The New Sudoku Players' Forum

[ronk]

Here is an interesting puzzle published on a French forum.
.........1....72...7..84.6...8....93.6..4..7.93....6...9.73..8...59....2.........

With the many SK loops I have studied there is a corresponding 4 digit rank zero row based multifish, whose base set corresponds to the digits in the pairs of linking digits in the boxes of the SK loop. Here we only have the 3 digits: 125. So far I can't construct a complementary multifish. Any thoughts?

I think the most likely answer is this puzzle does not contain an sk-loop.

[champagne]

I think the most likely answer is this puzzle does not contain an sk-loop.

This is the most surprising conclusion.

After easy moves, we are there

Code: Select all

A     B    C     |D    E     F     |G      H   I     
24568 2458 2469  |1256 12569 3     |145789 145 145789
1     458  3469  |56   569   7     |2      345 4589   
235   7    239   |125  8     4     |1359   6   159   
-----------------------------------------------------
457   145  8     |1256 12567 1256  |145    9   3     
25    6    12    |3    4     9     |158    7   158   
9     3    147   |8    157   15    |6      2   145   
-----------------------------------------------------
246   9    1246  |7    3     1256  |145    8   1456   
34678 148  5     |9    16    168   |1347   134 2     
23678 128  12367 |4    1256  12568 |13579  135 15679

with a clear SK Loop

r3c13 r3c79 r12c8 r89c8 r7c79 r7c13 r89c2 r12c2
25 39 15 34 15 46 12 48 25

to pjb did you work on the same PM??

I did not look at the multi fish issue

[pjb]

ronk wrote:
I think the most likely answer is this puzzle does not contain an sk-loop.

I too am puzzled. It has the usual sequence of logic shared by other unequivocal SK loops, as described above by champagne. I await with interest to see where the problem lies.

Phil

[champagne]

The corresponding XSUDO diagram for that sk loop

sk l [29,188] 52 Candidates,
16 Truths = {37N1 1289N2 37N3 37N7 1289N8 37N9}
16 Links = {39r3 46r7 3c8 4c28 8c2 1b379 2b17 5b139}
18 Eliminations --> r1c179<>5, r9c379<>1, r1c79<>1, r1c13<>2, r9c13<>2, r9c79<>5, r2c9<>5,
r4c2<>4, r7c6<>6, r8c7<>1,

[JC Van Hay]

Here is an interesting puzzle published on a French forum.
.........1....72...7..84.6...8....93.6..4..7.93....6...9.73..8...59....2.........

With the many SK loops I have studied there is a corresponding 4 digit rank zero row based multifish, whose base set corresponds to the digits in the pairs of linking digits in the boxes of the SK loop. Here we only have the 3 digits: 125. So far I can't construct a complementary multifish. Any thoughts?

I think the most likely answer is this puzzle does not contain an sk-loop.

The puzzle : .........1....72...7..84.6...8....93.6..4..7.93....6...9.73..8...59....2.........
contains a "Loop All Cells" the structure of which fits the ALS-XY Ring given above

Code: Select all

+------------------------+--------------------+-------------------------+
| 468-25  (25 48) 469-2  | 1256  12569  3     | 4789-15 (15  4) 4789-15 |
| 1       ( 5 48) 3469   | 56    569    7     | 2       ( 5 34) 489-5   |
| (25  3) 7       (2  39)| 125   8      4     | (15 39) 6       (15  9) |
+------------------------+--------------------+-------------------------+
| 457     15-4    8      | 1256  12567  1256  | 145     9       3       |
| 25      6       12     | 3     4      9     | 158     7       158     |
| 9       3       147    | 8     157    15    | 6       2       145     |
+------------------------+--------------------+-------------------------+
| ( 2 46) 9       (12 46)| 7     3      125-6 | (15 4 ) 8       (15 46) |
| 34678   (1  48) 5      | 9     16     168   | 347-1   (1  34) 2       |
| 3678-2  (12  8) 367-12 | 4     1256   12568 | 379-15  (15 3 ) 679-15  |
+------------------------+--------------------+-------------------------+
[29,188] 52 Candidates, Loop All Cells
16 Truths = {37N1 1289N2 37N3 37N7 1289N8 37N9}
16 Links = {39r3 46r7 3c8 4c28 8c2 1b379 2b17 5b139}
18 Eliminations --> r1c179<>5, r9c379<>1, r1c79<>1, r1c13<>2, r9c13<>2,
                    r9c79<>5, r2c9<>5, r4c2<>4, r7c6<>6, r8c7<>1

The simplest "complementary" multi-fish :

[29,188] 34 Candidates, Loop "2 Rows + 2 Columns"   
11 Truths = {125R3 125R7 125C2 15C8}
11 Links = {4n2 3n4 7n6 1b379 2b17 5b139}
18 Eliminations --> r1c179<>5, r9c379<>1, r1c79<>1, r1c13<>2, r9c13<>2,
                    r9c79<>5, r2c9<>5, r4c2<>4, r7c6<>6, r8c7<>1

While the "complementary" Loop All Rows is :

[29,188] 46 Candidates, Loop All Rows
13 Truths = {1R34567 2R357 5R34567}
13 Links = {1c379 2c13 5c179 4n2 3n4 7n6 15b5}
18 Eliminations --> r1c179<>5, r9c379<>1, r1c79<>1, r1c13<>2, r9c13<>2,
                    r9c79<>5, r2c9<>5, r4c2<>4, r7c6<>6, r8c7<>1

As to the "complementary" Loop All Columns, the situation is more complex due to cannibalism.
Here is the raw result !

[29,188] 49 Candidates, Cannibalized Logic
14 Truths = {1C24568 2C2456 5C24568}
22 Links = {1r1489 2r19 5r1249 1n248 3n4 4n2 7n6 125b2 125b5}
20 Eliminations --> r1c134<>2, r1c179<>5, r9c379<>1, r1c79<>1, r2c59<>5,
                    r9c13<>2, r9c79<>5, r4c2<>4, r7c6<>6, r8c7<>1

[David P Bird]

Pjb's puzzle is an interesting case of a multi-fish with only 3 digits (125) in the focus digit set rather than the usual 4

Code: Select all

Rows *-------------------------*-------------------------*-------------------------*  Boxes
  .  | 468-25  2458    469-2   | 1256    12569   3       | 4789-15 145     4789-15 | 25 -  -
  .  | 1       458     3469    | 56      569     7       | 2       345     489-5   | -  .  -
  39 | 235     7       239     | 125     8       4       | 1359    6       159     | -  -  15
     *-------------------------*-------------------------*-------------------------*
  .  | 457     15-4    8       | 1256    12567   1256    | 145     9       3       | .  -  -
  .  | 25      6       12      | 3       4       9       | 158     7       158     | -  .  -
  .  | 9       3       147     | 8       157     15      | 6       2       145     | -  -  .
     *-------------------------*-------------------------*-------------------------*
  46 | 246     9       1246    | 7       3       125-6   | 145     8       1456    | 12 -  -
  .  | 34678   148     5       | 9       16      168     | 347-1   134     2       | -  .  -
  .  | 3678-2  128     367-12  | 4       1256    12568   | 379-15  135     679-15  | -  -  15
     *-------------------------*-------------------------*-------------------------*
 Colms .       48      .         .       .       .         .       34      .

Multi-sector Locked Set:(39)r3,(46)r7,(48)c2,(34)c8,(25)b1,(15)b3,(12)b7,(15)b9 18 Eliminations in 12 cells

Although (7) is in the complementary set, the givens for it at r3c2, r5c8, & r7c4 effectively eliminate it from consideration. This suggests that for the truth balancing approach I use, adding it to the focus set to give (1257) would also produce a balance, but it doesn't!

I've mentioned somewhere before that one of my colouring schemes identifies intersecting mini-lines that hold 5 candidates (ie AAHSs). These abound in this puzzle, but it identifies r37c28 as worthy of checking as all these intersection cells hold singles.

From my POV, given <here>, this is therefore a Multi-fish that can be expressed as a hidden pair loop and is therefore an SK loop.

[ronk]

I think the most likely answer is this puzzle does not contain an sk-loop.

This is the most surprising conclusion. After easy moves, we are there

Code: Select all

24568 2458 2469  |1256 12569 3     |145789 145 145789
1     458  3469  |56   569   7     |2      345 4589   
235   7    239   |125  8     4     |1359   6   159   
-----------------------------------------------------
457   145  8     |1256 12567 1256  |145    9   3     
25    6    12    |3    4     9     |158    7   158   
9     3    147   |8    157   15    |6      2   145   
-----------------------------------------------------
246   9    1246  |7    3     1256  |145    8   1456   
34678 148  5     |9    16    168   |1347   134 2     
23678 128  12367 |4    1256  12568 |13579  135 15679

with a clear SK Loop

Not clear to me, because it doesn't have some usual markers of the sk-loop pattern, which include ...

• Three clues in each of boxes b1, b3, b7 and b9 versus the two clues of this puzzle,
• One clue in each of mini-lines r3b2, c2b4, c8b6 and r7b8 versus the two and three clues of this puzzle.

As to the "complementary" [...], the situation is more complex due to cannibalism.
Here is the raw result !

Code: Select all

14 Truths = {1C24568 2C2456 5C24568}
22 Links = {1r1489 2r19 5r1249 1n248 3n4 4n2 7n6 125b2 125b5}
20 Eliminations --> r1c134<>2, r1c179<>5, r9c379<>1, r1c79<>1, r2c59<>5,
                    r9c13<>2, r9c79<>5, r4c2<>4, r7c6<>6, r8c7<>1

There is no cannibalism via the 0-rank logic set:

Code: Select all

     14 Truths = {1C24568 2C2456 5C24568}
     14 Links = {1r189 2r19 5r129 4n2 3n4 7n6 125b5}
     18 Eliminations --> r1c179<>5, r9c379<>1, r1c79<>1, r1c13<>2, r9c13<>2, r9c79<>5, r2c9<>5,
     r4c2<>4, r7c6<>6, r8c7<>1

From my POV, given <here>, this is therefore a Multi-fish that can be expressed as a hidden pair loop and is therefore an SK loop.

I await your hidden pair loop expression.

champagne, you're getting overly generous with the white space again.

[David P Bird]

I await your hidden pair loop expression.

Arghh! I thought it would automatically translate into twin AIC loops of hidden pairs, but it doesn't as very few of the inferences are strong as this attempt shows:

(25-39)r3c12 = (39-15)r3c79 - (15-34)r12c8 = (34-15)r89c8 - (15-46)r7c79 - (46-12)r7c13 - (12-48)r89c2 - (48-25)r12c2 - Loop

From my hierarchy of definitions this would therefore only qualify as a multi-fish.

For the uninitiated*, hidden pair loops combine two different ways the alternating inferences can be followed. First assuming that blue pairs of digits hold 2 truths and the red ones hold >0 truths, and second assuming that the blue pairs hold >0 truths and the red ones 2 truths. If the inference can be properly alternated, these combine to give three possible divisions of truths: a) blue = 2, red = 0, b) blue = 0, red = 2, c) blue = 1 red = 1. Consequently any other instances of the linking digits in the same house can be eliminated.

*I don't want to be accused of preaching to the choir again.

[champagne]

Not clear to me, because it doesn't have some usual markers of the sk-loop pattern, which include ...

• Three clues in each of boxes b1, b3, b7 and b9 versus the two clues of this puzzle,
• One clue in each of mini-lines r3b2, c2b4, c8b6 and r7b8 versus the two and three clues of this puzzle.

It seems to me there is here some confusion between intrinsic property of the SK loop and the so called "usual markers".

The intrinsic property of a SK loop is to establish through a loop of bi locations that these bi locations are forming strong inferences. This leads to eliminations.

This is the case here, this is verified in the XSUDO corresponding diagram, all other contributors feel comfortable with that, so where is the problem!!!

Once more, I am lost

[ronk]

The intrinsic property of a SK loop is to establish through a loop of bi locations that these bi locations are forming strong inferences. This leads to eliminations.

Without knowing your meaning for "bi-location" in this context, I suggest that several patterns meet that definition. Are we to call all of them sk-loops?

This is the case here, this is verified in the XSUDO corresponding diagram, all other contributors feel comfortable with that, so where is the problem!!!

Not true. David P Bird withdrew his opinion that it' a sk-loop and JC Van Hay didn't say AFAICT.

[champagne]

From my POV, given <here>, this is therefore a Multi-fish that can be expressed as a hidden pair loop and is therefore an SK loop.

I know my English is very poor, but this seems to tell David sees that as a sk loop

[daj95376]

First puzzle:

Code: Select all

..8...5...6...3.7.4.......1.7..32......9.5......67..2.5.......4.2.3...6...1...8..

 +--------------------------------------------------------------------------------+
 |  12379   39+1   #8       |  1247    12469   14679   | #5       4+39    2369    |
 |  1+29    6       5+29    |  12458   124589  3       |  29+4    7       29+8    |
 | #4       39+5    23579   |  2578    25689   6789    |  2369    8+39   #1       |
 |--------------------------+--------------------------+--------------------------|
 |  1689    7       4569    |  148     3       2       |  1469    14589   5689    |
 |  12368   1348    2346    |  9       148     5       |  13467   1348    3678    |
 |  1389    134589  3459    |  6       7       148     |  1349    2       3589    |
 |--------------------------+--------------------------+--------------------------|
 | #5       8+39    3679    |  1278    12689   16789   |  12379   39+1   #4       |
 |  79+8    2       79+4    |  3       14589   14789   |  1+79    6       5+79    |
 |  3679    4+39   #1       |  2457    24569   4679    | #8       39+5    23579   |
 +--------------------------------------------------------------------------------+
 # 178 eliminations remain

 r2[(15=29)c13 - (29=48)c79] - c8[(48=39)r13 - (39=15)r79] -
 r8[(15=79)c79 - (79=48)c13] - c2[(48=39)r79 - (39=15)r13] - SK_loop  =>

 r2c5<>29; r6c2<>39; r8c6<>79; r1c1,r7c7<>1;
 r2c4<>2; r5c28<>3; r3c3,r9c9<>5; r4c8,r8c5<>9

 Cells marked with (#) prevent all-quad value SK-Cells.


French puzzle:

Code: Select all

.........1....72...7..84.6...8....93.6..4..7.93....6...9.73..8...59....2.........

 +--------------------------------------------------------------------------------+
 |  24568   48+25   2469    |  1256    12569  #3       |  145789  15+ 4   145789  |
 | #1       48+ 5   3469    |  56      569     7       | #2        5+34   4589    |
 |  25+3    7       2+39    |  125     8       4       |  39+15   6       9+15    |
 |--------------------------+--------------------------+--------------------------|
 |  457     145     8       |  1256    12567   1256    |  145     9      #3       |
 |  25      6       12      |  3       4       9       |  158     7       158     |
 | #9       3       147     |  8       157     15      | #6       2       145     |
 |--------------------------+--------------------------+--------------------------|
 |  46+2    9       46+12   |  7       3       1256    |  15+4    8       15+46   |
 |  34678   1 +48  #5       |  9       16      168     |  1347    34+1   #2       |
 |  23678   12+ 8   12367   | #4       1256    12568   |  13579   3 +15   15679   |
 +--------------------------------------------------------------------------------+
 # 135 eliminations remain

 r3[(25=39)c13 - (39=15)c79] - c8[(15=34)r12 - (34=15)r89] -
 r7[(15=46)c79 - (46=12)c13] - c2[(12=48)r89 - (48=25)r12] - SK_loop  =>

 r1c1<>25; r1c79,r9c79<>15; r9c3<>12; r8c7<>1; r1c3<>2; r4c2<>4; r2c9<>5; r7c6<>6

 Cells marked with (#) prevent all-quad value SK-Cells. r1c6 & r9c4 are solved cells.


In both puzzles, the (#) cells don't prevent the SK-Loop pair bonds from forming.

Maybe the lack of symmetry in the (#) cells of the second puzzle prevent the multi-fish pattern normally seen by pjb.

ADDENDUM: It's also possible that the presence of seven values in the (#) cells might be influencial.

[ronk]

A number of participants here seem to be unaware that the SK-loop is named for Steve Kurzhals or unaware that the AALS version of the SK-loop is a complementary version of an AAHS version that Steve first presented as a hidden-pair-loop.

The AALS version became popular because only 16 cells appeared in AIC and nice loop expressions, instead of 24 cells. However, this doesn't mean that the AAHS version has been deleted from the definition. IOW the AAHS version, i.e. the complement of AALS version, should still look like Steve's original hidden-pair-loop. For those who appreciate pictures, here is an example from an earlier post.

___ ___ ___ (Clickable images)

Logic sets for the three forms: Show

Code: Select all

..9.....6.1...8.3.2.....7.........1.....73..5.4.8.5.....6.....9.5.3...4.7.....2..;9698;TkP;3952
Hidden sets in 2 rows & 2 boxes (hidden-pair-loop)
     16 Truths = {2679R28 2679C28}
     16 Links = {45n2 2n4 28n5 8n6 56n8 29b37 67b19}

Cell sets in 2 rows & 2 boxes (or cell sets in 4 boxes)
     16 Truths = {28N1 1379N2 28N3 28N7 1379N8 28N9}
     16 Links = {45r2 18r8 3c2 5c8 8c28 2b37 6b19 7b19 9b37}

Hidden sets in 4 boxes
     16 Truths = {1B379 3B179 4B137 5B139 8B1379}
     16 Links = {45r2 18r8 3c2 5c8 8c28 17n1 39n3 17n7 39n9}

Exclusions for each of the three above:
     11 Eliminations --> r2c45<>4, r2c45<>5, r5c28<>8, r8c56<>1, r4c2<>38, r8c5<>8

From my POV, given <here>, this is therefore a Multi-fish that can be expressed as a hidden pair loop and is therefore an SK loop.

I know my English is very poor, but this seems to tell David sees that as a sk loop

See his following post.

[edit: '24' cells was '32' cells]

[David P Bird]

Woe is me! Yesterday I made a schoolboy error and missed half the strong inferences that exist in that French puzzle. It can be expressed as two continuous alternating inference loops of hidden pairs. As no-one picked up on this it seems sensible to go back to basics:

Loop 1: (25#2=39#1)r3c13 - (39#2=15#1)r3c79 - (15#2=34#1)r12c8 - (34#2=15#1)r89c8 - (15#2=46#1)r7c89 - (46#2=12#1)r7c13 - (12#2=48#1)r89c2 - (48#2-25#1)r12c2 – Loop

Loop 2: (25#1=39#2)r3c13 - (39#1=15#2)r3c79 - (15#1=34#2)r12c8 - (34#1=15#2)r89c8 - (15#1=46#2)r7c89 - (46#1=12#2)r7c13 - (12#1=48#2)r89c2 - (48#1-25#2)r12c2 – Loop

For each term #1 and #2 suffixes show how many truths the pairs are considered to hold.

In loop 1 (25#2=39#1)r3c13 says that the Boolean conditions [both (2) & (5) are true], and [at least one of (3) & (9) is true] can't both be false (an OR condition).
Then (39#1)r3c13 – (39#2)r3c79 says that the Booleans [at least one of the digits is true in r3c13] and [both digits are true in r3c79] can't both be true (a NAND condition).

As the chains form continuous loops, then and only then, every inference is proved to be conjugate to become an EITHER/OR (XOR) condition.

The first loop covers cases when the odd numbered terms contain two truths and the second when the even terms do. If both these cases are false, then the only remaining possibility is for every term to contain one true digit (without the same digit being true twice in any house). For all three outcomes the eliminations follow.

Champagne, I hope you could follow this.

The most common mistake is to consider strong inferences to be conjugate. This is usually true but not always and AICs must handle cases when both terms in a strong inference are true or both term in a weak inference are false.

I therefore return to my opening position, as these hidden pair loops can be constructed, this pattern does qualify as an SK loop.

ronk as I recall Steve Kurzahls called his loops 'hidden pair loops' which is what this puzzle contains. I don’t think he stipulated a required pattern of givens. That's something Myth Jellies identified for puzzles following the Easter Monster pattern.

[champagne]

Champagne, I hope you could follow this.

Hi David,

I express it in a slightly different way, but the underlying logic is exactly the same.

In fact, It's enough for the proof to establish the loop in one way. (as with a classical loop)