## "Potential Hardest" 12

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### "Potential Hardest" 12

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`+-------+-------+-------+| . . . | . 9 8 | 7 . . || . . . | 5 4 . | . 6 . || . . . | . . . | . . 5 |+-------+-------+-------+| . . 4 | . . . | . 5 3 || . . 7 | . . . | 2 . . || 1 . . | . . 2 | 8 . . |+-------+-------+-------+| 9 1 . | . 2 . | . . . || 7 . . | . 1 9 | . . . || . . 3 | 6 . . | . . . |+-------+-------+-------+....987.....54..6.........5..4....53..7...2..1....28..91..2....7...19.....36.....`

(As a reminder, these are puzzles which are rated 11+ by SE, but fall to some advanced technique not implemented by SE.)
mith

Posts: 369
Joined: 14 July 2020

### Re: "Potential Hardest" 12

Code: Select all
`*--------------------------------------------------------------------*|#3456-2 #3456-2 *1256 |*123     9     8     | 7    *1234   *124     || 238     23789   1289 | 5       4     137   | 139   6       1289    || 2348    234789  1289 | 127-3   367   1367  | 1349  1289-34 5       ||----------------------+---------------------+-----------------------|| 268     2689    4    | 1789    678   167   | 169   5       3       || 3568    35689   7    | 189-34  3568  13456 | 2     19-4    19-46   || 1      #356-9  *569  |*349    #356   2     | 8    *479    *4679    ||----------------------+---------------------+-----------------------|| 9       1      *568  |*3478    2    #345-7 |#3456 *3478   *4678    || 7      #456-28 *2568 |*348     1     9     |#3456 *2348   *2468    || 2458    2458    3    | 6       578   457   | 1459  12789-4 12789-4 |*--------------------------------------------------------------------*`

Multifish : Truths 16 ; 3456r1678; Links 16 ; 56c3, 34c48, 46c9, Cells r1c12, r6c25, r7c67, r8c27 => - 2 r1c12, - 3 r3c4, - 34 r3c8, - 34 r5c4, - 4 r5c8, - 46 r5c9, - 9 r6c2, - 7 r7c6, - 28 r8c2, - 4 r9c89; btte

Leren
Leren

Posts: 4006
Joined: 03 June 2012

### Re: "Potential Hardest" 12

Hi mith and Leren,
I find a msls on the subset r1678-c3489 for the partition 12789-3456 which eliminates the 16 candidates 2 r1c1, 2 r1c2, 9 r6c2, 7 r7c6, 2 r8c2, 8 r8c2, 3 r3c4, 3 r5c4, 4 r5c4, 3 r3c8, 4 r3c8, 4 r5c8, 4 r9c8, 4 r5c9, 6 r5c9, 4 r9c9 (see puzzle).
puzzle: Show
These are the same eliminations as those found by Leren, so what's the difference between this msls and this multifish, especially since Phil's solver finds the msls but not the multifish?
Cordialy
Robert
Mauriès Robert

Posts: 459
Joined: 07 November 2019
Location: France

### Re: "Potential Hardest" 12

Hi Robert,

Mauriès Robert wrote:I find a msls on the subset r1678-c3489 for the partition 12789-3456 which eliminates the 16 candidates 2 r1c1, 2 r1c2, 9 r6c2, 7 r7c6, 2 r8c2, 8 r8c2, 3 r3c4, 3 r5c4, 4 r5c4, 3 r3c8, 4 r3c8, 4 r5c8, 4 r9c8, 4 r5c9, 6 r5c9, 4 r9c9 (see puzzle).
These are the same eliminations as those found by Leren, so what's the difference between this msls and this multifish, especially since Phil's solver finds the msls but not the multifish?

I'm really surprised that Phil's solver doesn't find the multifish. It's usually really good at that, and Phil's documentation of multifishes is the best I know. This is a very basic multifish, too, so I suspect some weird bug.

The difference between your MSLS and Leren's Multifish is in the sets of truths and links they use to form the 16x16 Rank 0 structure. Yours uses 16 cells as truths and Leren's uses 4 rows for 4 digits (4x4 = 16). The candidates in those sets are slightly different, so they need partly different linksets to cover them. Yours uses rows and columns to cover the candidates in the truth cells, while Leren's uses columns and cells to cover the truth rows.

Despite the different structures they produce the same eliminations, which is very common in these situations. If a Rank 0 pattern exists, it usually has many equivalent forms. Here's how I see the two forms in this case (* marks truths and \ marks links):

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`                   \56     \34                          \34       \46.------------------------.----------------------.--------------------------.| 3456-2  3456-2   *1256 | *123     9     8     | 7     *1234     *124     | \12| 238     23789     1289 |  5       4     137   | 139    6         1289    || 2348    234789    1289 |  127-3   367   1367  | 1349   1289-34   5       |:------------------------+----------------------+--------------------------:| 268     2689      4    |  1789    678   167   | 169    5         3       || 3568    35689     7    |  189-34  3568  13456 | 2      19-4      19-46   || 1       356-9    *569  | *349     356   2     | 8     *479      *4679    | \79:------------------------+----------------------+--------------------------:| 9       1        *568  | *3478    2     345-7 | 3456  *3478     *4678    | \78| 7       456-28   *2568 | *348     1     9     | 3456  *2348     *2468    | \28| 2458    2458      3    |  6       578   457   | 1459   12789-4   12789-4 |'------------------------'----------------------'--------------------------' MSLS: 16x16 {1678N3489 \ 12r1 79r6 78r7 28r8 56c3 34c48 46c9} => 16 elims`

Code: Select all
`  \1n      \268n    \56    \34      \6n    \7n      \78n  \34      \46.-------------------------.-----------------------.-------------------------.| \3456-2  \3456-2   1256 | 123      9      8     |  7     1234     124     | *3456|  238      23789    1289 | 5        4      137   |  139   6        1289    ||  2348     234789   1289 | 127-3    367    1367  |  1349  1289-34  5       |:-------------------------+-----------------------+-------------------------:|  268      2689     4    | 1789     678    167   |  169   5        3       ||  3568     35689    7    | 189-34   3568   13456 |  2     19-4     19-46   ||  1       \356-9    569  | 349     \356    2     |  8     479      4679    | *3456:-------------------------+-----------------------+-------------------------:|  9        1        568  | 3478     2     \345-7 | \3456  3478     4678    | *3456|  7       \456-28   2568 | 348      1      9     | \3456  2348     2468    | *3456|  2458     2458     3    | 6        578    457   |  1459  12789-4  12789-4 |'-------------------------'-----------------------'-------------------------' MF(R): 16x16 {3456R1678 \ 56c3 34c48 46c9 1n1 268n2 6n5 7n6 78n7} => 16 elims`

Personally I prefer multifishes, because they're easier and quicker to find and use manually, at least for me.

--
Edit. Fixed a typo in the MF: 1689 -> 1678.
Last edited by SpAce on Fri Dec 18, 2020 5:35 am, edited 1 time in total.
-SpAce-: Show
Code: Select all
`   *             |    |               |    |    *        *        |=()=|    /  _  \    |=()=|               *            *    |    |   |-=( )=-|   |    |      *     *                     \  ¯  /                   *    `

"If one is to understand the great mystery, one must study all its aspects, not just the dogmatic narrow view of the Jedi."

SpAce

Posts: 2674
Joined: 22 May 2017

### Re: "Potential Hardest" 12

Another MSLS (the easiest to spot manually: just list givens per row, 3456 are the only givens in rows 2349)
Code: Select all
` +--------------------------+-------------------------+---------------------------+ |  3456-2  23456    1256   |  123     9      8       |  7      1234     124      | | <238     2789-3   1289   |  5       4     <137     | <139    6        1289     | 3  | <2348    2789-34  1289   |  127-3  <367   <1367    | <1349   1289-34  5        | 346 +--------------------------+-------------------------+---------------------------+ | <268     289-6    4      |  1789   <678   <167     | <169    5        3        | 6 |  356-8   35689    7      |  13489   356-8  3456-1  |  2      149      1469     | |  1       3569     569    |  349     356    2       |  8      479      4679     | +--------------------------+-------------------------+---------------------------+ |  9       1        568    |  3478    2      345-7   |  3456   3478     4678     |  |  7       24568    2568   |  348     1      9       |  3456   2348     2468     | | <2458    28-45    3      |  6      <578   <457     | <1459   12789-4  12789-4  | 45 +--------------------------+-------------------------+---------------------------+    28                                 78     17         19`

MSLS
15 cell truths r2c167, r349c1567; 15 links 3r2, 346r3, 6r4, 45r9, 28c1, 78c5, 17c6, 19c7
16 eliminations -2 r1c1, -8 r5c1, -8 r5c5, -1 r5c6, -7 r7c6, -3 r2c2, -34 r3c28, -3 r3c4, -6 r4c2, -45 r9c2, -4 r9c89; lclste
Cenoman
Cenoman

Posts: 1576
Joined: 21 November 2016
Location: Paris, France

### Re: "Potential Hardest" 12

Hi Robert,

If you want to view a multifish as a cell-based MSLS, it's quite easy. Just switch into a different space. For example, Leren's row-based multifish becomes an MSLS in the nr-space (and a different-looking multifish in the nc-space). If it were column-based, then it would be an MSLS in the nc-space (and a different-looking multifish in the nr-space). The conversion is straight-forward as long as boxes aren't used as bases or covers.

Note that to make the same coordinates work in every space, I'm not using Allan Barker's confusing set identifiers below. So 'n' actually means a digit and rc-cells are written normally as rYcX instead of yNx.

Leren's row-based multifish in the rc-space:

Code: Select all
`      c1       c2      c3     c4       c5     c6       c7    c8       c9   .------------------------.-----------------------.-------------------------.r1 | \3456-2  \3456-2  1256 | 123      9      8     |  7     1234     124     | *3456r2 |  238      23789   1289 | 5^       4^     137   |  139   6^       1289    |r3 |  2348     234789  1289 | 127-3    367    1367  |  1349  1289-34  5^      |   :------------------------+-----------------------+-------------------------:r4 |  268      2689    4^   | 1789     678    167   |  169   5^       3^      |r5 |  3568     35689   7    | 189-34   3568   13456 |  2     19-4     19-46   |r6 |  1       \356-9   569  | 349     \356    2     |  8     479      4679    | *3456   :------------------------+-----------------------+-------------------------:r7 |  9        1       568  | 3478     2     \345-7 | \3456  3478     4678    | *3456r8 |  7       \456-28  2568 | 348      1      9     | \3456  2348     2468    | *3456r9 |  2458     2458    3^   | 6^       578    457   |  1459  12789-4  12789-4 |   '------------------------'-----------------------'-------------------------'     \r1      \r168   \n56   \n34     \r6    \r7      \r78  \n34     \n46MF-R(rc): 16x16 {n3456r1678 \ n56c3 n34c4 n34c8 n46c9 r1c1 r168c2 r6c5 r7c6 r78c7} => 16 elimsThat's using 16 (nr-)rows as bases, and 8 (nc-)columns and 8 (rc-)cells as covers.`

The same in the nr-space:

Code: Select all
`      r1       r2    r3         r4     r5       r6        r7      r8      r9     .--------------------------.------------------------.--------------------------.n1 |  3489     3679  34678    | 467    4689     1      |  2       5       789     |n2 |  3489-12  1239  12348    | 12     7        6      |  5       389-2   1289    |n3 | *1248     1267  12567-48 | 9      1256-4  *245    | *4678   *478     3       | \c48n4 | *1289     5     127-8    | 3      6-489   *489    | *46789  *24789   1267-89 | \c489n5 | *123      4     9        | 8      1256    *235    | *367    *237     12567   | \c3n6 | *123      8     56       | 12567  1256-9  *2359   | *379    *2379    4       | \c39n7 |  7        26    2456     | 456    3        89     |  489-6   1       5689    |n8 |  6        1239  1238     | 1245   1245     7      |  3489    3489-2  12589   |n9 |  5        2379  2378     | 247    2489     3489-2 |  1       6       789     |   '--------------------------'------------------------'--------------------------'     \c12                                      \c25      \c67    \c27MSLS(nr): 16x16 {n3456r1678 \ c48n3 c489n4 c3n5 c39n6 c12r1 c25r6 c67r7 c27r8} => 16 elimsThat's using 16 (nr-)cells as bases, and 8 (cn-)rows and 8 (cr-)columns as covers.`

And lastly in the nc-space:

Code: Select all
`     c1      c2        c3      c4       c5    c6       c7     c8         c9   .------------------------.------------------------.---------------------------. n1 | 6       7         123  |  1345     8     2345   | 2349   1359       1259    |n2 | 2349-1  2349-18   1238 |  13       7     6      | 5      139        129     |n3 | 1235    12356     9    | \1678-35  356   2357   | 2378  \178-3      4       | *r1678n4 | 139     1389      4    | \678-5    2     579    | 3789  \1678-359  \1678-59 | *r1678n5 | 159     15689    \1678 |  2        569   579    | 789    4          3       | *r1678n6 | 1345    134568   \1678 |  9        3456  345    | 478    2         \678-5   | *r1678n7 | 8       23        5    |  3467     349   2349-7 | 1      679        679     |n8 | 23459   23459-8   2378 |  4578     459   1      | 6      3789       2789    |n9 | 7       2345-6    236  |  456      1     8      | 2349   3569       2569    |   '------------------------'------------------------'---------------------------'    \r1     \r168     \n56    \n34     \r6   \r7      \r78   \n34       \n46MF-R(nc): 16x16 {r1678n3456 \ r1c1 r168c2 r6c5 r7c6 r78c7 n56c3 n34c4 n34c8 n46c9} => 16 elimsThat's using 16 (rn-)rows as bases, and 8 (rc-)columns and 8 (nc-)cells as covers.`

Note that the actual base and cover sets are exactly the same in each case, just written slightly differently to reflect the different perspectives.

PS. Feel free to use rn- instead of nr-space if it feels more natural to keep the rows in the rows. I prefer to have identical orientations for the nr- and nc-spaces because it makes it easier to work with the boxes.
-SpAce-: Show
Code: Select all
`   *             |    |               |    |    *        *        |=()=|    /  _  \    |=()=|               *            *    |    |   |-=( )=-|   |    |      *     *                     \  ¯  /                   *    `

"If one is to understand the great mystery, one must study all its aspects, not just the dogmatic narrow view of the Jedi."

SpAce

Posts: 2674
Joined: 22 May 2017