The New Sudoku Players' Forum

[Havard]

Hi.

I have been trying to get my head around the concepts of a grouped nice loop. I was wondering if any of you could take a look at this one and tell me if I am on the right track?

(this is a very "academic" loop, with no real use (I think) in actually solving the puzzle. I just wanted to check if I have gotten the concepts right. I marked each node with a number to make it easier to follow. (for myself anyway...) It is nr. 24 from the top1465)

Code: Select all

3     168#6   7       | 59#4   5689- 4        | 19#3  2     569#3         
456   26#6    2456    | 1      2569  235679#1 | 8     3456  345679#2-
9     1268#6  14568   | 2357#8 2568  235678-  | 1347  13456 34567         
----------------------+-----------------------+----------------------------
1678- 12678#6 128     | 457#7  3     15       | 147-  9     4568           
167   5       39      | 8      1249  1279     | 12347 1346  3467           
178   4       39      | 6      1259  12579    | 1237  1358  3578           
----------------------+-----------------------+----------------------------
4678  36789-  468     | 2349   24689 23689    | 5     3478- 1             
2     1368#6  14568-  | 345#5  7     1568-    | 349#5 348#5 3489#5         
14578 13789-  1458    | 349    14589 13589    | 6     3478- 2             

[r2c6]=7=[r2c9]=9=[r1c79]-9-[r1c4]-5-[r8c4789]-3-[r1237c2]-7-[r4c4]=7=[r3c4]-7-[r2c6]

r2c9<>3,4,5,6
r1c5<>9
r8c3<>4,8
r79c8<>8
r8c6<>8
r9c2<>1,8
r7c2<>6,8
r4c1<>7
r3c6<>7

I like this one because (as far as I can tell) it has all the different connections and node-types. grouped / non grouped with strong / weak links

Havard

[Viggo]

Havard, this is a wonderfull Nice Loop you have found here!

I support here the uncommented puzzle (easy to load) and some graphics of the links.

Code: Select all

 3      168    7      | 59     5689   4      | 19     2      569   
 456    26     2456   | 1      2569   235679 | 8      3456   345679
 9      1268   14568  | 2357   2568   235678 | 1347   13456  34567
----------------------+----------------------+----------------------
 1678   12678  128    | 457    3      15     | 147    9      4568 
 167    5      39     | 8      1249   1279   | 12347  1346   3467 
 178    4      39     | 6      1259   12579  | 1237   1358   3578 
----------------------+----------------------+----------------------
 4678   36789  468    | 2349   24689  23689  | 5      3478   1     
 2      1368   14568  | 345    7      1568   | 349    348    3489 
 14578  13789  1458   | 349    14589  13589  | 6      3478   2     

[r2c6]=7=[r2c9]=9=[r1c79]-9-[r1c4]-5-[r8c4789]-3-[r1237c2]-7-[r4c4]=7=[r3c4]-7-[r2c6]

r2c9<>3,4,5,6
r1c5<>9
r8c3<>4,8
r79c8<>8
r8c6<>8
r9c2<>1,8
r7c2<>6,8
r4c1<>7
r3c6<>7

I think the Nice Loop is all right except for a typo in [r1237c2], which should be [r1238c2]. It is like two naked quads are "released" as part of the loop in r8 and in c2. I suppose they are called ALS? The following reduction should also be possible:

r34c4<>5
r4c7<>7 (I forgot this one in the graphics)

I think your "good question" is: Are the following reductions valid?

r8c3<>4,8
r79c8<>8
r8c6<>8
r9c2<>1,8
r7c2<>6,8

I think these reductions seems resonable, but I do not have the skills to sure about them.

/Viggo

[Carcul]

This is a puzzle from Top87, right? Anyway, good find Havard. However, although the following deductions may be correct, you cannot make them with your loop:

r8c3<>4,8; r79c8<>8; r8c6<>8.

All other eliminations you listed are correct with your loop. Also, you forgot to include the reduction r4c7<>7.

Carcul

[Viggo]

However, although the following deductions may be correct, you cannot make them with your loop:

r8c3<>4,8; r79c8<>8; r8c6<>8.

If you reverse the direction of the loop, it should look like this:

[r2c6]-7-[r3c4]=7=[r4c4]-7-[r1234c2]-16-[r8c2789]-5-[r1c4]-9-[r1c79]=9=[r2c9]=7=[r2c6]

I think that the two directions of this Nice Loop represent the two states this loop can be in. When both states imply the same candidates to be eliminated, then I think such a candidate elimination should be valid. Therefore:

The elimination, r79c8<>8 is NOT valid in the reversed direction.
The eliminations, r8c3<>4,8 and r8c6<>8 are valid.

and the other eliminations are valid.


/Viggo

[Havard]

This is a puzzle from Top87, right? Anyway, good find Havard. However, although the following deductions may be correct, you cannot make them with your loop:

r8c3<>4,8; r79c8<>8; r8c6<>8.

All other eliminations you listed are correct with your loop. Also, you forgot to include the reduction r4c7<>7.

Carcul

Hi Carcul and Viggo! Thanks for your feedback (and graphics! great! )

I understand from your comment that it is my ALS reductions that you don't agree with? I thought when you had an ALS in a "closed" loop, you can eliminate all other candidates that is not part of the connections to other ALS or SL/GSL. if you take the node out:

-5-[r8c4789]-3-

and write it as an ALS:
-5-[34589]-3-

subtract the 5 and 3:
-5-[489]-3-

You can now eliminate any 4, 8 or 9 that can see all the 4, 8's or 9's in that ALS. Hence those eliminations.

Don't you concider this part of the Nice Loop domain to do these eliminations? If not, I think it should be added, since after all these are well known when dealing with ALS. And it makes the Nice Loop even more powerful!

Havard

[ronk]

... although the following deductions may be correct, you cannot make them with your loop:

r8c3<>4,8; r79c8<>8; r8c6<>8.

The elimination, r79c8<>8 is NOT valid in the reversed direction.
The eliminations, r8c3<>4,8 and r8c6<>8 are valid.

All of Havard's deductions look valid to me. For the ALS set in dispute ... set A = {r8c4789} = {34589} ...

when 5 is removed {3489} remains
when 3 is removed {4589} remains.

Since the loop is continuous (closed), common digits {489} are locked in set A. This excludes {489} from all other cells in r8.

Digits {89} of A appear only in box 9. IOW {89} are locked in box 9 and may be excluded from all other cells of box 9. This is a valid deduction of the continuous loop not explicitly shown by Harvard's nice loop expression. Therefore, I suppose it's correct to say exclusions based on it are invalid, but "my pencil is not that sharp".

If you reverse the direction of the loop, it should look like ...

Nice loop expressions -- at least those for double implication streams -- are meant to be read both left-to-right and right-to-left, so there should not be a need to write a NL for the "reverse direction".

[ravel]

Yet another way to look at it (similar to Ron's):
The elimination of 8 in r79c8 does not follow directly from the loop as denoted, but it follows from the cells contained in the loop alone, because an 8 in r79c8 locks r8c789 to 349 => r8c4=5 and, following the loop, r8c2 would forced to be 3.

[Carcul]

If you reverse the direction of the loop, it should look like this:

[r2c6]-7-[r3c4]=7=[r4c4]-7-[r1234c2]-16-[r8c2789]-5-[r1c4]-9-[r1c79]=9=[r2c9]=7=[r2c6]

I know that all eliminations are correct, but I was referring to the rules of Continuous Nice Loops: you cannot eliminate any "8" in row 8 because, in the way you have written the loop you don't have any weak link "-8-". A better way, in my opinion, would be:

[r2c6]-7-[r3c4]=7=[r4c4]-7-[r4c2]=7|3=[r8c2](-3-[r8c4])-3-[r8c789]-8-
-[r8c4]-5-[r1c4]-9-[r1c79]=9=[r2c9]=7=[r2c6].

Nice loop expressions -- at least those for double implication streams -- are meant to be read both left-to-right and right-to-left, so there should not be a need to write a NL for the "reverse direction".

That is very true.

Carcul

[Viggo]

I have to agree, that the reduction, r79c8 is all right. Ravels deduction and Carsuls new way of writing the Nice loop makes it very clear to me.

If you reverse the direction of the loop, it should look like ...

Nice loop expressions -- at least those for double implication streams -- are meant to be read both left-to-right and right-to-left, so there should not be a need to write a NL for the "reverse direction".

Yes, you are right, but sometimes the other direction puts some other perspective on reduction for me, and when it becommes difficult. In this case the reverse direction change the cells of the two ALS involved. In the first (forward) direction, the two ALS is placed in r8c4789 and r1238c2. In the other direction they are placed in r1234c2 and r8c2789.

This makes me think about, if five cells should be included in each of the two ALS so no shift of cells in the ALS takes place, when you reverse the direction. Can you help me here?

/Viggo

[ronk]

In the first (forward) direction, the two ALS is placed in r8c4789 and r1238c2. In the other direction they are placed in r1234c2 and r8c2789.

This makes me think about, if five cells should be included in each of the two ALS so no shift of cells in the ALS takes place, when you reverse the direction. Can you help me here?

Maybe Havard's typo has you sidetracked a bit. The two chained ALSs are r8c4789 and r12348c2 with the chain fragment ...

-5-r8c4789-3-r12348c2-7-

Now that looks great, but it's too terse for me. I prefer a more verbose expression like ...

-5-{ALS():r8c4=5|489|3=r8c4789}-3-{ALS(r12348c2):r8c2=3|1268|7=r4c2}-7-

The set contents and implicaiton streams for left-to-right and right-to-left flows are:

Implication streams:
L->R: r8c4<>5 -> r8c4789=3 -> r8c2<>3 -> r4c2=7
L<-R: r8c4=5 <- r8c4789<>3 <- r8c2=3 <- r4c2<>7
(Note the change in arrow directions and the swap of "=" and "<>".)

Set contents: Where set A = {r8c4789} and set B = {r12348c2} ...
L->R: A = {4893} and B = {12687}
L<-R: A = {5489} and B = {31268}

When the chain fragment is part of a continuous loop, the ALS digits between the vertical bars ('|") in the verbose expression are locked. In this example then, it's rather easy to see that {489} and {1268} are locked in sets A and B, respectively, without needing to refer to the grid.

[edit: fixed typo]

[Viggo]

ronk, thank you for this very clear description on these two ALSs, and especially your comment about the locked part of the ALSs.

I do also recognize Carsuls comment on the missing weak link "-8-". However I think that Havards and Ronks comments makes it clear, that the locked parts of candidate numbers in the ALSs also are able to eliminate candidates in the units, they are placed and when they are a part of a continuous Nice Loop. Because of the ALS in r8c4789 has the locked candidate, 8 placed in r8 and box 9 then r8c236<>8 and r79c8<>8 (As Havard already explained). So I think that also r8c2<>8 - am I right?

/Viggo

[ronk]

So I think that also r8c2<>8 - am I right?

Of course, but don't tell Havard ... else he''ll start calling it a cannibalistic loop.

[Havard]

So I think that also r8c2<>8 - am I right?

Of course, but don't tell Havard ... else he''ll start calling it a cannibalistic loop.

lol! (too late now...)

Very interesting though! Are there any general rules that decides what eliminations that can "eat away" at the loop itself? My guess would be that this would only apply to eliminations of this kind: that the "extra" candidates of an ALS that ends up forming a Locked Set in a closed loop can eat at any other bits of the loop...? Anyone feel like trying to prove me wrong or right...?

Havard

[GreenLantern]

Very interesting grouped nice loop! I think that r34c4<>5 can also be added to your list of eliminations that follow from the continuous nice loop.

[ronk]

I think that r34c4<>5 can also be added to your list of eliminations ...

I agree and think that's the last of them ... unless you want to back up to Simple Sudoku's sticking point ...

Code: Select all

 3      B168     7       |*59     568(9)  4        |*19     2      *569
 456    B26      2456    | 1      2569   *235679   | 8      3456   *79(3456)
 9      B1268    124568  |*237(5) 2568    23568(7) | 1347   13456   34567
-------------------------+-------------------------+------------------------
 168(7) B12678   1268    |*47(5)  3       15(7)    | 14(7)    9     4568
 167     5       39      | 8      1249    1279     | 12347  1346    3467
 178     4       39      | 6      1259    12579    | 1237   1358    3578
-------------------------+-------------------------+------------------------
 4678    379(68) 468     | 2349   24689   23689    | 5      347(8)  1
 2      B136(8)  156(48) |A345    7       156(38)  |A349   A348    A3489
 14578   379(18) 1458    | 349(5) 14589   13589    | 6      347(8)  2

... in which case there are also exclusions r9c4<>5, r8c6<>3 and r4c6<>7. Here hopefully all 23 exclusions (based on the loop found by Havard) are shown in parentheses.

r2c6=7=r2c9=9=r1c79-9-r1c4-5-{A:r8c4=5|489|3=r8c4789} -3- {B(r12348c2):r8c2=3|1268|7=r4c2}-7-r4c4=7=r3c4-7-r2c6