The New Sudoku Players' Forum

by **David P Bird** » Sat Aug 29, 2015 6:26 pm

champagne wrote:I'll check in details later, but my first reaction is that we have no SK loop and no 'V' loop here.

IMO the only difference between the SK loop and the "V" loop is the constraint added by "ronk" to have 2 unknown cells in the boxes not belonging to the loop to qualify it of "sk loop".

To have a "V" loop, You must have the possibility to chain disjoints pairs of candidates in the sequence row-box-column-box-row-box-column-box-loop

in XSUDO, this is normally a 16 truths (4x4cells in 4 boxes) and 16 links (2x2 rows 2x2 columns 4x2 boxes).

Thanks for the quick response and I think you are confirming my memories. However in loops that Blue presented on the programmer's forum and also in some of yours I also seem to remember that some links in the loop didn't follow the regular SK Loop pattern. This makes me wonder if a sort of hybrid loop is possible where the link types switch in different houses.
.

by **ronk** » Sat Aug 29, 2015 6:39 pm

Leren wrote:Just for fun I ran a Multisector Locked set survey on this puzzle and came up with the following 29 MSLS arrangements for the same 16 eliminations.

MSLS 1 : Base 78; r1234 c5689 & r7c5689 : 20 Links; 78r3 7r4 & 8r7 ; 124c5 259c6 145c8 259c9 ; 36b2 36b3 ;
MSLS 2 : Base 78; r1237 c5689 & r4c5689 : 20 Links; 78r3 8r7 & 7r4 ; 124c5 259c6 145c8 259c9 ; 36b2 36b3 ;
MSLS 3 : Base 78; r12347 c5689 : 20 Links; 78r3 7r4 8r7 ; 124c5 259c6 145c8 259c9 ; 36b2 36b3 ;
MSLS 4 : Base 378; r1234 c5689 & r7c5689 : 20 Links; 78r3 7r4 & 8r7 ; 124c5 259c6 145c8 259c9 ; 36b2 36b3 ;
MSLS 5 : Base 378; r1237 c5689 & r4c5689 : 20 Links; 78r3 8r7 & 7r4 ; 124c5 259c6 145c8 259c9 ; 36b2 36b3 ;
MSLS 6 : Base 378; r12347 c5689 : 20 Links; 78r3 7r4 8r7 ; 124c5 259c6 145c8 259c9 ; 36b2 36b3 ;
...
I haven't run MSLS for a couple of years but I think they are all different from the 7 arrangements already posted. That's 36 ways to skin this particular cat - 27 more lives than a cat normally has !!!

Leren

Other than grouping and ordering, I see no difference in these first six "MSLS"; same strong inference sets and same weak inference sets. What difference do you see?

by **Leren** » Sat Aug 29, 2015 11:21 pm

ronk wrote : Other than grouping and ordering, I see no difference in these first six "MSLS"; same strong inference sets and same weak inference sets. What difference do you see?

A quick look at the links indicates that the following MSLS groups are the same apart from grouping and ordering : 1-6, 7-13, 4-16, 17-20, 21-25, 26-29. So that's 6 essentially different SLGs as far as I can see.

As I said before, I haven't run the MSLS code (or even thought about MSLS) for 2 or 3 years, so I'm obviously very rusty. One possible cause of the redundancy (at least as far as my rusty memory goes) is that my code allowed for cell grids that were not necessarily rectangular. They either included partial extra rows/columns, or solved cells within a rectangular array, or both. In cases where the "partial" extra row or column is actually a full one you can end up with a redundant set.

Obi-Whan found 4 essentially different SLGs in the group and he might be right, there may be more redundancy that I haven't picked up on. From his latest post he (apparently) now sees 35 different SLGs for the puzzle (so far), so I'm very happy for him. I'll bet that they all produce the same 16 eliminations !

Leren

by **Obi-Wahn** » Sun Aug 30, 2015 7:54 am

It turned out that for each of the numbers 12459 I have five options for the left band. I can either use columns 123 as truth sets or any combination of box 4 or row 4 and box 7 or row 7. Thats five to the the power of five combinations that can then be combined with either column or cell truths for column 4 and column 7. To illustrate this I used a different option for each number in the following diagram: 1c123, 2b47, 4b4 + 4r7, 5r4 + 5b7 and 9r47.

20 Truths = {4R7 5R4 9R47 1C123 12459C7 5689N4 2B47 4B4 5B7}
20 Links = {1r58 2r689 4r5 5r9 3c4 7n1 4n2 47n3 13n7 1b1 4b8 5b56 9b68}

The same works for the top band instead also. It's like some sort of SLG construction kit and adding in the possiblities to replace 12459b4 by 56n123, 12459r4 by 4n5689, 4n56 + 56n4 by 367b5 and the like the final tally rose to 26675 combinations. I gave up on the idea of listing them all.

[edit] I completely forgot about the possibilities to replace 3b89 by 3r89 or 7b58 by 7c56, so the actual count is even higher.

In order to understand all the possibilities I made a list of the relevant transformation rules and by doing so I realized why combinations of truths in box 23 and box 47 weren't possible, but I also realized that I went too far in ruling them all out, because some combinations are indeed possible and that will further rise the count.

This is what I believe to be the complete list of relevant transformations:

Code: Select all: 1) 4N56,56N4/125b5 <=> 367B5/5n6,6n5 2) 4N89,56N7/459b6 <=> 678B6/5n9,6n8 3) 7N56,89N4/249b8 <=> 378B8/8n6,9n5 4) 7N89,89N7/125b9 <=> 368B9/8n9,9n8 5) 6B56 <=> 6R56 6) 7B56/7r4 <=> 7R56/7b4 7) 3B89 <=> 3R89 8) 8B89/8r7 <=> 8R89/8b7 9) 3B58/3c4 <=> 3C56/3b2 10) 7B58 <=> 7C56 11) 6B69/6c7 <=> 6C89/6b3 12) 8B69 <=> 8C89 13) 4N5689/7r4 <=> 12459R4/4n23 14) 7N5689/8r7 <=> 12459R7/7n13 15) 5689N4/3c4 <=> 12459C4/23n4 16) 5689N7/6c7 <=> 12459C7/13n7 17) 1R4/1b5 <=> 1B4/1r5 18) 2R4/2b5 <=> 2B4/2r6 19) 4R4/4b6 <=> 4B4/4r5 20) 5R4/5b56 <=> 5B4/5r56 21) 9R4/9b6 <=> 9B4/9r6 22) 1R7/1b9 <=> 1B7/1r8 23) 2R7/2b89 <=> 2B7/2r89 24) 4R7/4b8 <=> 4B7/4r8 25) 5R7/5b9 <=> 5B7/5r9 26) 9R7/9b8 <=> 9B7/9r9 27) 1C4/1b5 <=> 1B2/1c5 28) 2C4/2b58 <=> 2B2/2c56 29) 4C4/4b8 <=> 4B2/4c5 30) 5C4/5b5 <=> 5B2/5c6 31) 9C4/9b8 <=> 9B2/9c6 32) 1C7/1b9 <=> 1B3/1c8 33) 2C7/2b9 <=> 2B3/2c9 34) 4C7/4b6 <=> 4B3/4c8 35) 5C7/5b69 <=> 5B3/5c89 36) 9C7/9b6 <=> 9B3/9c9 37) 12459B2/23n4 <=> 123N56/368b2 38) 12459B3/13n7 <=> 123N89/367b3 39) 12459B4/4n23 <=> 56N123/378b4 40) 12459B7/7n13 <=> 89N123/678b7 41) 1B23 <=> 1R123/1b1 42) 2B23 <=> 2R123/2b1 43) 4B23 <=> 4R12 44) 5B23 <=> 5R12 45) 9B23 <=> 9R23 46) 1B47 <=> 1C123/1b1 47) 2B47 <=> 2C123/2b1 48) 4B47 <=> 4C12 49) 5B47 <=> 5C13 50) 9B47 <=> 9C23

It should be possible to generate all the combinations by successively applying these transformations.

by **ronk** » Sun Aug 30, 2015 4:04 pm

Obi-Wahn wrote:This is what I believe to be the complete list of relevant transformations:
Code: Select all
... ... 41) 1B23 <=> 1R123/1b1 42) 2B23 <=> 2R123/2b1 43) 4B23 <=> 4R12 44) 5B23 <=> 5R12 45) 9B23 <=> 9R23 46) 1B47 <=> 1C123/1b1 47) 2B47 <=> 2C123/2b1 48) 4B47 <=> 4C12 49) 5B47 <=> 5C13 50) 9B47 <=> 9C23

It should be possible to generate all the combinations by successively applying these transformations.

While I believe your list to be correct, I think it overstates the case for several reasons. An easy observation is that 4B23 would tolerate additional candidates 4R3C58 (4R3) and thus can be said to be broader in scope than 4R12. [Of course, given r3c3=4 would not be possible as in this puzzle.]

That said, I have no idea how to easily filter such "breadths in scope."

by **ronk** » Mon Aug 31, 2015 9:17 pm

champagne wrote:
David P Bird wrote:The chain representing the MSLS is
(125#2=3)r56c4 - (3=249#2)r89c4 - (249#2=8)r7c56 - (8=125#2)r7c89 -
(125#2=6)r89c7 - (6=459#2)r56c7 - (459#2=7)r4c89 - (7=125#2)r4c56 – Loop => 16 Eliminations

The usual understanding for an ABC#2 term is that it will be true when it contains at least two truths and false otherwise.
With this understanding it is an AIC continuous loop which will allow all the eliminations for (3678) which appear as single digit terms.

However ......

So Champagne, is this a model that is true for all 'V loops' or not?

I'll check in details later, but my first reaction is that we have no SK loop and no 'V' loop here.

For ~~an AIC loop~~ a possible V-loop, I would choose this continuous loop of four ALSs in boxes b5689. [Both [b]blue[/b] and David P Bird believe my original chain (quoted in blue's post) is an invalid AIC, so I've stricken the term for this loop and switched to old-timey NL notation.]

Code: Select all: In NL notation: -3- r56c4 =3|125|7= r4c56 -7- r4c89 =7|459|6= r56c7 -6- r89c7 = 6|125|8= r7c89 -8- r7c56 =8|249|3= r89c4 - continuous loop

____

Details including the 16 exclusions: Show

by **blue** » Tue Sep 01, 2015 4:23 am

ronk wrote:For an AIC loop, I would choose this continuous loop of four ALSs in boxes b5689:

Code: Select all
(3125=1257)r56c4,r4c56 - (7459=4596)r4c89,r56c7 - (6125=1258)r89c7,r7c89 - (8249=2493)r7c56,r89c4 - loop

The problem with it, is that the strong links aren't valid "on thier own".
They can be seen to be valid, once a proper loop is formed, but they can't be used to establish that same loop.

How about one of these instead ?

Code: Select all: (3=7)r56c4,r4c56 - (7=6)r4c89,r56c7 - (6=8)r89c7,r7c89 - (8=3)r7c56,r89c4 - loop 3r56c4 = (1257-3125)r56c4,r4c56 = 7r4c56 - 7r4c89 = (4596-7459)r4c89,r56c7 = 6r56c7 - 6r89c7 = (1258-6125)r89c7,r7c89 = 8r7c89 - 8r7c56 = (2493-8249)r7c56,r89c4 = 3r89c4 - loop

by **David P Bird** » Tue Sep 01, 2015 9:29 am

blue wrote:How about one of these instead ?

Code: Select all
(3=7)r56c4,r4c56 - (7=6)r4c89,r56c7 - (6=8)r89c7,r7c89 - (8=3)r7c56,r89c4 - loop 3r56c4 = (1257-3125)r56c4,r4c56 = 7r4c56 - 7r4c89 = (4596-7459)r4c89,r56c7 = 6r56c7 - 6r89c7 = (1258-6125)r89c7,r7c89 = 8r7c89 - 8r7c56 = (2493-8249)r7c56,r89c4 = 3r89c4 - loop

Blue, your first suggestion doesn't make the eliminations for (2) and (5) in the boxes so still needs further explanations. I'm also personally not in favour of only partially identifying the digits in an ANS.

However your second suggestion is a perfectly sound AIC that makes all the eliminations with no further explanations necessary! For me it was quite a new way to view the workings of the loop, so I immediately tried it out on a regular SK loop but sad to say couldn't reduce it to an AIC.

Its only drawback is that it's rather long, but as the pattern will be very rare that's not such a hardship compared to using a shorthand notation which requires an explanation.

DPB

TAGdpbSKLoop

by **ronk** » Tue Sep 01, 2015 2:55 pm

blue wrote:
ronk wrote:For an AIC loop, I would choose this continuous loop of four ALSs in boxes b5689:
Code: Select all
(3125=1257)r56c4,r4c56 - (7459=4596)r4c89,r56c7 - (6125=1258)r89c7,r7c89 - (8249=2493)r7c56,r89c4 - loop
The problem with it, is that the strong links aren't valid "on thier own".
They can be seen to be valid, once a proper loop is formed, but they can't be used to establish that same loop.

How about one of these instead ?

I replaced it with NL notation with which I'm more familiar.

Code: Select all: -3- r56c4 =3|125|7= r4c56 -7- r4c89 =7|459|6= r56c7 -6- r89c7 = 6|125|8= r7c89 -8- r7c56 =8|249|3= r89c4 - continuous loop

The digits '|xxx|' between the '|' pairs are the digits locked in the respective ALS, when the ALS becomes an LS, of course.

[edit: Added missing labels on weak links in NL notation. Hmm, wonder what else I forgot.]

by **blue** » Tue Sep 01, 2015 4:16 pm

ronk wrote:I replaced it with NL notation with which I'm more familiar.
Code: Select all
- r56c4 =3|125|7= r4c56 - r4c89 =7|459|6= r56c7 - r89c7 = 6|125|8= r7c89 - r7c56 =8|249|3= r89c4 - continuous loop
The digits '|xxx|' between the '|' pairs are the digits locked in the respective ALS, when the ALS becomes an LS, of course.

I wrote something similar for a (continuous) ALS loop, in one of the puzzle threads ... using a notation that I made up on the fly.
For this one, it would have looked like:

Code: Select all: (3=7,125)r56c4,r4c56 - (7=6,459)r4c89,r56c7 - (6=8,125)r89c7,r7c89 - (8=3,249)r7c56,r89c4 - loop

[ It looks too confusing in this case, though ... with the commas also appearing in the ALS cell lists. ]

David P Bird wrote:However your second suggestion is a perfectly sound AIC
(...)

For me it was quite a new way to view the workings of the loop, so I immediately tried it out on a regular SK loop but sad to say couldn't reduce it to an AIC.

I had the same thoughts and hopes ... hopes that were quickly dashed

David P Bird wrote:Its only drawback is that it's rather long, ...

Yes, and after it gets going, it ends up looking like "3 steps forward, one step back, 3 steps forward, one step back ...".

by **ronk** » Tue Sep 01, 2015 4:42 pm

blue wrote:I wrote something similar for a (continuous) ALS loop, in one of the puzzle threads ... using a notation that I made up on the fly.
For this one, it would have looked like:
Code: Select all
(3=7,125)r56c4,r4c56 - (7=6,459)r4c89,r56c7 - (6=8,125)r89c7,r7c89 - (8=3,249)r7c56,r89c4 - loop

[ It looks too confusing in this case, though ... with the commas also appearing in the ALS cell lists. ]

I could live with that, and the commas are not really an issue IMO. They're lists in both cases. FWIW I had to fix an error in my NL notation above.

blue wrote:
David P Bird wrote:However your second suggestion is a perfectly sound AIC
(...)
For me it was quite a new way to view the workings of the loop, so I immediately tried it out on a regular SK loop but sad to say couldn't reduce it to an AIC.
I had the same thoughts and hopes ... hopes that were quickly dashed

Did you try the complementary A*LS loop, what champagne calls the V-loop?

by **ttt** » Tue Sep 01, 2015 4:43 pm

ronk wrote:
blue wrote:
ronk wrote:For an AIC loop, I would choose this continuous loop of four ALSs in boxes b5689:
Code: Select all
(3125=1257)r56c4,r4c56 - (7459=4596)r4c89,r56c7 - (6125=1258)r89c7,r7c89 - (8249=2493)r7c56,r89c4 - loop
The problem with it, is that the strong links aren't valid "on thier own".
They can be seen to be valid, once a proper loop is formed, but they can't be used to establish that same loop.

How about one of these instead ?

I replaced it with NL notation with which I'm more familiar.
Code: Select all
-3- r56c4 =3|125|7= r4c56 -7- r4c89 =7|459|6= r56c7 -6- r89c7 = 6|125|8= r7c89 -8- r7c56 =8|249|3= r89c4 - continuous loop
The digits '|xxx|' between the '|' pairs are the digits locked in the respective ALS, when the ALS becomes an LS, of course.

[edit: Added missing labels on weak links in NL notation. Hmm, wonder what else I forgot.]

Yes, you can hehe… Like me, I use Eureka notation because I'm more familiar with it…..

IMO, why the notation is very important: because we should speak the same language, we have mutual interest to get rid of the ego. So, that is reason I like all of you should use Eureka (AIC) notation.

Back to the past then remind me comment on AU Site:

Hi All and HAPPY NEW YEAR 2011 – quite late to say that
Just some of my opinions on your comments recently about Eureka/AIC and Alfred’s notation:
At first, I come here to enjoy with puzzles and happy to see you also enjoy with them but I don’t care HOW YOU ENJOY WITH PUZZLES – except when you post an invalid solution. So, I don’t know why you did comment too much about Eureka/AIC and Alfred’s notation. I’m not sure but I think that all or most of Alfred’s solutions could translate to Eureka/AIC and many of them with elegant solutions as Steve noticed one time (don’t remember that date).I have no idea about Alfred’s notion but Eureka/AIC can show you patterns then you can use them later...
I’m sorry by my poor English, just to say: come here enjoy with puzzles and that is enough...

Happy to see most of you still alive (online) here… hehehe…. :lol:

ttt

by **David P Bird** » Sun Sep 06, 2015 12:31 pm

I've been surveying the nine examples of V-loops given by daj < here > because I've never been clear what distinguishes them from SK Loops. What they all have is an SK Loop with a common digit in the digit pairs used to link the two rows and two columns in the loop. This digit will never appear as a given in the 8 houses that contain the loop which provides a way to recognise when an SK Loop is also a V-loop.

Example: 9.......1.3...4.7...6...2...5.3.2.......6........78.5...2...6...4.7...3.1.......9#8400;tax;tarek-ultra-0204

Code: Select all: *-----------------------*-----------------------*-----------------------* | <9> 278 4578 | 2568 2358 3567 | 3458 468 <1> | (12)b2 (58)r2 | 258 <3> 158 | 125689 12589 <4> | 589 <7> 568 | | 4578 178 <6> | 1589 13589 13579 | <2> 489 3458 | (69)b3 *-----------------------*-----------------------*-----------------------* | 4678 <5> 14789 | <3> 149 <2> | 14789 14689 4678 | | 23478 12789 134789 | 1459 <6> 159 | 134789 12489 23478 | | 2346 1269 1349 | 149 <7> <8> | 1349 <5> 2346 | *-----------------------*-----------------------*-----------------------* | 3578 789 <2> | 14589 134589 1359 | <6> 148 4578 | (69)b7 (58)r8 | 568 <4> 589 | <7> 12589 1569 | 158 <3> 258 | | <1> 678 3578 | 24568 23458 356 | 4578 248 <9> | (12)b9 *-----------------------*-----------------------*-----------------------* (78)c2 (48)c8

Alongside the grid the linking digits used in the SK loop in each house are shown where (8) can be seen to be the common digit in r28, c28. Note that there is only one given for (8) at r6c6
SK Loop:
(78=69)r79c2 - (69=58)r8c13 - (58=12)r8c79 - (12=48)r79c8 -
(48-69)r13c8 - (69=58)r1c79 - (59=12)r2c13 - (12=78)r13c2 - Loop
=> 13 Eliminations: (58)r2c4, (58)r2c5, (48)r4c8, (78)r5c2, (48)r5c8, (58)r8c5, (5)r8c6
The loop is not an AIC. Two successive doublet terms will together hold both digits allowing them to be eliminated from the other cells in the same house.

Proof of SK Loop Eliminations: Show

Alternative Loop
(7=689)r79c2 - (689 =5)r8c13 - (5=128)r8c79 - (128=4)r79c8 -
(4=689)r13c8 - (689=5)r2c79 - (5=128)r2c13 - (128=7)r13c2 - Loop
=> 15 Eliminations: (8)r1c3, (8)r1c7, (5)r2c4, (5)r2c5, (8)r3c1, (8)r3c9, (4)r4c8, (7)r5c2, (4)r5c8, (8)r7c1, (8)r7c9, (5)r8c5, (5)r8c6, (8)r9c3, (8)r9c7
Like the SK loop this is not a pure AIC as the triplet terms are considered true when they contain two truths and false when they contain one truth (the minimum they must hold). Two successive triplets will together hold all three digits allowing them to be eliminated from any other cells in the same house.

The two loops give a mix of common and unique eliminations and it appears that the alternative loop is more productive. However they produce different follow-on eliminations after which the end results are identical.
Follow-on eliminations after the SK Loop: (8)4Fish:c2458r1379 => r1c37,r3c19,r7c19,r9c37 <> 8 (8 eliminations)
Follow-on eliminations after the alternative loop: (8)4Fish: c2458r1379 => r2c45,r45c8,r5c2,r8c5 <> 8 (6 eliminations)

Code: Select all: Reduced grid: *-----------------------*-----------------------*-----------------------* | <9> 278 457 | 2568 2358 3567 | 345 468 <1> | | 258 <3> 158 | 1269 129 <4> | 589 <7> 568 | | 457 178 <6> | 1589 13589 13579 | <2> 489 345 | *-----------------------*-----------------------*-----------------------* | 4678 <5> 14789 | <3> 149 <2> | 14789 169 4678 | | 23478 129 134789 | 1459 <6> 159 | 134789 129 23478 | | 2346 1269 1349 | 149 <7> <8> | 1349 <5> 2346 | *-----------------------*-----------------------*-----------------------* | 357 789 <2> | 14589 134589 1359 | <6> 148 457 | | 568 <4> 589 | <7> 129 169 | 158 <3> 258 | | <1> 678 357 | 24568 23458 356 | 457 248 <9> | *-----------------------*-----------------------*-----------------------*

(12=69)r56c2 - (69)r79c2 = (69)r8c13 - (69=12)r8c56 - (12)r8c79 = (12)r79c8 -
(12=69)r45c8 - (69)r13c8 = (69)r2c79 - (69=12)r2c45 - (12)r2c13 = (12)r13c2 - Loop

This is a version of the SK loop with four key terms underlined. It uses only the four tell-tale digits that are givens in the pivot boxes.
If the terms on the left contain two truths then (9) is forced true in r8c5 and r2c5 because (6) must occupy r8c6 and r2c4
If the terms on the right contain two truths them (9) is forced true in r8c5 and r2c5 because (6) must occupy r6c2 and r4c8
Because of these contradictions the each doublet is a STP (Single Truth Pair) and this will also apply to the usual version of the loop.
Another consequence is that (1269)r2c45,r8c56 and (1269)r56c2,r45c8 must be PLQs (Pattern Locked Quads).
Together these derived inferences can simplify the rest of the solution considerably.

The givens in boxes 2,4,6 & 8 follow a classic pattern as no two of them are in sight of each other. Now in box 5, the remaining non-loop box there is a given for one of the tell-tale digits (6), which is out of sight of all of these which signals that STPs will exist (if two adjacent boxes in the loop contain all four of the tell-tell digits as givens). STPs also frequently arise when the puzzle contains a JExocet too, but the proof is different.

Champagne before I make any further observations I would like know to if you have any examples of V-Loops that can't also be considered as SK loops? There are a number of ways of composing alternative loops but they all produce the same reductions as far as I can see.

DPB

TAGdpbSKLoop
[Edit] Missing underlining in the third loop corrected, extra blue condition added for STPs to exist.

by **champagne** » Sun Sep 06, 2015 1:34 pm

David P Bird wrote:Champagne before I make any further observations I would like know to if you have any examples of V-Loops that can't also be considered as SK loops? There are a number of ways of composing alternative loops but they all produce the same reductions as far as I can see.

DPB

I already answered to that but I can repeat it.

If you consider that a sk loop must verify the "ronk constraint" to have 2-2-2 unknown cells in each row/column of the loop, then we have one example of a "V" loop that does not verify that constraint.

I introduced the "V" loop to avoid endless fights on a small point, but personally I quite accept a wider definition of the SK loop.

And again, I did not investigate to see whether other examples of "V" loops that are not SK- loops with the "ronk constraint" have been found.

by **David P Bird** » Sun Sep 06, 2015 3:15 pm

Champagne, once again thank you for your rapid response. Your earlier response confused me a bit though as you started it with

I'll check in details later, but my first reaction is that we have no SK loop and no 'V' loop here.

So I was expecting a follow-up from you.

Now I don't fully understand the "ronk constraint" as you call it because your descriptions of it are different in your two responses.

1) IMO the only difference between the SK loop and the "V" loop is the constraint added by "ronk" to have 2 unknown cells in the boxes not belonging to the loop to qualify it of "sk loop".

2) If you consider that a sk loop must verify the "ronk constraint" to have 2-2-2 unknown cells in each row/column of the loop, then we have one example of a "V" loop that does not verify that constraint.

An example map would help me considerably as none of the 9 examples I've looked at seem to fit either description.

I'm sorry to bother you but I'm just trying to tidy these issues up because every so often the debate is re-opened and no-one can define what a V-loop is.

David

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