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"Finned" X-wing rule

 
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Paladin



Joined: 10 Feb 2006
Posts: 15

PostPosted: Thu Apr 27, 2006 6:04 pm    Post subject: "Finned" X-wing rule Reply with quote

This puzzle appeared at www.onlysudoku.com.

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



After solving 32 cells, the remaining puzzle appears as follows:


Code:
+-------------+----------------+----------------+
|  6   4   7  |  5     1    9  |  38    2    38 |
|             |                |                |
|  2   3   8  |  4     7    6  |   1    5     9 |
|             |                |                |
|  1   5   9  |  8     3    2  |   4    7     6 |
+-------------+----------------+----------------+
|  9  12   4  |  3    26    8  |   5   16     7 |
|             |                |                |
|  3   7   5  | 19  2469   14  | 269    8    24 |
|             |                |                |
|  8  12   6  |  7   249    5  | 239   14  1234 |
+-------------+----------------+----------------+
|  4  68   1  | 26     5    3  |   7    9    28 |
|             |                |                |
|  5  68   2  | 19    49    7  |  68    3    14 |
|             |                |                |
|  7   9   3  | 26     8   14  |  26  146     5 |
+-------------+----------------+----------------+


At this point, I believe I located a “finned” horizontal X-wing of 1's at r4c28 - r6c289. Applying the “finned” X-wing rule (as I understand it), I removed the 1 that appears in cell r6c9; now the only remaining 1 for column 9 appears in cell r8c9. When r8c9 is set at 1, the puzzle collapses.

Question: (1) Is this a correct application of the “finned” X-wing rule?; (2) If not, how can the puzzle be solved from this point?

Thank you for your responses.

Paladin
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TKiel



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

PostPosted: Fri Apr 28, 2006 12:12 am    Post subject: Reply with quote

Paladin,

Not sure if that is the correct application of 'finned x-wing'. I think the exclusion must come from a cell that is not part of the pattern, though my understanding is a bit hazy.

The puzzle can also be solved from this point by colouring on digit 4, which allows exclusion of 4 at r5c9.
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szarn
Guest





PostPosted: Fri Apr 28, 2006 1:03 am    Post subject: easier way Reply with quote

Hi Paladin,
I don't know about finned x wing, but don't see how it can apply here, because the 6 in r9c8 is not real -- you already have a pair of 26's in that row, so really r9c8 is 14, which pairs up with the 14 in r6c8. Then r4c8 becomes 6, not 16, so you're not matching up the 1's to make an xwing..

I bet it all falls out from there.
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TKiel



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

PostPosted: Fri Apr 28, 2006 1:16 am    Post subject: Reply with quote

Scarn,

You're right about the 6's but that doesn't solve the puzzle. You still need to either colour on the 4's or use the fact that it is now a BUG+1 grid.
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David Bryant



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

PostPosted: Fri Apr 28, 2006 1:09 pm    Post subject: It's not a finned X-Wing ... yet Reply with quote

Paladin wrote:
Question: (1) Is this a correct application of the “finned” X-wing rule?; (2) If not, how can the puzzle be solved from this point?

Hi, Paladin!

Your application of the "finned" X-Wing rule is incorrect. Interestingly, however, if you follow up on the suggestion our guest szam made (enter a "6" at r4c8), you will soon arrive at this position, which does contain a "finned" X-Wing pattern.
Code:
  6     4     7     5     1     9    38     2    38
  2     3     8     4     7     6     1     5     9
  1     5     9     8     3     2     4     7     6
  9     1     4     3     2     8     5     6     7
  3     7     5    19     6    14    29     8    24
  8     2     6     7    49*    5    39    14    134*
  4    68     1    26     5     3     7     9    28
  5    68     2    19    49*    7    68     3    14*
  7     9     3    26     8    14    26    14     5

From this point there are several ways to proceed. We could use "coloring" on the digit "4" to show that r5c6, r8c5, and r9c8 must all contain "4", as Tracy has suggested. Or we could recognize the "BUG" pattern that would exist if r6c9 contains either "1" or "3", and avoid it by setting r6c9 = 4. We can also use a "finned" X-Wing at this point.

Notice the four cells marked with an asterisk above. These would form an X-Wing (on rows) if there were no "4" at r6c8; they would also form an X-Wing (on columns) if there were no "4" at r5c9. We reason as follows.

-- If r6c8 = 4, then r5c9 = 2 (both in the middle right 3x3 box).
-- If r6c8 <> 4 there's an X-Wing on rows at r6c5&9 / r8c5&9, and we can still conclude that r5c9 = 2.
-- With the "fin" at r5c9 eliminated we recognize an X-Wing on columns and eliminate the "4" at r6c8.

So the "finned" X-Wing rule is pretty simple -- if one corner of an "almost X-Wing" pattern lies in a box that has three or more instances of the digit forming the pattern, we may be able to reason as above -- either this digit lies here (and we can eliminate other instances of the digit within the same 3x3 box) or it doesn't lie here (and the X-Wing pattern really exists). If we can rule out a particular digit in one or more cells either way, then we can apply the "finned" X-Wing rule. dcb
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Marty R.



Joined: 12 Feb 2006
Posts: 5123
Location: Rochester, NY, USA

PostPosted: Fri Apr 28, 2006 5:03 pm    Post subject: Reply with quote

David, I want to make sure I understand this. Do we define a Finned X-Wing as an X-Wing except for one extra occurrence of the number in the row/column? Does the "fin" have to be in the same box or can it be anywhere in the row/column? And then, we only can apply the "rule" when one corner is in a box with 3+ possibilities?
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Paladin



Joined: 10 Feb 2006
Posts: 15

PostPosted: Fri Apr 28, 2006 5:50 pm    Post subject: "Finned" X-wing rule Reply with quote

Thank you all so very much for your responses.

I guess I was so anxious to find some exotic solution to this puzzle that I did not even notice either: the 2,6 pair in row 9; or the fact that r7c9, r8c7 and r9c7 represent a "2,6,8 chain" (which can also be used to exclude the 6 in r9c8). Live and learn.

Since this is a very quick and unstudied reply to all of your kind responses, perhaps in the next short while I will pursue the finer points of some of your comments through private messages.

Thank you, David, for your very comprehensive response to my question.

Paladin
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David Bryant



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

PostPosted: Fri Apr 28, 2006 9:54 pm    Post subject: Making sense of the "fin" Reply with quote

Marty R. wrote:
Do we define a Finned X-Wing as an X-Wing except for one extra occurrence of the number in the row/column? Does the "fin" have to be in the same box or can it be anywhere in the row/column? And then, we only can apply the "rule" when one corner is in a box with 3+ possibilities?

Well, I'm not an expert on this finny-business. But here's what I understand about it.

-- The "fin" can occur with an X-Wing pattern, or with a swordfish, or even with a "jellyfish", I suppose.

-- The "fin" is always one extra cell that mars the "fishy" pattern.

-- We can only apply the rule when there's a third "possible" cell in the same 3x3 box as the fin. And we do this by saying "if the fin contains this number, then this number can't lie anywhere else in this 3x3 box -- if the fin doesn't contain this number, then the fishy pattern exists."

That's the best explanation I can give, Marty. Here ... I'll draw another picture.
Code:
..X...X..
.........
..X...Xx.

This is a "finned" X-Wing, but all we can do with it is eliminate x at r3c8 ... in other words, it's just an ordinary X-Wing after all.
Code:
..X...X..
......x..
..X...Xx.

This is a useful "finned" X-Wing. If r3c8 is true for x, then r2c7 can't contain x. And if r3c8 does not contain x, then the X-Wing pattern exists, and r2c7 still can't contain x. And with r2c8 out of the way, we're back to the first case (which is really just an ordinary X-Wing). dcb
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Guest






PostPosted: Sat Apr 29, 2006 12:36 pm    Post subject: Reply with quote

Maybe this helps to understand it:

Suppose in all of this 3 samples there are only the shown 1's in rows 3 and 5.
The first one is a normal x-wing, the second a finned, the third a sashimi x-wing.

Code:
 . . x | . x . | . . .       . . . | . x . | . . .      . . . | . . . | . . .
 . . x | . x . | . . .       . . . | . x . | . . .      . . . | . . . | . . .
 - - 1 | - 1 - | - - -       - - 1 | 1 1 1 | - - -      - - 1 | - 1 x | - - -
-----------------------     -----------------------    -----------------------
 . . x | . x . | . . .       . . . | . . . | . . .      . . . | . x . | . . .
 - - 1 | - 1 - | - - -       - - 1 | - 1 - | - - -      - - 1 | - - 1 | - - -
 . . x | . x . | . . .       . . . | . . . | . . .      . . . | . x . | . . .
-----------------------     -----------------------    -----------------------
 . . x | . x . | . . .       . . . | . . . | . . .      . . . | . . . | . . .
 . . x | . x . | . . .       . . . | . . . | . . .      . . . | . . . | . . .
 . . x | . x . | . . .       . . . | . . . | . . .      . . . | . . . | . . .

Then in all examples you can eliminate 1 from cells x:
If there would be a 1 in x, then for the rows 3 and 5 it kills all but the two 1's in column 3 . But since they share a column, you then have not enough 1's for both rows.

You can apply exactly the same argument for a (finned) swordfish.
Code:
 . . . | . . . | . . .   
 . . . | . . . | . . .   
 - - 1 | - 1 - | 1 - -   
-----------------------   
 . . . | . . . | . . .   
 - - 1 | - 1 - | 1 - -     
 . . . | . . . | . . .     
-----------------------   
 . . . | . . . | x . .     
 . . . | . . . | x . .     
 - - 1 | - 1 - | 1 1 1     

If there would be a 1 in x, then for the rows 3, 5 and 8 it kills all but the 1's in columns 3 and 5. But since they lie in 2 columns, you then have not enough 1's for all 3 rows.

Of course the xwings or swordfish can be restricted to rows instead of columns also. Then the same applies, when you switch row and column above.
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ravel



Joined: 21 Apr 2006
Posts: 536

PostPosted: Sat Apr 29, 2006 12:41 pm    Post subject: Reply with quote

What helps the registration, when i do not log in ? Smile
I wanted to edit, that of course in the 1st sample the 1's in column 5 remain or the x's in col 3.
And for the swordfish it should say : rows 3, 5 and 9.
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TKiel



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

PostPosted: Sat Apr 29, 2006 1:06 pm    Post subject: Reply with quote

It seems that there are two 'finned X-wings' in this puzzle, using either r5c9 as the fin, which would eliminate r6c8 or using r6c8 as the fin, which would eliminate r5c9. This made me curious, so I backtracked in the puzzle to see if one of them showed up at an earlier point and I reached here: (I haven't checked to see if all exclusions/assignments have been made at this point, as I was only interested in seeing if the 'finned X-wing appeared at an earlier point.)
Code:
 
 *--------------------------------------------------------------------*
 | 6    4    789     | 5789    1       389     | 389     2    389     |
 | 2    3    789     | 4       679     689     | 1       5    89      |
 | 1    5    89      | 289     239     2389    | 4       7    6       |
 |-------------------+-------------------------+----------------------|
 | 89   12   489     | 3       24679#  124689  | 5       146# 12479#  |
 | 39   7    3459    | 1269    2469#   12469   | 2369    8    12349#  |
 | 389  12   6       | 12789   2479*   5       | 2379    14*  123479* |
 |-------------------+-------------------------+----------------------|
 | 4    68   1       | 256     2356    236     | 2678    9    2578    |
 | 5    68   2       | 169     469*    7       | 68      3    148*    |
 | 37   9    37      | 1256    8       12346   | 267     146  12457   |
 *--------------------------------------------------------------------*


The cells marked with an * form the same 'finned X-wing' as noted by David. Does this allow the exclusion of 4 from all cells marked with #?
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ravel



Joined: 21 Apr 2006
Posts: 536

PostPosted: Sat Apr 29, 2006 2:30 pm    Post subject: Reply with quote

TKiel wrote:
Does this allow the exclusion of 4 from all cells marked with #?

No, the 4 in r4c8 does not kill the 4 in r8c9 and the 4's in r45c5 dont kill the fin in r6c8 (so in these cases you still have two 4's in two columns left for the rows 6 and 8).
In other words, you can only eliminate the 4's that both "see" the fin(s) and the two 4's in the x-wing column.
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Marty R.



Joined: 12 Feb 2006
Posts: 5123
Location: Rochester, NY, USA

PostPosted: Sat Apr 29, 2006 4:14 pm    Post subject: Reply with quote

Thanks all, for the explanations.
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