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Looking for X-wings, Kites, and Skyscrapers (Turbot Fish)

 
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keith



Joined: 19 Sep 2005
Posts: 3184
Location: near Detroit, Michigan, USA

PostPosted: Fri Dec 18, 2009 4:40 am    Post subject: Looking for X-wings, Kites, and Skyscrapers (Turbot Fish) Reply with quote

Let's start with this puzzle:
Code:
Puzzle: BB120809sh
+-------+-------+-------+
| 6 . 4 | . 1 . | 9 . 5 |
| . . . | . . . | . . . |
| 5 . 3 | . 6 . | 2 . 7 |
+-------+-------+-------+
| . . . | 7 5 4 | . . . |
| 4 . 7 | 9 . 6 | 3 . 1 |
| . . . | 1 8 3 | . . . |
+-------+-------+-------+
| 3 . 1 | . 9 . | 6 . 2 |
| . . . | . . . | . . . |
| 9 . 2 | . 3 . | 5 . 8 |
+-------+-------+-------+

After basics:
Code:
+----------------+----------------+----------------+
| 6    27   4    | 238  1    278  | 9    38   5    |
| 78   1279 89   | 2345 47   2579 | 14   1346 346  |
| 5    19   3    | 48   6    89   | 2    148  7    |
+----------------+----------------+----------------+
| 1    3    69   | 7    5    4    | 8    2    69   |
| 4    8    7    | 9    2    6    | 3    5    1    |
| 2    569  569  | 1    8    3    | 47   4679 469  |
+----------------+----------------+----------------+
| 3    457  1    | 458  9    578  | 6    47   2    |
| 78   4567 568  | 2456 47   1257 | 147  39   39   |
| 9    467  2    | 46   3    17   | 5    147  8    |
+----------------+----------------+----------------+

Now, you need to go fishing. How do you do it?

Perhaps the more direct question is: How do you quickly eliminate candidates that cannot make X-wing type (2 strong link) eliminations? (I think the same logic applies to identify candidates that cannot make coloring or multi-coloring eliminations. Possibly, it applies to all single digit advanced moves, including all the fishes, and ER's.)

I think I must be a very slow learner, for I am just starting to figure this out. My method to now has been brute force: For each candidate, map out each and every strong link, and then stare at the link diagrams to find fishy and coloring eliminations. I think there is a better way!

Keith
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daj95376



Joined: 23 Aug 2008
Posts: 3855

PostPosted: Fri Dec 18, 2009 4:03 pm    Post subject: Reply with quote

Quite by accident, it appears that Puzzle NR 09/12/18 (A) might be a good source of examples. I'll let Keith decide.

BTW: Although Asellus and I have a difference of opinion, I consider the Empty Rectangle to be part of the Turbot Fish family.
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Marty R.



Joined: 12 Feb 2006
Posts: 5179
Location: Rochester, NY, USA

PostPosted: Fri Dec 18, 2009 4:28 pm    Post subject: Reply with quote

Quote:
BTW: Although Asellus and I have a difference of opinion, I consider the Empty Rectangle to be part of the Turbot Fish family.

I'll stay away from that as I have no clue. But ERs are so simple and mechanical I could accept it if they were considered part of the "basics."
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keith



Joined: 19 Sep 2005
Posts: 3184
Location: near Detroit, Michigan, USA

PostPosted: Fri Dec 18, 2009 11:08 pm    Post subject: Reply with quote

So, here's the idea.

In the following diagrams, X shows a box / block in the puzzle (posted above) where the candidate is not solved.
Code:

Diagram A

1
+---+---+---+
| X | . | X |
+---+---+---+
| . | . | . | 
+---+---+---+
| . | X | X |
+---+---+---+

2
+---+---+---+
| X | X | . |
+---+---+---+
| . | . | . |
+---+---+---+
| . | X | . |
+---+---+---+

3
+---+---+---+
| . | X | X |
+---+---+---+
| . | . | . |
+---+---+---+
| . | . | X |
+---+---+---+

4
+---+---+---+
| . | X | X |
+---+---+---+
| . | . | X |
+---+---+---+
| X | X | X |
+---+---+---+

5
+---+---+---+
| . | X | . |
+---+---+---+
| X | . | . |
+---+---+---+
| X | X | . |
+---+---+---+

6
+---+---+---+
| . | . | X |
+---+---+---+
| X | . | X |
+---+---+---+
| X | X | . |
+---+---+---+

7
+---+---+---+
| X | X | . |
+---+---+---+
| . | . | X |
+---+---+---+
| X | X | X |
+---+---+---+

8
+---+---+---+
| X | X | X |
+---+---+---+
| . | . | . |
+---+---+---+
| X | X | . |
+---+---+---+

9
+---+---+---+
| X | X | . |
+---+---+---+
| X | . | X |
+---+---+---+
| . | . | X |
+---+---+---+

Let me be bold:

You will only find two-link single-digit advanced eliminations (X-wings, Turbot fish, skyscrapers, ERs, ...) if the pattern in Diagram A above contains four X's in a rectangle, like, for example:
Code:

Diagram B
+---+---+---+
| X | X | . |
+---+---+---+
| . | . | . |
+---+---+---+
| X | X | . |
+---+---+---+

By this rule, only 4, 7, and 8 are worth looking at.

(That 2 and 3, and even 1 are not worth looking at might be obvious. But, I have wasted a lot of time looking for stuff in patterns like 5, 6, and 9.)

Sure enough, there are X-wings on 4 and 7.

I am well aware that X-wings can occur in two boxes (that is the basis of a Type 6 UR) but I don't recall ever seeing one that made an actual elimination. Theoretically, the pattern for 2 could contain an X-wing in B28 (strong links in the columns) making eliminations in B1. Does this ever occur?

edit: I withdraw my statement for and higher fish, coloring and multi-coloring.

I think you can go further: For ER's, Turbot fish, and kites, all the action occurs only in the four boxes on the rectangle X in Diagram B.

Best wishes,
Keith

(I could spend my weekend drawing stencils of all possible patterns for advanced single-digit eliminations, and then my office would look like the set of "A Beautiful Mind". Or, I can just wait for all of you to prove me wrong, or to point out that everyone knows this already.)
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keith



Joined: 19 Sep 2005
Posts: 3184
Location: near Detroit, Michigan, USA

PostPosted: Sat Dec 19, 2009 12:57 am    Post subject: Reply with quote

daj95376 wrote:
I consider the Empty Rectangle to be part of the Turbot Fish family.

Even when the hinge pin cell contains the candidate?

Keith
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daj95376



Joined: 23 Aug 2008
Posts: 3855

PostPosted: Sat Dec 19, 2009 1:13 am    Post subject: Reply with quote

keith wrote:
daj95376 wrote:
I consider the Empty Rectangle to be part of the Turbot Fish family.

Even when the hinge pin cell contains the candidate?

Yes, it's the only example I know that isn't impacted by a hinge pin/ERI cell.

You must take the hinge pin/ERI cell into consideration when looking for a grouped 2-String Kite!


Last edited by daj95376 on Sat Dec 19, 2009 1:26 am; edited 3 times in total
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daj95376



Joined: 23 Aug 2008
Posts: 3855

PostPosted: Sat Dec 19, 2009 1:16 am    Post subject: Reply with quote

keith wrote:
You will only find two-link single-digit advanced eliminations (X-wings, Turbot fish, skyscrapers, ERs, ...) if the pattern in Diagram A above contains four X's in a rectangle, ...

An observation that has long slipped my mind. Thanks for the reminder.
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daj95376



Joined: 23 Aug 2008
Posts: 3855

PostPosted: Sat Dec 19, 2009 1:20 am    Post subject: Reply with quote

keith wrote:
I am well aware that X-wings can occur in two boxes (that is the basis of a Type 6 UR) but I don't recall ever seeing one that made an actual elimination. Theoretically, the pattern for 2 could contain an X-wing in B28 (strong links in the columns) making eliminations in B1. Does this ever occur?

Let's consider an X-Wing in two boxes ... and it's potential eliminations.

Code:
 X-Wing r46\c25 w/eliminations
 +-----------------------------------+
 |  .  *  .  |  .  *  .  |  .  .  .  |
 |  .  *  .  |  .  *  .  |  .  .  .  |
 |  .  *  .  |  .  *  .  |  .  .  .  |
 |-----------+-----------+-----------|
 |  /  X  /  |  /  X  /  |  /  /  /  |
 |  .  *  .  |  .  *  .  |  .  .  .  |
 |  /  X  /  |  /  X  /  |  /  /  /  |
 |-----------+-----------+-----------|
 |  .  *  .  |  .  *  .  |  .  .  .  |
 |  .  *  .  |  .  *  .  |  .  .  .  |
 |  .  *  .  |  .  *  .  |  .  .  .  |
 +-----------------------------------+

Now, let's assume that you must perform basic eliminations first.

Code:
 Locked Candidate 2 in r5b6   ... then

 Locked Candidate 2 in c2b4
 Locked Candidate 2 in c5b5
 +-----------------------------------+
 |  .  /  .  |  .  /  .  |  .  .  .  |
 |  .  /  .  |  .  /  .  |  .  .  .  |
 |  .  /  .  |  .  /  .  |  .  .  .  |
 |-----------+-----------+-----------|
 |  /  X  /  |  /  X  /  |  /  /  /  |
 |  /  /  /  |  /  /  /  |  .  .  .  |
 |  /  X  /  |  /  X  /  |  /  /  /  |
 |-----------+-----------+-----------|
 |  .  /  .  |  .  /  .  |  .  .  .  |
 |  .  /  .  |  .  /  .  |  .  .  .  |
 |  .  /  .  |  .  /  .  |  .  .  .  |
 +-----------------------------------+

No eliminations remain for X-Wing in two boxes Exclamation

Regards, Danny
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keith



Joined: 19 Sep 2005
Posts: 3184
Location: near Detroit, Michigan, USA

PostPosted: Sat Dec 19, 2009 1:25 am    Post subject: Reply with quote

daj95376 wrote:
keith wrote:
You will only find two-link single-digit advanced eliminations (X-wings, Turbot fish, skyscrapers, ERs, ...) if the pattern in Diagram A above contains four X's in a rectangle, ...

An observation that has long slipped my mind. Thanks for the reminder.

Danny,

I found something you have forgotten? Very Happy

Seriously, I did not imagine this would be useful to programmers, but it is an easy visual check for those who solve by pencil & paper.

Keith
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keith



Joined: 19 Sep 2005
Posts: 3184
Location: near Detroit, Michigan, USA

PostPosted: Sat Dec 19, 2009 1:29 am    Post subject: Reply with quote

daj95376 wrote:

Let's consider an X-Wing in two boxes ... and it's potential eliminations.
No eliminations remain for X-Wing in two boxes Exclamation

Danny,

Thank you for that! Idea
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keith



Joined: 19 Sep 2005
Posts: 3184
Location: near Detroit, Michigan, USA

PostPosted: Sat Dec 19, 2009 4:13 am    Post subject: Reply with quote

Quote:
Are you planning to discuss recognizing some patterns by starting in a box and working out in two directions?

No, you should start that thread.

But, I am planning to add the corollary: If any of the boxes "X" has candidates only in one row or one column, that box can be dropped from the pattern.

After the elimination of that box, the rectangle rule still applies to the remaining boxes.

Keith
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keith



Joined: 19 Sep 2005
Posts: 3184
Location: near Detroit, Michigan, USA

PostPosted: Sun Dec 20, 2009 3:18 am    Post subject: Reply with quote

keith wrote:
I am planning to add the corollary: If any of the boxes "X" has candidates only in one row or one column, that box can be dropped from the pattern.

After the elimination of that box, the rectangle rule still applies to the remaining boxes.
Keith

In the current example, the pattern for 8 is:
Code:
8
+---+---+---+
| X | X | X |
+---+---+---+
| . | . | . |
+---+---+---+
| X | X | . |
+---+---+---+

which meets the requirement for a rectangle of boxes 1378.

However, the detailed pattern for 8 is
Code:
+-------+-------+-------+
| . . . | x . x | . x . |
| x . x | . . . | . . . |
| . . . | x . x | . x . |
+-------+-------+-------+
| . . . | . . . | . . . |
| . . . | . . . | . . . |
| . . . | . . . | . . . |
+-------+-------+-------+
| . . . | x . x | . . . |
| x . x | . . . | . . . |
| . . . | . . . | . . . |
+-------+-------+-------+

Boxes 1, 3, 7, 8, have candidates only in one row or column. When these boxes are excluded, the digit 8 clearly does not qualify for possible X-wings, Turbot fish, ER's, ....

Keith
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keith



Joined: 19 Sep 2005
Posts: 3184
Location: near Detroit, Michigan, USA

PostPosted: Sun Dec 20, 2009 3:33 am    Post subject: Reply with quote

daj95376 wrote:
Quite by accident, it appears that Puzzle NR 09/12/18 (A) might be a good source of examples.


Danny,

Yes, it is. But, I don't think I can make additional points. Maybe, we'll leave it up to the readers as an exercise for extra credits?

http://www.dailysudoku.com/sudoku/forums/viewtopic.php?t=4123

Keith
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Pat



Joined: 23 Feb 2010
Posts: 165

PostPosted: Tue Feb 23, 2010 11:38 am    Post subject: re: an X-Wing in 2 boxes Reply with quote

daj95376 wrote:

    Let's consider an X-Wing in 2 boxes
    and its potential eliminations.

      No eliminations remain for X-Wing in 2 boxes


so many discussions seem to have 3x3 hard-wired

Pat (2006.Nov.28) wrote:

  • boxes-to-rows ( or boxes-to-columns )
  • rows-to-boxes ( or columns-to-boxes )
example X-wing
b1 , b3 can have the digit only in c1 , c2;
exclude the digit elsewhere in those Columns

Code:

 X X / / | . . . .
 X X / / | . . . .
---------+---------
 X X / / | . . . .
 X X / / | . . . .
---------+---------
 * * . . | . . . .
 * * . . | . . . .
---------+---------
 * * . . | . . . .
 * * . . | . . . .


equivalent example X-wing
c3 , c4 can have the digit only in b5 , b7;
exclude the digit elsewhere in those Boxes

Code:

 . . / / | . . . .
 . . / / | . . . .
---------+---------
 . . / / | . . . .
 . . / / | . . . .
---------+---------
 * * X X | . . . .
 * * X X | . . . .
---------+---------
 * * X X | . . . .
 * * X X | . . . .




~ Pat
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