The New Sudoku Players' Forum

[David P Bird]

Let's run this hierarchy of MSLSs up the flagpole and see who salutes it!

A Multi-Sector Locked Set is a set of N cells distributed over several houses that together contain N confined digits where each digit is considered to be confined in just one of its covering houses.

A Multi-fish is a MSLS with two families of houses, one which confines a focus set of digits and the other which confines the complementary digit set.

An SK Loop is a Multi-fish where the confined digits in each house can be ordered to form a loop of potential [Hidden edit] Locked Sets.

No doubt the wording can be improved, but at this stage I'm more interested in whether there is any support for this classification system. It could help to standardise how we report and discuss our findings. For example using truth and link sets, by convention it could be agreed to use cell truth sets to identify a MSLS cell set (I believe there are always at least two alternative sets that can be used).

These are all rank 0 patterns that can be found using different search methods, but which method is used should be relatively unimportant. The question is whether or not there are other rank 0 patterns we've not yet explored that will affect this viewpoint.

[Edit] The fact that no-one responded either positively or negatively to this post indicates how interested people are in reaching any form of consensus. The change from Hidden to Locked sets is the result of a later wrangle in this thread.

TAGdpbMSLS

[edit Mar 2017 tag added]

[ronk]

..1...5...2.4...6.3....7....6.28........9..2.......4.65.....1...9.8...4...7.....3;54;col;H2

19 Truths = {1R28 1C28 3R28 3C28 5R28 5C28 7R28 7C28 6N456 }
19 Links = {1r6 1b37 3r6 3b37 5r6 5b19 7r6 7b139 2n56 4n8 5n2 8n56 }[/hidden]

This is what I call a "0-rank almost sk-loop." To avoid comparing apples and oranges, if "a helper ALS" is allowed for the 2-row/2-col A*HS form, then "a helper AHS" should be allowed for the A*LS form in 4 boxes (or 2-rows/2-cols, if you prefer).

___ _ _ (Clickable thumbnails)

logic sets and exclusion list: Show

Code: Select all

2-row/2-col A*HS with helper ALS
     19 Truths = {1357R28 1357C28 6N456}
     19 Links = {1357r6 5n2 28n56 4n8 13b37 5b19 7b139}

A*LS in four boxes with helper AHS
     19 Truths = {289R6 28N1 1379N2 28N3 28N7 1379N8 28N9}
     19 Links = {89r2 26r8 4c2 8c28 9c8 6n13 1b37 3b37 5b19 7b139}

A*HS in four boxes with helper AHS
     19 Truths = {289R6 2B379 4B137 6B179 8B1379 9B139}
     19 Links = {89r2 26r8 4c2 8c28 9c8 169n1 367n3 39n7 137n9}

For each:
     20 Eliminations --> r1c19<>7, r8c56<>2, r8c56<>6, r69c1<>1, r67c3<>3, r36c3<>5, r2c6<>89,
     r5c2<>48, r3c9<>1, r4c8<>9, r6c1<>7, r7c9<>7

Exploring manually the entire file is a big task

If it's more of the above, you sure don't have to continue searching on my account. Thanks for this one though.

[edit: For completeness sake, add 3rd logic set "A*HS in four boxes with helper AHS"]

[champagne]

2-row/2-col A*HS with helper ALS
A*LS in four boxes with helper AHS

As far as I could see, in a huge majority of them, my solver found also a pure row and a pure column solution

After Leren last post and knowing

that my solver is not looking for more than 4 digits when something has been found
that already a huge majority of puzzles having a sk loop or an Almost SK loop have shown such rank 0 logic

I am wondering whether it is not granted that such a logic exists for any SK loop or Almost SK loop.

Exploring manually the entire file is a big task

If it's more of the above, you sure don't have to continue looking on my account. Thanks for this one though.

Impossible to know before the job has been done, but I'll try later to filter using a temporary change in the code;

[JC Van Hay]

Basic loops of all the cells at the intersection of N rows and N columns in a 9x9 latin square

Code: Select all

   XWing of cells

   xa ya
   xb yb

   Swordfish of cells

   xab yab zab
   xcd ycd zcd
   xef yef zef

   Jellyfish of cells Type I

   xabc yabc zabc uabc
   xcde ycde zcde ucde
   xefg yefg zefg uefg
   xhij yhij zhij uhij

   Jellyfish of cells Type II

   xyab zuab vwab stab
   xycd zucd vwcd stcd
   xyef zuef vwef stef
   xygh zugh vwgh stgh

Properties

1. What is said about rows is also true for columns and vice-versa.
2. The cover is made up of a total of 4,9,16,16 lines repectively.
   They are therefore rank 0 patterns.
3. They are the only rank 0 patterns at the intersection of N rows and N columns, N=1 to 4.
4. There is no starfish of cells simply because the above loops need 4,6,8,8 different digits.
   A starfish of cells would therefore need 10 different digits which is impossible.
5. The solutions of these loops are equivalent to the following respective solution :
   
   

   Code: Select all

      x a
   b y

   x  ab ab
   cd y  cd
   ef ef z

   x   abc abc abc
   cde y   cde cde
   efg efg z   efg
   hij hij hij u

   xy zu ab ab
   xy zu cd cd
   ef ef vw st
   gh gh vw st

   
   
6. The set of digits in the row cover sets and the set of digits in the column cover sets are disjoint.
   They are contained in the loop of columns and in the loop of rows equivalent to the loop of cells, respectively.

Structure of loops of cells in a 9x9 latin square.

Obviously,

1. The basic loops can contain less cells provided that each time a cell is removed a cover set is removed.
   But this also means that a given can take the place of a removed cell.
2. Conversely, cells can be added in the cover set of the basic loops provided that each time a cell is added a cover set is added.

Looking for a loop of cells in a latin square

1. Selection of all the cells devoid of given at the intersection of N rows and N columns;
2. Analysis of the covering of these cells;
3. if necessary, removal or addition of cells as explained above until a loop can be found.

Of course, except for N=2, the amount of work needed is inversely proportional to N.

Loops of cells in a 9x9 sudoku puzzle.

Box constraints bring new features :

1. If a box contain 2 or 3 cells of an XWing of cells, these cells acquire freedom of position in the box.
2. The other basic loops remain essentially the same, except that a box cover can appear, at the expense of a line cover, each time 4 cells of the loop are in a box.
   In this case, an equivalent loop for one of the lines could be impossible to find.
3. There are now 2 more basic Jellyfishes : the Jellyfish of cells Type III and Type IV where the cells at the intersection of 2 rows and 2 columns are in 4 boxes, 2 adjacent boxes being crossed by 1 or 2 cover lines, as in a "ronk-loop" and an "sk-loop of cells, respectively . In these cases, there is a loop of 2 rows and 2 columns and a loop of boxes equivalent to the loop of cells. As to the existence of an equivalent loop of rows or of columns, all depends apparently on the properties of an eventual equivalent Jellyfish of type I or II, extended by adding cells or not, in the puzzle.
4. A Starfish of cells Type I or II might exist as an extension of the Jellyfishes of cells Type III and Type IV by adding the 2 cells, ouside the 4 boxes, at the intersections of the generating lines with another appropriate line. To achieve this, 1 box cover must be added in a box that can be compensated by these 2 cells and 1 cover line over them.
   In this case, some of the expected equivalent loops could be impossible to find.

Example of application : puzzle 55 as analyzed by David

After drawing on the back of an envelope a 9x9 square of cells and putting an x in all the cells containing a given, it is easily found that the cells at the intersection of R2578 and C15678 do not contain a given.
A Jellyfish of cells could therefore be found in this rectangle of 4x5 cells. The analysis of the candidates in each of these cells leads to a covering as indicated in the puzzle below.

Code: Select all

....56.8...71.....6.....4.......85...3......29...4..6..1.7.......2.....38...9..5.;55;elev;30;G1
+--------------------------+------------------------+-----------------------+
| 123-4   249     1349     | 2349  5       6        | 1237-9   8       179  |
| (45 23)  4589-2  7       | 1     (8  23) (4 9 23) | (6 9 23) ( 9 23) 569  |
| 6       2589    13589    | 2389  237-8   237-9    | 4        123-9   159  |
+--------------------------+------------------------+-----------------------+
| 127-4   2467    146      | 2369  1237-6  8        | 5        137-49  1479 |
| (45 17)  3       4568-1  | 569   (6  17) ( 59 17) | ( 89 17) (49 17) 2    |
| 9       2578    158      | 235   4       1237-5   | 137-8    6       178  |
+--------------------------+------------------------+-----------------------+
| (45 3)   1       4569-3  | 7     (68 23) (45  23) | ( 89 23) (49  2) 4689 |
| (45 7)   4569-7  2       | 4568  (68  1) (45   1) | (6 9 17) (49 17) 3    |
| 8       467     346      | 2346  9       123-4    | 127-6    5       1467 |
+--------------------------+------------------------+-----------------------+

Puzzle 55 [21,231] 73 Candidates,
20 Truths = {2N15678 5N15678 7N15678 8N15678}
20 Links = {1r58 2r27 3r27 7r58 4c168 5c16 6c57 8c57 9c678}
17 Eliminations --> r3c68<>9, r4c18<>4, r1c1<>4, r1c7<>9, r2c2<>2, r3c5<>8, r4c5<>6, r4c8<>9,
                    r5c3<>1, r6c6<>5, r6c7<>8, r7c3<>3, r8c2<>7, r9c6<>4, r9c7<>6

The columns of cells are covered by 45C1, 68C5, 459C6, 689C7, 49, while
the rows of cells are covered by 23R2, 17R5, 23R7, 17R8,
that is a total of 20 cover sets.

There is no Jellyfish of cells but the addition of a column of cells in the rows is just what is needed to get a loop of cells.
It is to be noted that the digits are 45689 in the column cover sets and 1237 in the row cover sets.
It happens here that these 2 set of digits are making a partition of the digits 1 to 9.

Associated to this loop of cells : a loop of rows (45689)R2578 and a loop of columns (1237)C15678

Code: Select all

+----------------------------+------------------------+-------------------------+
| 123-4   249       1349     | 2349    5       6      | 1237-9   8       179    |
| 23(45)  -2(4589)  7        | 1       23(8)   23(49) | 23(69)   23(9)   (569)  |
| 6       2589      13589    | 2389    237-8   237-9  | 4        123-9   159    |
+----------------------------+------------------------+-------------------------+
| 127-4   2467      146      | 2369    1237-6  8      | 5        137-49  1479   |
| 17(45)  3         -1(4568) | (569)   17(6)   17(59) | 17(89)   17(49)  2      |
| 9       2578      158      | 235     4       1237-5 | 137-8    6       178    |
+----------------------------+------------------------+-------------------------+
| 3(45)   1         -3(4569) | 7       23(68)  23(45) | 2(689)   2(49)   (4689) |
| 7(45)   -7(4569)  2        | (4568)  1(68)   1(45)  | 17(689)  17(49)  3      |
| 8       467       346      | 2346    9       123-4  | 127-6    5       1467   |
+----------------------------+------------------------+-------------------------+

Puzzle 55 [21,231] 69 Candidates,
20 Truths = {45689R2 45689R5 45689R7 45689R8}
20 Links = {4c168 5c16 6c57 8c57 9c678 28n2 57n3 58n4 27n9}
17 Eliminations --> r3c68<>9, r4c18<>4, r1c1<>4, r1c7<>9, r2c2<>2, r3c5<>8, r4c5<>6, r4c8<>9,
                    r5c3<>1, r6c6<>5, r6c7<>8, r7c3<>3, r8c2<>7, r9c6<>4, r9c7<>6

+-------------------------+--------------------------+--------------------------+
| -4(123)  249     1349   | 2349  5         6        | -9(1237)  8         179  |
| 45(23)   4589-2  7      | 1     8(23)     49(23)   | 69(23)    9(23)     569  |
| 6        2589    13589  | 2389  -8(237)   -9(237)  | 4         -9(123)   159  |
+-------------------------+--------------------------+--------------------------+
| -4(127)  2467    146    | 2369  -6(1237)  8        | 5         -49(137)  1479 |
| 45(17)   3       4568-1 | 569   6(17)     59(17)   | 89(17)    49(17)    2    |
| 9        2578    158    | 235   4         -5(1237) | -8(137)   6         178  |
+-------------------------+--------------------------+--------------------------+
| 45(3)    1       4569-3 | 7     68(23)    45(23)   | 689(2)    49(2)     4689 |
| 45(7)    4569-7  2      | 4568  68(1)     45(1)    | 689(17)   49(17)    3    |
| 8        467     346    | 2346  9         -4(123)  | -6(127)   5         1467 |
+-------------------------+--------------------------+--------------------------+

Puzzle 55 [21,231] 73 Candidates,
20 Truths = {1C15678 2C15678 3C15678 7C15678}
20 Links = {1r58 2r27 3r27 7r58 14n1 34n5 369n6 169n7 34n8}
17 Eliminations --> r3c68<>9, r4c18<>4, r1c1<>4, r1c7<>9, r2c2<>2, r3c5<>8, r4c5<>6, r4c8<>9,
                    r5c3<>1, r6c6<>5, r6c7<>8, r7c3<>3, r8c2<>7, r9c6<>4, r9c7<>6

[ronk]

Associated to this loop of cells : a loop of rows (45689)R2578 and a loop of columns (1237)C15678

Except for a structure similar to a 2x2x2 swordfish, I cannot visualize a "loop" of rows or a "loop" of columns. Would you please clarify what you mean, preferably with an illustration.

[JC Van Hay]

Except for a structure similar to a 2x2x2 swordfish, I cannot visualize a "loop" of rows or a "loop" of columns. Would you please clarify what you mean, preferably with an illustration.

Loop = valid rank 0 pattern of non-overlapping base sets
In a loop of rows (columns, cells or boxes), the native SIS in the base are all rows (columns, cells or boxes) of candidates.
So, as illustrated in my post, the loop of rows (45689)R2578 is the valid rank 0 pattern of all the candidates for the digits 45689 in the rows R2578 (the non-verlapping of the base sets being obvious).

[ronk]

Except for a structure similar to a 2x2x2 swordfish, I cannot visualize a "loop" of rows or a "loop" of columns. Would you please clarify what you mean, preferably with an illustration.

Loop = valid rank 0 pattern of non-overlapping base sets

Ugh! Surely there is a better term that doesn't trample on the well established usage of loop.

[JC Van Hay]

Except for a structure similar to a 2x2x2 swordfish, I cannot visualize a "loop" of rows or a "loop" of columns. Would you please clarify what you mean, preferably with an illustration.

Loop = valid rank 0 pattern of non-overlapping base sets

Ugh! Surely there is a better term that doesn't trample on the well established usage of loop.

As you know, I am using Allan Barker's vocabulary . Perhaps you could suggest a better short term

[ronk]

When I posted the 3 puzzles above, I was hoping that someone could come up with a multifish with a row only based truth set. However, only 2row 2column truth set solutions have been given. You hinted my goal might be achieved using the central box for box links. Try as I may I haven't achieved it.

I'm sure you've seen Lerner's solutions. Here are mine, which use four base digits and have box covers in one box. That the cell covers are in the same locations as for the 2-row/2-col sk-loops is appealing as well.

#9698, #9681 and #9650 logic sets: Show

9698;TkP;3952
009000006010008030200000700000000010000073005040805000006000009050300040700000200
19 Truths = {269R24568 7R2468}
19 Links = {27c39 69c17 45n2 2n4 28n5 8n6 56n8 269b5}
11 Eliminations --> r2c45<>4, r2c45<>5, r5c28<>8, r8c56<>1, r4c2<>38, r8c5<>8

9681;TkP;3642
001000008030007040600000900000000072000015000040203000009000600020300050800000001
19 Truths = {1R2468 689R24568}
19 Links = {1c17 6c39 8c37 9c19 45n2 2n4 28n5 8n6 56n8 689b5}
13 Eliminations --> r2c45<>5, r8c56<>4, r1c1<>9, r2c5<>2, r3c3<>8, r4c2<>5, r5c8<>3, r5c2<>7, r7c1<>1, r8c5<>7, r9c3<>6

9650;TkP;5008
003000001020004070600000800000007090000260000050300007001000006090500040800000300
18 Truths = {18R24568 3R2458 6R2468}
18 Links = {1c17 3c19 6c37 8c39 45n2 2n4 28n5 8n6 56n8 18b5}
15 Eliminations --> r2c45<>9, r8c56<>2, r45c2<>4, r1c7<>6, r2c5<>5, r3c9<>3, r5c8<>5, r5c2<>7, r6c8<>2, r7c1<>3, r8c5<>7, r9c3<>6

[ronk]

Loop = valid rank 0 pattern of non-overlapping base sets

Ugh! Surely there is a better term that doesn't trample on the well established usage of loop.

As you know, I am using Allan Barker's vocabulary . Perhaps you could suggest a better short term

I have no suggestion right now, but I do know we've survived for quite some time without a special term.

BTW Allan Barker's usage of loop included many situations that were not 0-rank overall, so IMO you are not "using Allan Barker's vocabulary." That said, I could only guess his meaning for the term.

[pjb]

Many thanks Leren & Ronk.

Your solutions are very instructive. I didn't realize one could have so many base set givens not participating in the 'multifish' (4 in Leren's first example), so back to the drawing board again.

Phil

[JC Van Hay]

...Ugh! Surely there is a better term that doesn't trample on the well established usage of loop.

As you know, I am using Allan Barker's vocabulary . Perhaps you could suggest a better short term

I have no suggestion right now, but I do know we've survived for quite some time without a special term.

BTW Allan Barker's usage of loop included many situations that were not 0-rank overall, so IMO you are not "using Allan Barker's vocabulary." That said, I could only guess his meaning for the term.

I must admit that I shortened Allan Barker's vocabulary : Continuous All Row Loop -> Loop of Rows; and the same for Column(s) and Box(es).
Therefore, a Loop of Cells is the shortening of Continuous All Cell Loop, even if he doesn't use this last expression, except in "SK Loop All Cells".

On the other hand, he uses Continuous Loop and Loop indifferently, except when he is explicitly mentioning Discontinuous Nice Loop (DNL). So there is no confusion at all, even when he calls a piece of logic "Finned Chain" or "Finned Loop", for example.
What may be a confusion, however, is his willingness to refer to classifications of others (Wings, Rings, Fishes, Kraken anything, ...) as well as his own (Rank N Logic, Overlap Truths, Cannibalized Logic, Illegal Logic, ...)

Finally, if I am understanding him well, every piece of logic is a Chain[N] : that is from N native SIS, it can be logically implied N' derived SIS; a Loop[N] is only a special but highly sought-for Chain[N] that can be interpreted as : the N native SIS are non-overlapping, contains exactly N truths and is a subset of N other non-overlapping native SIS; therefore, N derived SIs can be implied.

That said, I understand very well what you are refering to when you are using the expression "well established usage of a loop". I only have to say the following : don't expect from me a splitting of chains into particular chains and networks as well as a splitting of loops into particular continuous loops and continuous networks. I say particular because the frontier would vary from one player to another (... there is no accounting for taste ...). In any case, and this is only a personal POV, I do find irrational this splitting as well as any discussion about it.

[JC Van Hay]

Previous posts : I and II

1. The structure of Jellyfishes of cells in boxes : ALS XY-Ring
The simplest one : the XY-Ring in 4 boxes

Code: Select all

+----------+----------+
| xa  .  . | ya  .  . |
| .   .  . | .   .  . |
| .   .  . | .   .  . |
+----------+----------+
| xb  .  . | yb  .  . |
| .   .  . | .   .  . |
| .   .  . | .   .  . |
+----------+----------+

Loop : (x=a)-(a=y)-(y=b)-(b=x) :=> eliminations in rows and columns
|
V
The most general one : the ALS XY-Ring

Code: Select all

+-------------+-------------+
| XA.  A.  A. | YA.  A.  A. |
| X.   .   .  | Y.   .   .  |
| X.   .   .  | Y.   .   .  |
+-------------+-------------+
| XB.  B.  B. | YB.  B.  B. |
| X.   .   .  | Y.   .   .  |
| X.   .   .  | Y.   .   .  |
+-------------+-------------+

where each of X,Y,A,B is either a single digit or a set of at least 2 digits
and the "5 cells" in each box is a Locked Subset in absence of one of the "letter"
enabling to "write"

Loop : (X=A)-(A=Y)-(Y=B)-(B=X) :=> eliminations in rows, columns and boxes

or, if the notation looks too odd, enabling eliminations according to the solutions

Code: Select all

+------------+------------+     +------------+-----------+
| A.  A.  A. | Y.  .   .  |     | X.  .   .  | A.  A.  A |
| .   .   .  | Y.  .   .  |     | X.  .   .  | .   .   . |
| .   .   .  | Y.  .   .  |     | X.  .   .  | .   .   . |
+------------+------------+ and +------------+-----------+
| X.  .   .  | B.  B.  B. |     | B.  B.  B. | Y.  .   . |
| X.  .   .  | .   .   .  |     | .   .   .  | Y.  .   . |
| X.  .   .  | .   .   .  |     | .   .   .  | Y.  .   . |
+-------------+-----------+     +------------+-----------+

2. Two further examples of application

2a. From "66 An Easy Monster, 9 Truth Loop &&" in xsudon.sud
.2.9...76.......49.....3.28.4.7...6...8...7..1...5..............6.4....7..3.156..
There is a 5x5 square matrix of cells at the intersection of R12478 and C13567, as it is shown below.

Code: Select all

+--------------------------+--------------------------+---------------------+
| (3458 )  2      (145   ) | 9      (48    )  (148  ) | (135 )  7     6     |
| (35678)  13578  (1567  ) | 12568  (2678  )  (12678) | (135 )  4     9     |
|  45679   1579    145679  | 156     467       3      |  15     2     8     |
+--------------------------+--------------------------+---------------------+
| (2359 )  4      (259   ) | 7      (2389  )  (1289 ) | (289 )  6     1235  |
|  23569   359     8       | 1236    23469     12469  |  7      1359  12345 |
|  1       379     2679    | 2368    5         24689  |  249    389   234   |
+--------------------------+--------------------------+---------------------+
| (24589)  1589   (12459 ) | 2368   (236789)  (26789) | (2489)  135   135   |
| (2589 )  6      (1259  ) | 4      (2389  )  (289  ) | (289 )  135   7     |
|  24789   789     3       | 28      1         5      |  6      89    24    |
+--------------------------+--------------------------+---------------------+

Inside that matrix, the most interesting pattern lies at the intersection of R148 and C567.
Excluding the cell 1N7, the other 8 cells would reduce to 3 locked sets for the digits 48,289,289 in R148, respectively, if there were no candidates for the digits 1 and 3.
Therefore, the set of cells 1N56,4N567,8N567 is a Rank 2 pattern as it is covered by 48R1, 289R4, 289R8 and 3C5 and 1C6.
Adding the 6 cells 148N13 to the pattern yields a Rank 0 Pattern as they are already covered by the previous cover and by 35C1 and 15C3.

Code: Select all

+---------------------------+--------------------------+---------------------+
| (35 48 )  2      (15   4) | 9      (48    )  ( 48 1) | (135 )  7     6     |
| (35678 )  13578  (1567  ) | 12568  (2678  )  (12678) | (135 )  4     9     |
|  45679    1579    145679  | 156     467       3      |  15     2     8     |
+---------------------------+--------------------------+---------------------+
| (35 29 )  4      ( 5  29) | 7      (289  3)  (289 1) | (289 )  6     1235  |
|  23569    359     8       | 1236    23469     12469  |  7      1359  12345 |
|  1        379     2679    | 2368    5         24689  |  249    389   234   |
+---------------------------+--------------------------+---------------------+
| (24589 )  1589   (12459 ) | 2368   (236789)  (26789) | (2489)  135   135   |
| ( 5 289)  6      (15  29) | 4      (289  3)  (289  ) | (289 )  135   7     |
|  24789    789     3       | 28      1         5      |  6      89    24    |
+---------------------------+--------------------------+---------------------+

This loop all cells can be viewed as the Swordfish of Cells

Code: Select all

354 154  84 
352  52 382
 59 159 389

to which cells has been added in the 3 rows.

The eliminations are then easily determined.

Code: Select all

+------------------------+------------------------+--------------------+
| (3458)  2      (145)   | 9      (48)     (148)  | 135    7     6     |
| 678-35  13578  67-15   | 12568  2678     2678-1 | 135    4     9     |
| 4679-5  1579   4679-15 | 156    467      3      | 15     2     8     |
+------------------------+------------------------+--------------------+
| (2359)  4      (259)   | 7      (2389)   (1289) | (289)  6     135-2 |
| 269-35  359    8       | 1236   2469-3   2469-1 | 7      1359  12345 |
| 1       379    2679    | 2368   5        24689  | 249    389   234   |
+------------------------+------------------------+--------------------+
| 2489-5  1589   249-15  | 2368   26789-3  26789  | 2489   135   135   |
| (2589)  6      (1259)  | 4      (2389)   (289)  | (289)  135   7     |
| 24789   789    3       | 28     1        5      | 6      89    24    |
+------------------------+------------------------+--------------------+
An Easy Monster [23,219] 46 Candidates,  Loop All Cells
14 Truths = {1N1356 4N13567 8N13567}
14 Links = {2r48 4r1 8r148 9r48 1c36 3c15 5c13}17 Eliminations --> r2357c1<>5, r237c3<>1, r237c3<>5, r5c15<>3, r25c6<>1, r2c1<>3, r4c9<>2, r7c5<>3

As there is no box cover, the number of straightforward equivalent "Multi-Fishes" is 40 as the cells of the loop in any combination of rows or columns
(2 to the power 3 or 2 to the power 5) can be replaced by digits in rows or columns.
The simplest equivalent "Multi-Fish" : the loop 135R148

Code: Select all

+------------------------+------------------------+-----------------------+
| 48(35)  2      4(15)   | 9      48       48(1)  | (135)  7      6       |
| 678-35  13578  67-15   | 12568  2678     2678-1 | 135    4      9       |
| 4679-5  1579   4679-15 | 156    467      3      | 15     2      8       |
+------------------------+------------------------+-----------------------+
| 29(35)  4      29(5)   | 7      289(3)   289(1) | 289    6      -2(135) |
| 269-35  359    8       | 1236   2469-3   2469-1 | 7      1359   12345   |
| 1       379    2679    | 2368   5        24689  | 249    389    234     |
+------------------------+------------------------+-----------------------+
| 2489-5  1589   249-15  | 2368   26789-3  26789  | 2489   135    135     |
| 289(5)  6      29(15)  | 4      289(3)   289    | 289    (135)  7       |
| 24789   789    3       | 28     1        5      | 6      89     24      |
+------------------------+------------------------+-----------------------+
An Easy Monster [23,221] 23 Candidates, Loop All Rows
9 Truths = {1R148 3R148 5R148}
9 Links = {1c36 3c15 5c13 1n7 8n8 4n9}
17 Eliminations --> r2357c1<>5, r237c3<>1, r237c3<>5, r5c15<>3, r25c6<>1, r2c1<>3, r4c9<>2, r7c5<>3

while the most complex one is the loop 249C13567+67C1356

Code: Select all

+-----------------------------+-----------------------------+----------------------+
| 358(4)    2       15(4)     | 9       8(4)       18(4)    | 135      7     6     |
| 358(67)   1358-7  15(67)    | 158-26  8(267)     18(267)  | 135      4     9     |
| -5(4679)  1579    -15(4679) | 156     (467)      3        | 15       2     8     |
+-----------------------------+-----------------------------+----------------------+
| 35(29)    4       5(29)     | 7       38(29)     18(29)   | 8(29)    6     135-2 |
| -35(269)  359     8         | 1236    -3(2469)   -1(2469) | 7        1359  12345 |
| 1         379     (2679)    | 2368    5          -8(2469) | (249)    389   234   |
+-----------------------------+-----------------------------+----------------------+
| -58(249)  1589    -15(249)  | 2368    -38(2679)  -8(2679) | -8(249)  135   135   |
| 58(29)    6       15(29)    | 4       38(29)     8(29)    | 8(29)    135   7     |
| -8(2479)  789     3         | 28      1          5        | 6        89    24    |
+-----------------------------+-----------------------------+----------------------+
An Easy Monster [23,221] 88 Candidates,  Loop All Columns
23 Truths = {2C13567 4C13567 6C1356 7C1356 9C13567}
23 Links = {2r48 4r1 6r2 7r2 9r48 3n135 5n156 6n367 7n13567 9n1 2b2}
21 Eliminations --> r7c1567<>8, r357c1<>5, r5c15<>3, r37c3<>1, r37c3<>5, r2c4<>26, r2c2<>7
                    r4c9<>2, r5c6<>1, r6c6<>8, r7c5<>3, r9c1<>8

In any case, after the eliminations and "basics" (elimination of all the candidates not in any solution of a unit or of a digit), the PM is reduced to

Code: Select all

+------------------+------------------+------------------+
| 3458  2     145  | 9     48    148  | 135  7     6     |
| 678   135   67   | 15    2678  2678 | 135  4     9     |
| 4679  1579  4679 | 156   467   3    | 15   2     8     |
+------------------+------------------+------------------+
| 2359  4     259  | 7     2389  1289 | 289  6     135   |
| 269   359   8    | 1236  2469  2469 | 7    1359  12345 |
| 1     379   2679 | 2368  5     2469 | 249  389   234   |
+------------------+------------------+------------------+
| 249   158   249  | 38    67    67   | 249  135   135   |
| 25    6     125  | 4     2389  289  | 28   135   7     |
| 2479  789   3    | 28    1     5    | 6    89    24    |
+------------------+------------------+------------------+

The rating of the puzzle is now SER=8.4 from an initial SER=9.5.

2b. Puzzle 539
The puzzle contains 3 NxM rectangular matrices of cells at the intersection of N Rows and M Colums, where N and/or M > 3 : 12357N5689=R12357xC5689, 4689N2347=R4689xC2347 and 1247N3689=R1247xC3689.
The first contains the Jellyfish of cells Type II 1257N5689

Code: Select all

+-----------------------+--------------------------+--------------------------+
| 9       8       234-1 | 7       (34 16)  (25 16) | 2345-6  (23 16)  (45 16) |
| 7       6       234-1 | 2345-1  (34 19)  (25 19) | 8       (23 19)  (45 19) |
| 1234    1234    5     | 12348   1689-34  1689-2  | 23469   1679-23  1679-4  |
+-----------------------+--------------------------+--------------------------+
| 6       12379   1237  | 128     5        178-2   | 39      4        189     |
| 2345-1  2345-1  8     | 9       ( 4 16)  (2  16) | 7       ( 3 16)  ( 5 16) |
| 145     14579   147   | 148     1678-4   3       | 569     1689     2       |
+-----------------------+--------------------------+--------------------------+
| 2345-8  2345-7  9     | 6       ( 3 78)  ( 5 78) | 1       (2  78)  (4  78) |
| 1458    1457    1467  | 158     2        1789-5  | 469      6789     3      |
| 1238    1237    12367 | 138     1789-3   4       | 269      5        6789   |
+-----------------------+--------------------------+--------------------------+
539  [22,236] 56 Candidates,  Jellyfish of Cells Type II   
16 Truths = {1257N5 1257N6 1257N8 1257N9}
16 Links = {1r125 6r15 7r7 8r7 9r2 2c68 3c58 4c59 5c69}
18 Eliminations --> r2c34<>1, r3c68<>2, r3c58<>3, r3c59<>4, r5c12<>1, r1c3<>1, r1c7<>6,
                    r4c6<>2, r6c5<>4, r7c2<>7, r7c1<>8, r8c6<>5, r9c5<>3

The second needs 3 more cells to be a loop all cells as 4689N2347 is a Rank 2 pattern.
Replacing the cover sets 17C23 by 17B47 don't change the Rank; adding the base cells 789N1 and the cover set 8B7 give a Rank 0 pattern.

Code: Select all

+--------------------------------+-------------------------+------------------------+
| 9         8          1234      | 7        1346    1256   | 2345-6   1236    1456  |
| 7         6          1234      | 2345-1   1349    1259   | 8        1239    1459  |
| 1234      1234       5         | 234-18   134689  12689  | 234-69   123679  14679 |
+--------------------------------+-------------------------+------------------------+
| 6         (17 9 23)  (17   23) | (18 2 )  5       178-2  | ( 9  3)  4       189   |
| 2345-1    2345-1     8         | 9        146     126    | 7        136     156   |
| (1   45)  (17 9 45)  (17   4 ) | (18 4 )  1678-4  3      | (69  5)  1689    2     |
+--------------------------------+-------------------------+------------------------+
| 2345-8    2345-7     9         | 6        378     578    | 1        278     478   |
| (1 8 45)  (17   45)  (17 6 4 ) | (18  5)  2       1789-5 | (69 4 )  6789    3     |
| (1 8 23)  (17   23)  (17   23) | (18  3)  1789-3  4      | (69 2 )  5       6789  |
+--------------------------------+-------------------------+------------------------+
539  [22,236] 68 Candidates, Loop All Cells     
19 Truths = {4N2347 6N12347 8N12347 9N12347}
19 Links = {2r49 3r49 4r68 5r68 18c4 69c7 179b4 1678b7}
 14 Eliminations --> r5c12<>1, r23c4<>1, r13c7<>6, r3c4<>8, r3c7<>9, r4c6<>2, r6c5<>4,
                    r7c2<>7, r7c1<>8, r8c6<>5, r9c5<>3

The third doesn't give a loop all cells by adding cells to its rows and/or columns.
Finally, looking for an ALS XY-Ring in all the squares of 4 boxes, the following sk-loop all cells is straightforwardly found with the help of the B/B-plot :

Code: Select all

+-----------------------+------------------------+-----------------------+
| 9       8       1234  | 7       1346    1256   | 2345-6  1236    1456  |
| 7       6       1234  | 2345-1  1349    1259   | 8       1239    1459  |
| 1234    1234    5     | 234-18  134689  12689  | 234-69  123679  14679 |
+-----------------------+------------------------+-----------------------+
| 6       12379   1237  | (128)   5       178-2  | (39)    4       189   |
| 2345-1  2345-1  8     | 9       (146)   (126)  | 7       (136)   (156) |
| 145     14579   147   | (148)   1678-4  3      | (569)   1689    2     |
+-----------------------+------------------------+-----------------------+
| 2345-8  2345-7  9     | 6       (378)   (578)  | 1       (278)   (478) |
| 1458    1457    1467  | (158)   2       1789-5 | (469)   6789    3     |
| 1238    1237    12367 | (138)   1789-3  4      | (269)   5       6789  |
+-----------------------+------------------------+-----------------------+
539  [22,236] 47 Candidates, SK-Loop All Cells   
16 Truths = {4689N4 57N5 57N6 4689N7 57N8 57N9}
16 Links = {16r5 78r7 18c4 69c7 2b59 3b68 4b59 5b68}
14 Eliminations --> r5c12<>1, r23c4<>1, r13c7<>6, r3c4<>8, r3c7<>9, r4c6<>2, r6c5<>4,
                    r7c2<>7, r7c1<>8, r8c6<>5, r9c5<>3

In any case, after the eliminations and "basics" (elimination of all the candidates not in any solution of a unit or of a digit), the PM is reduced to

Code: Select all

+--------------------+------------------+------------------+
| 9     8      234   | 7     1346  1256 | 2345  1236  1456 |
| 7     6      234   | 2345  1349  1259 | 8     1239  1459 |
| 1234  1234   5     | 234   689   689  | 234   679   679  |
+--------------------+------------------+------------------+
| 6     12379  1237  | 128   5     178  | 39    4     189  |
| 2345  2345   8     | 9     146   126  | 7     136   156  |
| 145   14579  147   | 148   1678  3    | 569   1689  2    |
+--------------------+------------------+------------------+
| 2345  2345   9     | 6     378   578  | 1     278   478  |
| 1458  1457   1467  | 158   2     1789 | 469   6789  3    |
| 1238  1237   12367 | 138   1789  4    | 269   5     6789 |
+--------------------+------------------+------------------+

3. Questions

3a. How does all this compare to David's method ?
3b. Does a puzzle exist where 2 loops all cells give different PM after eliminations and "basics"?
3c. Does there exist any other rank 0 pattern not equivalent to a loop of cells ?

To the 2 last questions, as I may be wrong, my conjecture is NO for both.

[David P Bird]

JC Van Hay, I haven't had time to follow your two previous posts properly as I have home improvements in progress, but definitely side with ronk that you should not be overloading the 'loop' term the way you do. Alternative short collective nouns might be 'nest' or 'clan' but I'm not sure how they fit your mental picture.

I first thought that your approach was limited to when the Multi-Sector Locked Set cells happen to fall on the intersections of sets of rows and columns (which they often don't) and my method of looking for concentrations of houses holding givens for a focus set of digits would be quicker, but there may be other tricks you can play which might give us something new – I'm not sure.

Not having time to work each or your examples I checked your results for puzzle 539 against mine.
I found the SK loop you identified and once those elimination were made the (1234)Naked Set in r3 produced the rest – a common occurrence with SK loops.

I can compose the MSLS that gives all your eliminations in one go but only by breaking a rule of mine that keeps all the focus digits together in any cover set.

I like to make SK loop eliminations on their own as then I can run (and notate) a quick check to see if the linking pairs in the loop can alternately hold 2 and 0 truths or not. Here a UR in r57c12 would be formed if that was so and consequently they must all hold 1 truth.

[champagne]

I entered on my website a full path for the puzzle 10 of the data base of potential hardest

Code: Select all

2.......6.5..8..1...4...9...7.3.1......82.......7.5.3...9...4...8..1..5.6.......2
ER=11.8/11.8/11.4;tarek-ultra-0203;10


That solution can be improved, but contains interesting points.
It is derived from the path proposed by the current code of my new solver.

The SK loop is seen and solved.
Then, the 2 main belts invalidity is established
At that point, the solver fails in dynamic plus mode and a specific process recently coded is applied with the following key rules:

The properties of a SK loop with the main belts killed is translated in weak and strong inferences
The dynamic plus process is extended to UR’s.

2 keys eliminations have been found using that set of rules

<6> r3c5
<6> r7c5

The rest has been done with not more than the dynamic process equivalent to Sudoku Explainer corresponding set of rules (rating around 9.0). For sure, including the SK loop properties at that point would lead to a shorter path.

I cut the solver solution at a point where the sk loop was nearly solved, using the sk loop properties

puz10 path