## Nice Loops instead of Forcing Chains

Everything about Sudoku that doesn't fit in one of the other sections

### Nice Loops instead of Forcing Chains

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  2348   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    37 133    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

: fixed PM grid (was missing row 1)
Last edited by Draco on Sat Apr 19, 2008 7:40 pm, edited 1 time in total.
Draco

Posts: 143
Joined: 14 March 2008

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`

dan
dan

ArkieTech

Posts: 3355
Joined: 29 May 2006
Location: NW Arkansas USA

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 /_______________________________________________________________________________________________`
daj95376
2014 Supporter

Posts: 2624
Joined: 15 May 2006

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<>3r7c3 -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  |+-------------------+-------------------+----------------+`
Mike Barker

Posts: 458
Joined: 22 January 2006

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
Draco

Posts: 143
Joined: 14 March 2008

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
Draco

Posts: 143
Joined: 14 March 2008

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.

dan
dan

ArkieTech

Posts: 3355
Joined: 29 May 2006
Location: NW Arkansas USA

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.
Mike Barker

Posts: 458
Joined: 22 January 2006

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.
daj95376
2014 Supporter

Posts: 2624
Joined: 15 May 2006

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
Draco

Posts: 143
Joined: 14 March 2008