Nice Loops instead of Forcing Chains

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Nice Loops instead of Forcing Chains

Postby Draco » Sat Apr 19, 2008 7:39 pm

I have a test case that I am hoping one of you Nice Loop experts will look at and tell me how to convert do the same thing with a NL (instead of a forcing chain).

From the Chicago Tribune online for 4-18-08:

Code: Select all
1 . 4 | . . . | . . 8
. . 7 | 8 6 . | 2 1 .
. 5 . | . . . | . . .
------+-------+------
. . . | . 1 8 | . 6 .
. . . | 6 . 2 | . . 7
. 6 . | 3 4 . | . . .
------+-------+------
. . . | . . . | . 4 .
. 4 2 | . 8 6 | 9 . .
3 . . | . . . | 8 . 5


STSS brings us to:
Code: Select all
1    2    4    | 79   379  379  | 6    5  8
9    3    7    | 8    6    5    | 2    1  4
6    5    8    | 14   2    14   | 37   37 9
---------------+----------------+-----------
2457 79   359  | 579  1    8    | 345  6  23
48   189  1359 | 6    59   2    | 1345 89 7
2578 6    159  | 3    4    79   | 15   89 12
---------------+----------------+-----------
578  1789 159  | 2    3579 1379 | 17   4  6
57   4    2    | 157  8    6    | 9    37 13
3    179  6    | 1479 79   1479 | 8    2  5


One way to crack the puzzle is to attack r8c9 to see if it can be forced to a 1 or a 3 (nice color chain involving 3's is what attracted me to this spot). My solver found:

r5c8=9=r6c8=8=r6c1=2=r6c9=1=r8c9 and
r5c8-9-r5c5-5-r8c4=1=r8c9
ergo r8c9<>3, r7c7<>1, r6c9<>1

SSTS to solve.

I played with this a bit and tried to find a Nice Loop that would show r8c9<>3 (or, of course, r8c9=1) but don't see it.

Pointers appreciated!

Cheers & thanks...

- drac

[edit]: fixed PM grid (was missing row 1)
Last edited by Draco on Sat Apr 19, 2008 7:40 pm, edited 1 time in total.
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Postby ArkieTech » Sat Apr 19, 2008 9:00 pm

I am not a nice guy but
Code: Select all
 *-----------------------------------------------------------*
 | 1     2     4     | 79    379   379   | 6     5     8     |
 | 9     3     7     | 8     6     5     | 2     1     4     |
 | 6     5     8     | 14    2     14    | 37    37    9     |
 |-------------------+-------------------+-------------------|
 | 2457  79    359   | 579   1     8     | 345   6     23    |
 | 48    189   1359  | 6     59    2     | 1345  89    7     |
 | 2578  6     159   | 3     4     79    | 15    89    12    |
 |-------------------+-------------------+-------------------|
 | 578   1789  159   | 2     3579  1379  | 17    4     6     |
 | 57    4     2     | 157   8     6     | 9     37    13    |
 | 3     179   6     | 1479  79    1479  | 8     2     5     |
 *-----------------------------------------------------------*
ur(14) causes a hidden pair(79)in b7=>r7c56,b9c2,r8c4<>79


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Postby daj95376 » Sat Apr 19, 2008 9:24 pm

ArkieTech's UR is interesting because it forces the three cells [r9c456] to reduce to two possibe Naked Triples.

Code: Select all
UR => [r9c456]=179|479 => ArkieTech's eliminations

Same cells but ...

Code: Select all
UR Type 4 => [r9c46]<>1 => [r9c2]=1 => SSTS

As for [r8c9]<>3, all I found was a forcing net.

Code: Select all
         / [r4c9]=2 [r6c9]=1 [r6c7]=5 [r6c3]=9 [r7c3]<>9 \
[r8c9]=3                                                   [r7c3]=5 [r8c1]=7 [r8c8]=3 [r8c9]<>3
         \ [r8c8]=7 [r7c7]=1                   [r7c3]<>1 /
_______________________________________________________________________________________________
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Postby Mike Barker » Sat Apr 19, 2008 10:21 pm

Here's a couple of ways for a nice guy to look at it:

As a simple nice loop:
Code: Select all
7-element Nice Loop: r8c9 =1= r8c4 =5= r4c4 -5- r5c5 -9- r5c8 =9= r6c8 =8= r6c1 =2= r4c1 -2- r4c9 ~3~ r8c9 => r8c9<>3
+-------------------+-------------------+----------------+
|    1     2     4  |   79   379   379  |    6   5    8  |
|    9     3     7  |    8     6     5  |    2   1    4  |
|    6     5     8  |   14     2    14  |   37  37    9  |
+-------------------+-------------------+----------------+
| 2457*   79   359  |  579*    1     8  |  345   6   23* |
|   48   189  1359  |    6    59*    2  | 1345  89*   7  |
| 2578*    6   159  |    3     4    79  |   15  89*  12  |
+-------------------+-------------------+----------------+
|  578  1789   159  |    2  3579  1379  |   17   4    6  |
|   57     4     2  |  157*    8     6  |    9  37  1-3* |
|    3   179     6  | 1479    79  1479  |    8   2    5  |
+-------------------+-------------------+----------------+

As a grouped nice loop:
Code: Select all
5-element Grouped Nice Loop: r4c9 -2- ALS:r6c379 -9- r4c23 =9= r4c4 =5= r8c4 =1= r8c9 ~3~ r4c9 => r8c9<>3
+-------------------+-------------------+----------------+
|    1     2     4  |   79   379   379  |    6   5    8  |
|    9     3     7  |    8     6     5  |    2   1    4  |
|    6     5     8  |   14     2    14  |   37  37    9  |
+-------------------+-------------------+----------------+
| 2457   79*c 359*c |  579*    1     8  |  345   6   23* |
|   48   189  1359  |    6    59     2  | 1345  89    7  |
| 2578     6  159*b |    3     4    79  |  15*b 89  12*b |
+-------------------+-------------------+----------------+
|  578  1789   159  |    2  3579  1379  |   17   4    6  |
|   57     4     2  |  157*    8     6  |    9  37  1-3* |
|    3   179     6  | 1479    79  1479  |    8   2    5  |
+-------------------+-------------------+----------------+

As a multi-inference nice loop (trivalue cell / Kraken Blossom):
Code: Select all
Kraken Blossom: r7c3=159 => r8c9<>3
r7c3 -1- r9c2 =1= r9c46 -1- r8c4 =1= r8c9 -1-
||
r7c3 -5- r7c5 =5= r8c4 =1= r8c9 -1-
||
r7c3 -9- r6c37 -1- r6c9 =1= r8c9 -1- )
+-------------------+-------------------+------------------+
|    1     2     4  |   79   379   379  |    6   5      8  |
|    9     3     7  |    8     6     5  |    2   1      4  |
|    6     5     8  |   14     2    14  |   37  37      9  |
+-------------------+-------------------+------------------+
| 2457    79   359  |  579     1     8  |  345   6     23  |
|   48   189  1359  |    6    59     2  | 1345  89      7  |
| 2578     6   159d |    3     4    79  |   15d 89     12d |
+-------------------+-------------------+------------------+
|  578  1789   159* |    2  3579c 1379  |   17   4      6  |
|   57     4     2  | 157bc    8     6  |    9  37  1-3bcd |
|    3   179b    6  | 1479b   79  1479b |    8   2      5  |
+-------------------+-------------------+------------------+

As a multi-inference nice loop (trilocal unit / Kraken unit)
Code: Select all
Kraken Row: r4c234=9 => r8c9<>3:
r4c3|r4c2 -9- r6c37 -1- r6c9 =1= r8c9 -1-
||
r4c4 -9|5- r4c4 =5= r8c4 =1= r8c9 -1-
+-------------------+-------------------+-----------------+
|    1     2     4  |   79   379   379  |    6   5     8  |
|    9     3     7  |    8     6     5  |    2   1     4  |
|    6     5     8  |   14     2    14  |   37  37     9  |
+-------------------+-------------------+-----------------+
| 2457    79*  359* | 579*c    1     8  |  345   6    23  |
|   48   189  1359  |    6    59     2  | 1345  89     7  |
| 2578     6   159b |    3     4    79  |   15b 89    12b |
+-------------------+-------------------+-----------------+
|  578  1789   159  |    2  3579  1379  |   17   4     6  |
|   57     4     2  |  157c    8     6  |    9  37  1-3bc |
|    3   179     6  | 1479    79  1479  |    8   2     5  |
+-------------------+-------------------+-----------------+

As a t-chain:
Code: Select all
6-element NRCT chain: r8c9 =1= r6c9 -1- r6c7 -5- (1)r6c3 -9- (r4c3)r4c2 =9= r4c4 =5= r8c4 =1= r8c9 ~1~  => r8c9=1
+-------------------+-------------------+----------------+
|    1     2     4  |   79   379   379  |    6   5    8  |
|    9     3     7  |    8     6     5  |    2   1    4  |
|    6     5     8  |   14     2    14  |   37  37    9  |
+-------------------+-------------------+----------------+
| 2457    79*  359  |  579*    1     8  |  345   6   23  |
|   48   189  1359  |    6    59     2  | 1345  89    7  |
| 2578     6   159* |    3     4    79  |   15* 89   12* |
+-------------------+-------------------+----------------+
|  578  1789   159  |    2  3579  1379  |   17   4    6  |
|   57     4     2  |  157*    8     6  |    9  37  1-3* |
|    3   179     6  | 1479    79  1479  |    8   2    5  |
+-------------------+-------------------+----------------+
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Postby Draco » Sat Apr 19, 2008 11:48 pm

daj95376 wrote:ArkieTech's UR is interesting because it forces the three cells [r9c456] to reduce to two possibe Naked Triples.

Code: Select all
UR => [r9c456]=179|479 => ArkieTech's eliminations

Same cells but ...

Code: Select all
UR Type 4 => [r9c46]<>1 => [r9c2]=1 => SSTS

Ok NL's aside, I think I ge tthe UR's but want to be sure. I see a Deadly Pattern (UR type 1) in r39c46 with (14), leaving a quantum (79) in r9c36, which forms an open pair with r9c5 to force r9c2<>79. Then STSS solves.

Is this just another way of stating ArkiTech's UR (and, in essence, the same as your UR type 4 in the same way that an Hidden Pattern is just a shortcut to a larger Open Pattern)?

Cheers...

- drac
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Postby Draco » Sat Apr 19, 2008 11:56 pm

Mike Barker wrote:Here's a couple of ways for a nice guy to look at it:

As a simple nice loop:
Code: Select all
7-element Nice Loop: r8c9 =1= r8c4 =5= r4c4 -5- r5c5 -9- r5c8 =9= r6c8 =8= r6c1 =2= r4c1 -2- r4c9 ~3~ r8c9 => r8c9<>3


Thanks Mike. I see it now that you've laid it out but I doubt I ever would've found that myself. What do you use to queue up squares for a NL like this? Do you stick with bi-values (based on pairs or colors)? Or do you just start working the puzzle in one (or two) directions, or...? I've found a different way to view forcing chains that makes them easier to find; it hasn't helped me find NL's though.

As for the additional loops -- well clearly I need to be better at visualizing simple NL's before I'll ever have a hope of finding these. Thank you for the examples!

Cheers...

- drac
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Postby ArkieTech » Sun Apr 20, 2008 12:30 am

Draco said:
Ok NL's aside, I think I ge tthe UR's but want to be sure. I see a Deadly Pattern (UR type 1) in r39c46 with (14), leaving a quantum (79) in r9c36, which forms an open pair with r9c5 to force r9c2<>79. Then STSS solves.


You got it. It is what I like about Sudoku, everyone sees a puzzle differently yet the final result is the same. My problem is when I see one solution -- that is all I see-- until someone shows me another (better) one.

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Postby Mike Barker » Sun Apr 20, 2008 2:01 am

Drac, I didn't mean to imply that I found those loops my myself - I used my solver thinking that you were interested in existence not discovery. I think finding nice loops, as with many things, is a matter of practice. I use Jeff's b/b plot to find them by hand. Maxberan has come up with a more systematic approach which you may find useful.
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Postby daj95376 » Sun Apr 20, 2008 3:52 am

Draco wrote:Is this just another way of stating ArkiTech's UR (and, in essence, the same as your UR type 4 in the same way that an Hidden Pattern is just a shortcut to a larger Open Pattern)?

I'm not sure how ArkieTech perceived his Hidden Pair conclusion. While observing cells [r9c456], I alternately forced cell [r3c4] to 1 and 4. The results were always the same -- [r9c456] alternated between Naked Triples once the UR restriction was applied. So, I posted my observation as a similar perspective to ArkieTech's approach. The common 79 eliminations left [r9c2]=1 as a side-effect.

The UR Type 4 is completely different. It uses cells [r39c46], but it relies on an X-Wing in 4 to force eliminations in 1, and also left [r9c2]=1 as a side-effect.

The common side-effect of [r9c2]=1 was reached from two different UR perspectives!
Last edited by daj95376 on Sun Apr 20, 2008 5:41 am, edited 1 time in total.
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Postby Draco » Tue Apr 22, 2008 3:03 am

Mike - More than fair and thanks for the explaination. I've seen the b/b plots but had not seen Maxberan's article before. The references are appreciated (note that I also use my solver to find chains... it just doesn't do NL's cuz I have not figured them out well enuf yet to create a resonable implementation).

Danny - Thank you for your follow-up explanation. Coincidences, it seems, abound in this puzzle. I'll have to keep checking the Tribune on Friday's now that I have a downloader for it working in my solver:) .

Cheers...

- drac
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