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Kicking the x-wing up a notch...

 
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AZ Matt



Joined: 03 Nov 2005
Posts: 63
Location: Hiding under my desk in Phoenix AZ USA

PostPosted: Fri Aug 25, 2006 9:31 pm    Post subject: Kicking the x-wing up a notch... Reply with quote

Here's a good puzzle. Plenty of advanced techniques to work through. It really helped me work out my advanced x-wing solving techniques, and I will use it as an example of the technique I have used to solve many extreme puzzles.

Code:
1..8.239.
3...4...2
..6......
.7..3.9..
.6.4.5.7.
..3.1..5.
......6..
6...7...1
.851.6..7


So using basic techniques, I get to here:

Code:
1    4  7     8    6  2    3    9   5
3    5  89    79   4  179  178  6   2
289  29 6     3    5  179  178  6   48

5    7  124   26   3  8    9    124 46
289  6  1289  4    29 5    128  7   3
2489 29 3     2679 1  79   248  5   486

7    1  24    5    8  3    6    24  9
6    3  29    29   7  4    5    8   1
249  8  5     1    29 6    24   3   7


I first look for x-wings, mainly because I think of it as the next "basic" solving technique, and as a consequence I am also looking for swordfish and what I call imbalanced patterns -- patterns that don't permit an organized placement of one number in the cells for which it is a candidate.

The swordfish on the <2> in columns 4, 7, and 8 is fairly easy to spot (the r4c34 and the r8c34 are just crying out to be an x-wing), but doesn't yield much.

Now if I were one who looked for every solving technique all the time, I would have spotted potential in the <29> swirling around in boxes 5, 7 and 8. But I am not that person. I didn't find any x-wings, but I am still fishing around (so to speak), and here's what I see.

I look for strong links and I see that the <9> has only two candidate cells in each of rows 8 and 9, columns 2 and 5, and boxes 7 and 8. Looks like a good candidate for fishing. On a grid it looks like this (there are no canditates in boxes 3, 6, and 9):

Code:
... ...
..9 9.9
99. ..9

... ...
9.9 .9.
99. 9.9
... ...
..9 9..
9.. .9.


Now, I will try to simplify my fishing technique. I look for potential x-wings in the pattern and spot many (at least six). The one thing I note is that only the r5c3 has insufficient partners to form a potential x-wing. An imbalance. And sure enough, simply by looking at the interplay of row 5 and rows 8 and 9, it is easy to see that r5c3 cannot be <9>. Testing it out, I also discover that I can eliminate the nine as a candidate in r3c1 and r6c1.

Usually, when I make a discoverylike this, the puzzle is nearly over. Not even close here. As the downhill mountain road sign says -- "Truckers -- You are not down yet!!!"

I enjoyed this puzzle so much I did it twice, and discovered you can get the same information I got from a remote pair on the <29>, removing the <9> from r5c3. This makes a swordfish possible on <9>, eliminating it from r3c1 and r6c1.

Anyway, I have caught a lot of fish this way. I have given up on keeping track of the names, though I would be curious if this one has a name ("Nisho"?).

And I thought I'd share the puzzle.
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Marty R.



Joined: 12 Feb 2006
Posts: 5770
Location: Rochester, NY, USA

PostPosted: Sat Aug 26, 2006 12:34 am    Post subject: Reply with quote

Matt, you've got some sort of typo what with two 6s in box 3.

Code:
1    4  7     8    6  2    3    9   5
3    5  89    79   4  179  178  6   2
289  29 6     3    5  179  178  6   48

5    7  124   26   3  8    9    124 46
289  6  1289  4    29 5    128  7   3
2489 29 3     2679 1  79   248  5   486

7    1  24    5    8  3    6    24  9
6    3  29    29   7  4    5    8   1
249  8  5     1    29 6    24   3   7
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David Bryant



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

PostPosted: Sat Aug 26, 2006 12:37 pm    Post subject: Here's a corrected grid ... Reply with quote

Here's the corrected version of Matt's grid, Marty.
Code:
1    4  7     8    6  2    3    9   5
3    5  89    79   4  179  178  6   2
289  29 6     3    5  179  1478 14  48

5    7  124   26   3  8    9    124 46
289  6  1289  4    29 5    128  7   3
2489 29 3     2679 1  79   248  5   486

7    1  24    5    8  3    6    24  9
6    3  29    29   7  4    5    8   1
249  8  5     1    29 6    24   3   7

Matt:

I'm also very fond of the technique you used on the "9"s in this puzzle. The classical name for this technique is "Nishio", and it's usually explained as a sort of proof by contradiction.
Code:
1    4   7     8    6   2    3    9   5
3    5   89    79   4   179  178  6   2
289* 29  6     3    5   179  1478 14  48

5    7   124   26   3   8    9    124 46
289  6   1289  4    29  5    128  7   3
2489 29* 3     2679 1   79   248  5   486

7    1   24    5    8   3    6    24  9
6    3   29*   29   7   4    5    8   1
249  8   5     1    29* 6    24   3   7

r3c1 = 9 ==> r6c2 = 9
r3c1 = 9 ==> r8c3 = 9 ==> r9c5 = 9

See the cells marked "*" above -- clearly we cannot place a "9" in box 5 if these 4 cells contain "9", and therefore r3c1 cannot be a "9".

I noticed a "UR" pattern in the "9"s while I was performing the Nishio analysis -- that was kind of cute, but I think I'll wait untiil later to explain it ... this post is already a bit long. dcb
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AZ Matt



Joined: 03 Nov 2005
Posts: 63
Location: Hiding under my desk in Phoenix AZ USA

PostPosted: Sun Aug 27, 2006 4:21 pm    Post subject: Oops! Reply with quote

Sorry about the slip-up Embarassed .

I usually finish a puzzle a forget about it, but I have done this one three times now, and it seems endless how many (very difficult) ways there are to solve it.
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keith



Joined: 19 Sep 2005
Posts: 3355
Location: near Detroit, Michigan, USA

PostPosted: Sun Aug 27, 2006 10:21 pm    Post subject: Reply with quote

Or, there is a remote naked pair on <29>, an X-wing, a Swordfish ...

Keith
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ravel



Joined: 21 Apr 2006
Posts: 536

PostPosted: Mon Aug 28, 2006 9:21 am    Post subject: Reply with quote

... or you can solve it with the remote pair, 2 x-wings, 2 xy-wings, xyz-wing.
Really interesting puzzle.
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David Bryant



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

PostPosted: Mon Aug 28, 2006 4:15 pm    Post subject: Combining "Nishio" with a "UR" Reply with quote

Here's how I solved this puzzle, starting from the position Matt posted.
Code:
1    4   7     8    6   2    3    9   5
3    5   89    79+  4   179  178  6   2
289  29+ 6     3    5   179  1478 14  48
5    7   124   26   3   8    9    124 46
289+ 6   1289  4    29- 5    128  7   3
2489 29  3     2679 1   79+  248  5   486
7    1   24    5    8   3    6    24  9
6    3   29+   29-  7   4    5    8   1
249- 8   5     1    29+ 6    24   3   7

I started my analysis of the "9"s from r9c5.

r9c5 = 9 ==> r8c3 = 9
r9c5 = 9 ==> "9" in box 5 lies in row 6 ==> r5c1 = 9
(r5c1 = 9 & r8c3 = 9) ==> r3c2 = 9

These cells are marked with a "+" sign in the grid. Here's how we find a "UR" by following a chain.

r9c5 = 9 ==> r5c5 = 2 ==> r4c4 = 6 ==> {7, 9} at r6c4

But if this is true we almost have the "deadly pattern" in r26c46, so that r2c6 must be a "1". Therefore

r9c5 = 9 ==> r2c4 = 9 ==> r6c6 = 9

What if r9c5 is not 9?

r9c5 <> 9 ==> (r5c5 = 9 & r8c4 = 9 & r9c1 = 9)

These cells are marked "-" in the grid ... we can apparently eliminate "9" from r3c1, r5c3, r6c1, and r6c4, revealing an X-Wing in c34r28 that also allows us to eliminate the "9" at r2c6. (Keith should like this -- it combines "Nishio" / coloring with a "UR"). Now the grid looks like this.
Code:
1    4   7     8    6  2    3    9   5
3    5   89a   79   4  17   178  6   2
28A  29  6     3    5  179  1478 14  48B
5    7   124a  26   3  8    9    124 46*
289  6   128a  4    29 5    128  7   3
248  29a 3     267  1  79   248  5   486
7    1   24    5    8  3    6    24  9
6    3   29    29   7  4    5    8   1
249  8   5     1    29 6    24   3   7

A straightforward double-implication chain finishes it off.

A1. r3c1 = 2 ==> r6c2 = 2 ==> {1, 4} at r4c3 & {1, 8} at r5c3
A2. r3c1 = 2 ==> r2c3 = 8 ==> r5c3 = 1 ==> r4c3 = 4 ==> r4c9 = 6
B. r3c1 = 8 ==> r3c9 = 4 ==> r4c9 = 6

So r4c9 = 6, and the rest is easy. dcb
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