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AZ Matt
Joined: 03 Nov 2005
Posts: 63
Location: Hiding under my desk in Phoenix AZ USA
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 Posted: Fri Aug 25, 2006 9:31 pm Post subject: Kicking the x-wing up a notch... |
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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
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| Posted: Sat Aug 26, 2006 12:34 am Post subject: |
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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
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| Posted: Sat Aug 26, 2006 12:37 pm Post subject: Here's a corrected grid ... |
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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
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| Posted: Sun Aug 27, 2006 4:21 pm Post subject: Oops! |
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Sorry about the slip-up  .
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
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| Posted: Sun Aug 27, 2006 10:21 pm Post subject: |
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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
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| Posted: Mon Aug 28, 2006 9:21 am Post subject: |
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... 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
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| Posted: Mon Aug 28, 2006 4:15 pm Post subject: Combining "Nishio" with a "UR" |
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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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