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Paladin
Joined: 10 Feb 2006 Posts: 15

Posted: Tue May 02, 2006 5:52 pm Post subject: 4/30/06 Puzzle: Aha! The Infamous "Finned" Xwing? 


This puzzle appeared on April 30, 2006 at www.dailysudoku.com:
Code:  ++++
 . 7 8  4 . .  . . . 
   
 . . .  . 9 .  7 . . 
   
 . 2 .  . . .  . 8 3 
++++
 . . .  5 . .  8 6 . 
   
 1 . .  . . .  . . 9 
   
 . 5 9  . . 2  . . . 
++++
 8 1 .  . . .  . 2 . 
   
 . . 2  . 7 .  . . . 
   
 . . .  . . 8  9 3 . 
++++ 
After solving 25 cells, the remaining puzzle appears as follows:
Code:  ++++
 35 7 8  4 2 1356  156 9 56 
   
 345 346 13456  8 9 1356  7 14 2 
   
 9 2 1456  67 56 1567  1456 8 3 
++++
 2 34 7  5 34 9  8 6 1 
   
 1 8 34  367 346 467  2 5 9 
   
 6 5 9  1 8 2  3 7 4 
++++
 8 1 3456  9 3456 456  456 2 7 
   
 345 9 2  36 7 456  1456 14 8 
   
 7 46 456  2 1 8  9 3 56 
++++ 
At this point, I believe I located a “finned” horizontal Xwing of 3's at r5c345  r7c35. Since r4c5 can not be “3" if r5c4 (the “fin”) is “3", and it also cannot be “3" if the “finned” Xwing really turns out to be just an Xwing, in this case r4c5 must be set at 4. When r4c5 is set at 4, the puzzle collapses.
Questions: (1) Have I finally applied the “finned” Xwing rule (or at least perhaps a form of it) properly?; or (2) Did anyone use an alternate solution path to solve this puzzle?
Thank you for your responses.
Paladin 

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Angel
Joined: 26 Mar 2006 Posts: 31

Posted: Tue May 02, 2006 6:37 pm Post subject: 


I have no idea about 'finned xwings', but from the position you posted I would note that the 4 in row 9 must be in box 7. It cannot be elsewhere in box 7, in particular it cannot be in r8c1. This leads to r2c1 being 4, from
here... well as you will see a lot of numbers fall immediately into place.
I'm sorry if this is not what you were hoping for, but I don't think it qualifies for some fancy name. 

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

Posted: Tue May 02, 2006 7:06 pm Post subject: The Infamous "Finned" Xwing? 


Paladin
You have the finned Xwing bang to rights. Good work!
Unfortunately I did not use it, solving the puzzle in the same way as Angel.
Steve 

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Marty R.
Joined: 12 Feb 2006 Posts: 5197 Location: Rochester, NY, USA

Posted: Thu May 04, 2006 8:00 pm Post subject: 


Let's see if I've got this right. There is a Finned XWing situation when:
1) An XWing would exist were it not for a third occurrence of the number in one row/column.
And
2) One of the wing's corners must be in a box with at least three occurrences of the number.
I assume it's that third occurrence which is called the "Fin", is that correct?
Does the Fin have to be in one of the boxes occupied by the wing's corners? For example, if the XWing were in boxes 4 and 5, could the Fin be in box 6?
Once the Finned Wing is recognized, it appears that we do some DIC's or something akin to them. Is there something in particular that should be looked for or some particular cell that should be a starting point? In other words, can this be reduced to some sort of rule, as in, once the pattern is observed, the next step must be suchandsuch? Or is it a more openended thing? 

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David Bryant
Joined: 29 Jul 2005 Posts: 559 Location: Denver, Colorado

Posted: Thu May 04, 2006 10:29 pm Post subject: The"FiletO'Fish" rule 


Marty R wrote:  Let's see if I've got this right. There is a Finned XWing situation when:
1) An XWing would exist were it not for a third occurrence of the number in one row/column.
And
2) One of the wing's corners must be in a box with at least three occurrences of the number.
I assume it's that third occurrence which is called the "Fin", is that correct? 
Well, it's mostly right, Marty. But the "fin" doesn't have to be just one cell ... it can be two cells long. The logic is substantially similar in either case.
Code:  Case A Case B
......... .........
..X...X.. ..X...X..
......... .........
......... .........
......*.. ......*..
..X...X#. ..X...X##
......... .........
......... .........
..#...... ..#...... 
A. If r6c8 = "x", then r5c7 <> "x"; if r6c8 <> "x" then XWing on rows exists, and we still have r5c7 <> "x"
B. If r6c8 = "x" or r6c9 = "x", then r5c7 <> "x"; if r6c8 <> "x" and r6c9 <> "x", then XWing on rows exists, and we still have r5c7 <> "x"
In the special case where the "fin" is the only distortion of the basic XWing pattern we may be able to make a double (or triple, or even quadruple!) elimination.
Code:  Case C Case D Case E
......... ......... .........
..X...X.. ..X...X.. ..X...X..
......... ......... .........
......... ......*.. ......*..
......*.. ......*.. ......*..
..X...X#. ..X...X#. ..X...X##
......... ......... .........
......... ......... .........
......... ......... ......... 
C. Same logic as "A" above, but since there's no complication at r9c3, once we eliminate the "x" at r5c7 there's an XWing on columns that also allows us to eliminate the fin itself, at r6c8.
D. Same logic as "A" above, but since there's no complication at r9c3, once we eliminate "x" at r4c7 and r5c7 there's an XWing on columns that also allows us to eliminate the fin itself, at r6c8.
E. Same logic as "B" above, but since there's no complication at r9c3, once we eliminate "x" at r4c7 and r5c7, there's an XWing on columns that also allows us to eliminate both cells' worth of "fin", at r6c8 & r6c9.
Many other variations are possible, but I think these examples give you the flavor of the general argument.
Marty R wrote:  Does the Fin have to be in one of the boxes occupied by the wing's corners? For example, if the XWing were in boxes 4 and 5, could the Fin be in box 6? 
Yes, the "fin" must be in the same 3x3 box as one of the "corners" on the XWing. And it has to be in the same row (XWing on rows) or column (XWing on columns) as one leg of the XWing itself.
So the answer to your second question is no  if the XWing lies in boxes 4 and 5, the "fin" must lie either in box 4 or in box 5; it cannot lie in box 6.
Marty R wrote:  ... can this be reduced to some sort of rule, as in, once the pattern is observed, the next step must be suchandsuch? Or is it a more openended thing? 
Myth Jellies has reduced it to a general rule  you can read about that in another forum.
Personally I find this rule to be of little practical utility. It's helpful if one is writing a computer program, but it's a little too much for my poor little brain to latch onto all at once. So in practice I tend to look for the general form of the pattern (an "almost XWing") and then go through the reasoning from first principles. But that's just me. dcb 

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Marty R.
Joined: 12 Feb 2006 Posts: 5197 Location: Rochester, NY, USA

Posted: Fri May 05, 2006 5:03 pm Post subject: 


As always, David, thank you very much for taking the time to explain things. I've printed out the reply, as I generally have difficulty following reasoning on a monitor, but can do so much better on paper.
Update: it's now a little over an hour since I posted the above and I've read through it once.
1) In all five examples, ABCDE, the chain of reasoning started with a finned cell, not one of the wing corners. Could that be one of the general "rules" I was looking for?
2) When this finned pattern exists, is it generally successful (eliminates at least one candidate) or does it often result in no action being taken? 

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David Bryant
Joined: 29 Jul 2005 Posts: 559 Location: Denver, Colorado

Posted: Fri May 05, 2006 9:19 pm Post subject: More questions ... 


Marty R wrote:  1) In all five examples, ABCDE, the chain of reasoning started with a finned cell, not one of the wing corners. Could that be one of the general "rules" I was looking for? 
I suppose so. In general, I'm not real good at formulating rules. So if it works for you, go with it.
Marty R wrote:  2) When this finned pattern exists, is it generally successful (eliminates at least one candidate) or does it often result in no action being taken? 
Well, I don't have any real statistical evidence, but I imagine it's about 50  50, just by thinking about the possibilities.
1. The general case, where there's an "external complication," looks something like this.
Code:  . . . . . . . . .
. X . . . . . X .
. . . . . . . . .
. . . . . . . * .
. . . . . . . * .
. X . . . . # X #
. . . . . . . . .
. . . . . . . . .
. * . . . . . . . 
Here the "*" in r9c2 is the external complication, the hash marks are the "fin", and the two asterisks at r4c8 & r5c8 represent possible eliminations in box 6.
Since there are six cells in box 6 lying outside of the "finned XWing" pattern, and two of those are ripe for exclusion because of the pattern, it looks like this version will be successful onethird of the time.
2. The general case, where there is no external complication, looks something like this.
Code:  . . . . . . . . .
. X . . . . . X .
. . . . . . . . .
. . . . . . . * .
. . . . . . . * .
. X . . . . # X #
. . . . . . . . .
. . . . . . . . .
. . . . . . . . . 
Here we are guaranteed that we can either make the "dual exclusion" (first eliminate the target digit at r4c8 and at r5c8, followed by the "fin") or else we just have a regular XWing on columns (in which case the "fin" can be eliminated). So that's a 100% chance.
I'm not sure how much more likely case 1 is than case 2, but it appears that the overall probability of getting something useful out of the "finned" pattern is somewhere between onethird and 100%  a 50% chance is probably a good estimate. dcb 

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SL Guest

Posted: Fri May 05, 2006 11:33 pm Post subject: 


I think the easier way to solve it is 4 in Box 7 should only appear in r9, r2c1 should be 4,... 

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