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xyz-wing or finned xy-wing

 
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tlanglet



Joined: 17 Oct 2007
Posts: 2461
Location: Northern California Foothills

PostPosted: Wed Mar 17, 2010 3:52 pm    Post subject: xyz-wing or finned xy-wing Reply with quote

Code:
*-----------------------------------------------------------*
 | 2     39    5     | 39    4     8     | 6     1     7     |
 | 349   8     479   | 1379  6     137   | 359   2     359   |
 | 6     1     79    | 5     379   2     | 389   4     389   |
 |-------------------+-------------------+-------------------|
 | 49    459   3     | 8     1     6     | 579   579   2     |
 | 1     59    8     | 2     357   357   | 359   6     4     |
 | 7     2     6     | 4     35    9     | 1     8     35    |
 |-------------------+-------------------+-------------------|
 | 8     379   2     | 1379  3579  1357  | 4     379   6     |
 | 39    6     1     | 379   2     4     | 5789  3579  589   |
 | 5     3479  49    | 6     8     37    | 2     379   1     |
 *-----------------------------------------------------------*


This is the code after basic for daj's "Puzzle 10/03/14 ___ BBDB as VH+"

First look at the xyz-wing with hinge 379 in r8c4, 39 in r1c4 and 37 in r9c6; this deletes the 3 in r7c4. Nice, straight forward xyz-wing.

Now look at a finned xy-wing 379 with vertex 79 in r8c4, pincer 39 in r1c4, pincer 37 in r9c6 and fin 3 in r8c4.
If the xy-wing is true, then the 3 in r2c6 & r7c4 will be deleted.
If the fin is true: (3)r8c4; r7c4<>3
If the fin is true: (3)r8c4 - r8c1 = r79c2 - r1c2 = r1c4; r2c6<>3.
Thus the 3 in both r2c6 and r7c4 is deleted by both conditions.

I wonder if this has some practical use for achieving additional deletions for xyz wings?

Ted
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tlanglet



Joined: 17 Oct 2007
Posts: 2461
Location: Northern California Foothills

PostPosted: Fri Mar 19, 2010 1:09 pm    Post subject: Reply with quote

On March 18, Nataraj posted his solution to the puzzle "Free March 18, 2010 (Thursday)" found in the Other Puzzles thread of this forum. He used a simple xyz-wing that made a single deletion as a one step solution. But, if viewing this pattern as a finned xy-wing, several additional deletions are possible (which could be useful if the single xyz-wing deletion had not been adequate to complete the puzzle).


Code:
 *--------------------------------------------------------------------*
 | 6      34     9      | 48     7      248    | 1      25     35     |
 | 8      2      7      | 5     #36     1      | 4      69     39     |
 | 5      34     1      | 46    #236    9      | 7     #26     8      |
 |----------------------+----------------------+----------------------|
 | 3      1      4      | 69     269    26     | 5      8      7      |
 | 7      6      2      | 3      8      5      | 9      1      4      |
 | 9      5      8      | 1      4      7      | 2      3      6      |
 |----------------------+----------------------+----------------------|
 | 1      78     56     | 46789  569    468    | 3      49     2      |
 | 4      9      56     | 2      56     3      | 8      7      1      |
 | 2      78     3      | 4789   1      48     | 6      459    59     |
 *--------------------------------------------------------------------*

The xyz-wing 36-236-26 is in box2/row3, marked #, and deletes 6 in r3c4.

Now view this pattern as a finned xy-wing 23-6 with vertex 23 in r3c5, pincer 36 in r2c5, pincer 26 in r3c8 and fin 6 in r3c5.
If the xy-wing is true, then r2c8 & r3c4 <>6.
But the strong link on 6 in row2 & col8 force both pincer to equal 6 so additional deletions are possible: (6)r2c5 - r478c5 = r8c3 - r7c3.
In summary, if the xy-wing is true then the 6 in six cells is deleted: r2c8, r3c4, r478c5, & r7c3.

If the fin is true:
(6)r3c5; r2c5<>6
(6)r3c5; r3c48<>6,
(6)r3c5 - r478c5 = r8c3 - r7c3.
In summary, if the fin is true then the 6 in seven cells is deleted:
r2c5, r3c48, r478c5, & r7c3.

The resulting set of deletions common to both conditions are in five cells: r3c4, r478c5, & r7c3.

Ted
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