Joined: 23 Aug 2008
Posts: 3855

PostPosted: Mon Dec 13, 2010 8:09 pm    Post subject: Reply with quote

Ted: Here's how ronk taught me to handle two chains with contradicting conclusions:

Code:
* write one in the forward direction

abc: (4)r2c9-r2c6=(4)r9c6

Code:
* write the other in the reverse direction (and pray there isn't an ERI)

zyxwv: (4)r9c3=r4c3-r4c8=r2c8-(4)r2c9

Code:
* joint them using a strong/weak link as is appropriate (probably always weak)

abc+zyxwv: (4)r2c9-r2c6=(4)r9c6 - (4)r9c3=r4c3-r4c8=r2c8-(4)r2c9

Code:
* (daj addendum) convert discontinuous loop to an AIC (loop)

abc+zyxwv: (4)r2c6=r9c6-r9c3=r4c3-r4c8=(4)r2c8 -loop  =>  r2c9<>4

Regards, Danny

Other possible eliminations from the loop not considered for this example.

Note: this is essentially the reverse of the logic that I used here.

BTW: ronk just kept rewriting my forcing chains as discontinuous loops until the _ Idea _ came on over my head.
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tlanglet



Joined: 17 Oct 2007
Posts: 2468
Location: Northern California Foothills

PostPosted: Mon Dec 13, 2010 11:21 pm    Post subject: Reply with quote

Thanks Peter and Danny for your feedback. I attempted several AIC approaches and separate paths, but I seemed to have had a mental block that ...(4)r9c6 - (4)r9c3.... so my chains were fruitless.

Ted
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