Nice Loops: some exercises

Advanced methods and approaches for solving Sudoku puzzles

Nice Loops: some exercises

Postby Carcul » Tue Dec 20, 2005 11:36 pm

In this post I present six exercises for all those who like to use nice loops or just want to try out their use and apply the principles of the bilocation-bivalue plot technique. The purpose of the exercises is the same: to solve each puzzle using only nice loops (and, of course, the more basic techniques up to X-Wings and Swordfishes, for example; XY-Wings and anything beyond that is “not allowed”). The six puzzles presented here have been already posted in this forum.

Exercise 1:

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

This is an “easy” puzzle, useful for those who are making the “first steps” in the bilocation-bivalue plot technique. The solution can be reached easily.

Exercise 2:

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

Another easy puzzle. As an additional exercise, the reader may want to try to solve this grid with a single advanced nice loop (after the basic steps).

Exercise 3:

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

This one can be easily solved using uniqueness, but using the bilocation-bivalue plot is also "easy".

Exercise 4:

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

Menneske.no Very Hard #1751664

This one is a little harder but the solution is also straightforward.

Exercise 5:

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

This puzzle is similar to the "third toughest puzzle", but it is somewhat easier. See the Note.

Exercise 6:

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

This puzzle was posted in this forum in the thread entitled “Super Tough Sudoku from Crazy Dad...would love just a nudge” (16th November). It is the hardest of the six (at least from my viewpoint) and it is possible to find a lot of loops in it. See the Note.

Note: The aim of the last two exercises is also to solve the grids using only nice loops: however, contrary to exercises 1-4, in exercises 5 and 6 the nice loops can be not only simple, but also strong, weak, triple, or nice loops including disjoint subset links (I use the expression “triple nice loop” to designate a triple implication chain ‘composed’ of three simple nice loops).

Have fun.

Carcul
Last edited by Carcul on Mon Mar 06, 2006 12:27 pm, edited 5 times in total.
Carcul
 
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Postby Hud » Tue Dec 20, 2005 11:51 pm

Carcul, I dubbed #3 into the Pappocom format and it wouldn't accept it. That's pretty common (I think) on puzzles requiring advanced techniques. I guess I'll try it on paper and get my ego totally destroyed.
By the way, I don't know if English is your mother tongue, but you express yourself very well.
Hud
 
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Re: Nice Loops: some exercises

Postby Jeff » Wed Dec 21, 2005 1:24 am

Carcul wrote:The aim of this last exercise is also to solve the puzzle using only nice loops: however, in this case, these nice loops can be not only simple, but also strong, weak, triple, or nice loops including conditional disjoint subset links (I use the expression “triple nice loop” to designate a triple implication chain ‘composed’ of three simple nice loops).

This is great. Could you give an example of a CDSL (conditional disjoint subset link)?
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Postby Jeff » Wed Dec 21, 2005 4:10 am

Carcul,

I have completed excercise 1 using just a bilocation plot. I could post it (without solution) if anyone wants to see what my diagram looks like.

I am having a go at exercise 3 and only able to identify one simple nice loop so far. It is pretty tough.
Jeff
 
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Joined: 01 August 2005

Postby Hud » Wed Dec 21, 2005 5:20 am

Jeff, I have had a go at #3 also and think I'm in over my head. I did a couple basic moves and tried a swordfish on 7s but think it didn't work. I really need more swordfish work. I guess I'll go back to example 1 and really get humiliated lol.
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Joined: 29 October 2005

Postby Jeff » Wed Dec 21, 2005 6:06 am

Hud, here are the candidate grids after all the basic moves.

Exercise 1
Code: Select all
 *--------------------------------------------------------------------*
 | 1      8      45     | 2      349    6      | 359    7      39     |
 | 7      9      3      | 8      1      5      | 2      4      6      |
 | 45     2      6      | 349    7      34     | 8      1      359    |
 |----------------------+----------------------+----------------------|
 | 2      5      89     | 6      49     17     | 14     3      78     |
 | 3      7      1      | 45     8      2      | 6      9      45     |
 | 6      4      89     | 39     359    17     | 15     2      78     |
 |----------------------+----------------------+----------------------|
 | 459    13     2      | 345    6      34     | 7      8      1349   |
 | 8      6      7      | 1      234    9      | 34     5      234    |
 | 49     13     45     | 7      2345   8      | 349    6      12349  |
 *--------------------------------------------------------------------*

Exercise 2
Code: Select all
 *--------------------------------------------------------------------*
 | 456    3      45     | 1      2      79     | 479    8      467    |
 | 1      48     9      | 78     678    5      | 3      467    2      |
 | 7      2      68     | 389    368    4      | 1      69     5      |
 |----------------------+----------------------+----------------------|
 | 2      1      78     | 6      4      78     | 5      3      9      |
 | 9      457    4567   | 357    1      37     | 467    2      8      |
 | 4568   4578   3      | 578    9      2      | 47     1      467    |
 |----------------------+----------------------+----------------------|
 | 458    45789  2      | 3789   378    36789  | 4678   4567   1      |
 | 58     5789   1      | 4      78     6789   | 2      567    3      |
 | 3      6      478    | 2      5      1      | 4789   479    47     |
 *--------------------------------------------------------------------*

Exercise 3
Code: Select all
 *-----------------------------------------------------------*
 | 25    6     9     | 1     4     3     | 8     25    7     |
 | 1235  2358  7     | 9     26    58    | 16    25    4     |
 | 4     258   18    | 7     26    58    | 16    9     3     |
 |-------------------+-------------------+-------------------|
 | 9     24    24    | 8     5     7     | 3     6     1     |
 | 7     38    38    | 2     1     6     | 9     4     5     |
 | 6     1     5     | 3     9     4     | 2     7     8     |
 |-------------------+-------------------+-------------------|
 | 23    7     36    | 5     8     9     | 4     1     26    |
 | 8     249   24    | 6     7     1     | 5     3     29    |
 | 15    59    16    | 4     3     2     | 7     8     69    |
 *-----------------------------------------------------------*

Exercise 4
Code: Select all
 *-----------------------------------------------------------*
 | 2     3     67    | 5     4     167   | 9     17    8     |
 | 9     4     567   | 16    2678  2678  | 1267  257   3     |
 | 1     8     567   | 9     2367  2367  | 267   4     257   |
 |-------------------+-------------------+-------------------|
 | 67    5     4     | 36    1     236   | 8     9     27    |
 | 67    9     3     | 8     26    4     | 5     27    1     |
 | 8     2     1     | 7     9     5     | 3     6     4     |
 |-------------------+-------------------+-------------------|
 | 3     1     29    | 4     567   679   | 27    8     257   |
 | 4     6     28    | 13    3578  378   | 127   2357  9     |
 | 5     7     89    | 2     38    1389  | 4     13    6     |
 *-----------------------------------------------------------*

Exercise 5
Code: Select all
 *-----------------------------------------------------------------------------*
 | 29      29      1       | 57      8       3       | 6       57      4       |
 | 4       3       7       | 6       25      129     | 129     1589    12589   |
 | 5       6       8       | 1279    247     1249    | 1239    1379    1279    |
 |-------------------------+-------------------------+-------------------------|
 | 12379   124789  349     | 2789    2367    5       | 12349   134679  12679   |
 | 2379    2479    6       | 279     1       29      | 8       34579   2579    |
 | 12379   12789   5       | 4       2367    2689    | 1239    13679   12679   |
 |-------------------------+-------------------------+-------------------------|
 | 1679    1479    49      | 1258    2456    12468   | 149     14689   3       |
 | 1369    149     349     | 18      46      7       | 5       2       1689    |
 | 8       5       2       | 3       9       146     | 7       146     16      |
 *-----------------------------------------------------------------------------*

Exercise 6
Code: Select all
 *--------------------------------------------------------------------*
 | 7      156    456    | 2      458    146    | 3      458    9      |
 | 19     3      459    | 14589  7      149    | 2      458    6      |
 | 269    269    8      | 3      45     69     | 14     7      15     |
 |----------------------+----------------------+----------------------|
 | 3      2678   267    | 1478   9      1247   | 5      1246   128    |
 | 2689   4      25679  | 1578   258    127    | 1689   3      128    |
 | 289    2589   1      | 458    6      3      | 49     249    7      |
 |----------------------+----------------------+----------------------|
 | 12689  12689  269    | 469    24     5      | 7      1269   3      |
 | 4      2679   3      | 679    1      279    | 689    2569   258    |
 | 5      12679  2679   | 679    3      8      | 169    1269   4      |
 *--------------------------------------------------------------------*
Last edited by Jeff on Thu Dec 22, 2005 8:35 am, edited 3 times in total.
Jeff
 
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Postby Carcul » Wed Dec 21, 2005 10:23 am

Hi Hud.

Hud wrote:By the way, I don't know if English is your mother tongue, but you express yourself very well.


Thanks. In Portugal we all see a lot of movies speaked in English, and I guess that help.:D I think sometimes I make errors, but the important is that you and the other users understand what I write. My mother tongue is Portuguese, and sometimes I need to have the English dictionary "at hand" to help me write these posts.
Have fun with these puzzles, because that is the most important.:D

Regards, Carcul
Last edited by Carcul on Wed Dec 21, 2005 7:10 am, edited 2 times in total.
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Postby Carcul » Wed Dec 21, 2005 10:35 am

Hi Jeff.

Jeff wrote:This is great. Could you give an example of a CDSL (conditional disjoint subset link)?


Sorry. My intention was to write "disjoint subset link", a situation where we have an almost locked set (ALS) in which there are 2 candidates "a" and "b" that appear each in just one cell of the ALS, and so we can consider the link =a|b=. I have already edited the original post.

Jeff wrote:I have completed excercise 1 using just a bilocation plot. I could post it (without solution) if anyone wants to see what my diagram looks like.


Good work. However, forgive me to ask you to not post your plot, because I would like that as much people as possible can reach the solution just by himselfs, without any influence from the plot or solution from others.

Jeff wrote:I am having a go at exercise 3 and only able to identify one simple nice loop so far. It is pretty tough.


But, how about Exercise 2?:D

Regards, Carcul.
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Postby Jeff » Wed Dec 21, 2005 12:25 pm

Carcul wrote:But, how about Exercise 2?:D

I started with exercise 3 and went back to exercise 1 in despair. I haven't had a chance to try exercise 2 yet.:(
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Postby Carcul » Wed Dec 21, 2005 4:48 pm

I have added two more exercises in the original post.

Carcul
Last edited by Carcul on Wed Dec 21, 2005 9:00 pm, edited 1 time in total.
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Postby Hud » Wed Dec 21, 2005 10:13 pm

Jeff wrote:
Carcul wrote:But, how about Exercise 2?:D

I started with exercise 3 and went back to exercise 1 in despair. I haven't had a chance to try exercise 2 yet.:(


Jeff, I got the same candidate grid on puzzle 3 as you did. I still don't know if I did the swordfish correctly though. When I get time, I'll post my probably erroneous method.

Carcul, when I was in the army nearly 50 years ago, I had a friend who was a native of Brazil. As you know, they also speak Portugese which seemed similar to Spanish and a bit like French if I recall correctly.
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Postby Bob Hanson » Thu Dec 22, 2005 1:39 am

Exercise #5

I'll be interested to see how people do this one.

Sudoku Assistant reports: W2MwP+ or (with almost-locked sets) AaW3MwP+

The P+ here implies we need hypothesis and proof (trial and error) with at
least hidden singles "look-ahead". Some people refer to this as "bifurcation
tabling." Mw is weakly-linked chain analysis; W2/W3 are X-wings/Swordfish,
and A/a is almost-locked set analysis.

Without the hypothesis/proof business, we get here without almost-locked set analysis:

Code: Select all
   |---c1--|---c2--|---c3--||---c4--|---c5--|---c6--||---c7--|---c8--|---c9--
-----------------------------------------------------------------------------
r1 |     7 |   156 |   456 ||     2 |   458 |   146 ||     3 |   458 |     9
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r2 |    19 |     3 |   459 || 14589 |     7 |   149 ||     2 |   458 |     6
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r3 |   269 |   269 |     8 ||     3 |    45 |    69 ||    14 |     7 |    15
===========================||=======================||=======================
r4 |     3 |  2678 |   267 ||  1478 |     9 |  1247 ||     5 |  1246 |   128
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r5 |  2689 |     4 | 25679 ||  1578 |   258 |   127 ||  1689 |     3 |   128
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r6 |   289 |  2589 |     1 ||   458 |     6 |     3 ||    49 |   249 |     7
===========================||=======================||=======================
r7 | 12689 | 12689 |   269 ||   469 |    24 |     5 ||     7 |  1269 |     3
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r8 |     4 |  2679 |     3 ||   679 |     1 |   279 ||   689 |  2569 |   258
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r9 |     5 | 12679 |  2679 ||   679 |     3 |     8 ||   169 |  1269 |     4
-----------------------------------------------------------------------------


and here, just a bit better, with almost-locked set analysis:

Code: Select all
   |---c1--|---c2--|---c3--||---c4--|---c5--|---c6--||---c7--|---c8--|---c9--
-----------------------------------------------------------------------------
r1 |     7 |   156 |   456 ||     2 |   458 |    16 ||     3 |   458 |     9
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r2 |    19 |     3 |    45 || 14589 |     7 |   149 ||     2 |   458 |     6
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r3 |   269 |   269 |     8 ||     3 |    45 |    69 ||    14 |     7 |    15
===========================||=======================||=======================
r4 |     3 |  2678 |   267 ||  1478 |     9 |  1247 ||     5 |  1246 |   128
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r5 |  2689 |     4 | 25679 ||  1578 |   258 |   127 ||  1689 |     3 |   128
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r6 |   289 |  2589 |     1 ||   458 |     6 |     3 ||    49 |   249 |     7
===========================||=======================||=======================
r7 | 12689 | 12689 |   269 ||   469 |    24 |     5 ||     7 |  1269 |     3
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r8 |     4 |  2679 |     3 ||   679 |     1 |   279 ||   689 |  2569 |   258
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r9 |     5 | 12679 |  2679 ||   679 |     3 |     8 ||   169 |  1269 |     4
-----------------------------------------------------------------------------


good luck!:D
Bob Hanson
 
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Postby Hud » Thu Dec 22, 2005 5:42 am

Jeff, in reply to your candidate grid on puzzle 1, how did you eliminate the 5s at R1C9, R6C4, and R9C1? Pappocom rated it as a very hard but I appear to be stuck. I know I'm in over my head, but I guess this is the only way to learn.
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Postby Jeff » Thu Dec 22, 2005 7:53 am

Due to a swordfish, Hud.:D
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Postby Carcul » Thu Dec 22, 2005 12:18 pm

I have edited the original post (one more exercise added) for the last time.

Carcul
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