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xyz-wing or finned xy-wing

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tlanglet

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

Posted: Wed Mar 17, 2010 3:52 pm Post subject: xyz-wing or finned xy-wing

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

tlanglet

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

Posted: Fri Mar 19, 2010 1:09 pm Post subject:

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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