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wapati
Joined: 10 Jun 2008
Posts: 472
Location: Brampton, Ontario, Canada.
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 Posted: Fri Dec 10, 2010 4:20 pm Post subject: Knotty Puzzle. |
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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
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| Posted: Fri Dec 10, 2010 5:23 pm Post subject: Re: Knotty Puzzle. |
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Many steps, but none harder than UR, m-wing or w-wing. Krampus  type ...
| wapati wrote: | ... backdoors ... Carcul convinced me that most all puzzles may have at least one so good luck all!
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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" 
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
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| Posted: Fri Dec 10, 2010 5:51 pm Post subject: Re: Knotty Puzzle. |
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| 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... 
| Quote: | | r3c6<>1 (Courtesy of Hoduku) |
Unfortunately I have no logical basis for this elimination  - yet! |
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keith
Joined: 19 Sep 2005
Posts: 3355
Location: near Detroit, Michigan, USA
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| Posted: Fri Dec 10, 2010 9:34 pm Post subject: |
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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: 3854
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| Posted: Fri Dec 10, 2010 11:43 pm Post subject: |
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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
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| Posted: Sat Dec 11, 2010 12:08 am Post subject: |
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I missed the swordfish 
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
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| Posted: Sat Dec 11, 2010 2:28 am Post subject: |
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| 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
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| Posted: Mon Dec 13, 2010 7:14 pm Post subject: |
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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 |
*--------------------------------------------------------------------*
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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
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| Posted: Mon Dec 13, 2010 7:58 pm Post subject: |
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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: 3854
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| Posted: Mon Dec 13, 2010 8:09 pm Post subject: |
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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
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| Code: | * write the other in the reverse direction (and pray there isn't an ERI)
zyxwv: (4)r9c3=r4c3-r4c8=r2c8-(4)r2c9
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| 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
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| Code: | * (daj addendum) convert discontinuous loop to an AIC (loop)
abc+zyxwv: (4)r2c6=r9c6-r9c3=r4c3-r4c8=(4)r2c8 -loop => r2c9<>4
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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 _  _ came on over my head. |
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tlanglet
Joined: 17 Oct 2007
Posts: 2468
Location: Northern California Foothills
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| Posted: Mon Dec 13, 2010 11:21 pm Post subject: |
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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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