The New Sudoku Players' Forum

[mith]

Code: Select all

+-------+-------+-------+
| . 1 2 | . 3 . | . . . |
| 9 . . | . . 8 | . 7 . |
| . . . | . . . | . . . |
+-------+-------+-------+
| . . . | . . . | . . . |
| 6 . . | . . 7 | . . . |
| . 4 . | . 5 . | . . 1 |
+-------+-------+-------+
| . . . | . . . | 3 . . |
| 8 . . | . . 9 | . 6 . |
| 7 . . | . 4 . | 1 . 5 |
+-------+-------+-------+
.12.3....9....8.7...................6....7....4..5...1......3..8....9.6.7...4.1.5

[Leren]

Code: Select all

*------------------------------------------------------------------------*
| 45    1      2      | 7        3      *456    |#689-45 *4589   #689-4  | 689
| 9     356    3456   | 1245      12     8      | 245     7       234    |
| 345   78     78     | 245-69    269    245-6  | 2456    1       2346   |
|---------------------+-------------------------+------------------------|
| 1235  235789 135789 | 1234-689  12689  1234-6 | 2456789 2345-89 246789 |
| 6     23589  13589  | 1234-89   1289   7      | 24589   2345-89 2489   |
| 23    4     #789-3  |*689-23   5      *236    |#6789-2 *2389    1      | 6789
|---------------------+-------------------------+------------------------|
| 1245 *2569  *14569  |*68-125  #678-12 *1256   | 3      *2489   #789-24 | 6789
| 8     235    1345   | 1235     127     9      | 247     6       247    |
| 7    *2369  *369    |*68-23    4      *236    | 1      *289     5      | 689
*------------------------------------------------------------------------*
\---------69-b7-------/ 689              6                89

Multifish : 14 Truths = { 689R1 6789R6 6789R7 689R9 } 14 Links = { 689c4 6c6 89c8 1n79 6n37 7n59 69b7 }.

29 Eliminations : r1c7 <> 45, r1c9 <> 4, r3c4 <> 69, r3c6 <> 6, r4c4 <> 689, r4c6 <> 6, r4c8 & r5c48 <> 89, r6c3 <> 3, r6c4 <> 23, r6c7 <> 2, r7c4 <> 125, r7c5 <> 12, r7c9 <> 24, r9c4 <> 23; stte

Leren

<edit> To make it clearer I put the eliminations into the PM and corrected a typo in the elimination list. Leren

[denis_berthier]

.
After a trivial start:

start: Show

***********************************************************************************************
*** SudoRules 20.1.s based on CSP-Rules 2.1.s, config = S
*** Using CLIPS 6.32-r779
*** Download from: https://github.com/denis-berthier/CSP-Rules-V2.1
***********************************************************************************************
hidden-single-in-a-column ==> r3c8 = 1
hidden-single-in-a-row ==> r1c4 = 7
266 candidates, 2032 csp-links and 2032 links. Density = 5.77%
whip[1]: c8n3{r6 .} ==> r5c9 ≠ 3, r4c9 ≠ 3
whip[1]: r1n9{c9 .} ==> r3c9 ≠ 9, r3c7 ≠ 9
whip[1]: r1n8{c9 .} ==> r3c9 ≠ 8, r3c7 ≠ 8
hidden-pairs-in-a-row: r3{n7 n8}{c2 c3} ==> r3c3 ≠ 6, r3c3 ≠ 5, r3c3 ≠ 4, r3c3 ≠ 3, r3c2 ≠ 6, r3c2 ≠ 5, r3c2 ≠ 3
whip[1]: b1n6{r2c3 .} ==> r2c4 ≠ 6, r2c5 ≠ 6, r2c7 ≠ 6, r2c9 ≠ 6
x-wing-in-columns: n1{c1 c6}{r4 r7} ==> r7c5 ≠ 1, r7c4 ≠ 1, r7c3 ≠ 1, r4c5 ≠ 1, r4c4 ≠ 1, r4c3 ≠ 1

we reach the following resolution state:

Code: Select all

RESOLUTION STATE:
   45        1         2         7         3         456       45689     4589      4689     
   9         356       3456      1245      12        8         245       7         234       
   345       78        78        24569     269       2456      2456      1         2346     
   1235      235789    35789     234689    2689      12346     2456789   234589    246789   
   6         23589     13589     123489    1289      7         24589     234589    2489     
   23        4         3789      23689     5         236       26789     2389      1         
   1245      2569      4569      2568      2678      1256      3         2489      24789     
   8         235       1345      1235      127       9         247       6         247       
   7         2369      369       2368      4         236       1         289       5       

From that point, it is solved in two steps of Forcing-T&E:

FORCING-T&E applied to bivalue candidates n7r7c5 and n7r7c9 :
===> 0 values decided in both cases:
===> 7 candidates eliminated in both cases: n8r4c4 n2r4c5 n8r5c4 n1r5c5 n2r5c5 n2r7c5 n6r7c5

Code: Select all

CURRENT RESOLUTION STATE:
   45        1         2         7         3         456       45689     4589      4689     
   9         356       3456      1245      12        8         245       7         234       
   345       78        78        24569     269       2456      2456      1         2346     
   1235      235789    35789     23469     689       12346     2456789   234589    246789   
   6         23589     13589     12349     89        7         24589     234589    2489     
   23        4         3789      23689     5         236       26789     2389      1         
   1245      2569      4569      2568      78        1256      3         2489      24789     
   8         235       1345      1235      127       9         247       6         247       
   7         2369      369       2368      4         236       1         289       5         

FORCING-T&E applied to bivalue candidates n6r3c5 and n6r4c5 :
===> 14 values decided in both cases: n7r7c5 n4r7c8 n4r8c3 n1r7c1 n1r4c6 n1r5c3 n1r8c5 n2r2c5 n1r2c4 n8r6c8 n9r4c7 n4r4c4 n6r2c2 n2r3c7
===> 129 candidates eliminated in both cases: n4r1c7 n5r1c7 n9r1c7 n4r1c8 n8r1c8 n4r1c9 n6r1c9 n3r2c2 n5r2c2 n4r2c3 n6r2c3 n2r2c4 n4r2c4 n5r2c4 n1r2c5 n2r2c7 n2r2c9 n8r3c2 n2r3c4 n4r3c4 n6r3c4 n2r3c5 n2r3c6 n6r3c6 n4r3c7 n5r3c7 n6r3c7 n2r3c9 n4r3c9 n1r4c1 n3r4c1 n2r4c2 n3r4c2 n5r4c2 n9r4c2 n7r4c3 n8r4c3 n9r4c3 n2r4c4 n3r4c4 n6r4c4 n9r4c4 n9r4c5 n2r4c6 n3r4c6 n4r4c6 n6r4c6 n2r4c7 n4r4c7 n5r4c7 n6r4c7 n7r4c7 n8r4c7 n4r4c8 n5r4c8 n8r4c8 n9r4c8 n2r4c9 n4r4c9 n8r4c9 n9r4c9 n2r5c2 n3r5c2 n5r5c2 n3r5c3 n5r5c3 n8r5c3 n9r5c3 n1r5c4 n4r5c4 n9r5c4 n2r5c7 n8r5c7 n9r5c7 n2r5c8 n4r5c8 n8r5c8 n9r5c8 n8r5c9 n9r5c9 n3r6c3 n8r6c3 n3r6c4 n6r6c4 n8r6c4 n2r6c6 n2r6c7 n8r6c7 n9r6c7 n2r6c8 n3r6c8 n9r6c8 n2r7c1 n4r7c1 n5r7c1 n6r7c2 n9r7c2 n4r7c3 n5r7c3 n2r7c4 n5r7c4 n8r7c5 n1r7c6 n6r7c6 n2r7c8 n8r7c8 n9r7c8 n2r7c9 n4r7c9 n7r7c9 n2r8c2 n1r8c3 n3r8c3 n5r8c3 n1r8c4 n2r8c4 n2r8c5 n7r8c5 n2r8c7 n4r8c7 n4r8c9 n2r9c2 n6r9c2 n3r9c3 n9r9c3 n2r9c4 n3r9c4 n6r9c6 n8r9c8

Code: Select all

CURRENT RESOLUTION STATE:
   45        1         2         7         3         456       68        59        89       
   9         6         35        1         2         8         45        7         34       
   345       7         78        59        69        45        2         1         36       
   25        78        35        4         68        1         9         23        67       
   6         89        1         23        89        7         45        35        24       
   23        4         79        29        5         36        67        8         1         
   1         25        69        68        7         25        3         4         89       
   8         35        4         35        1         9         7         6         27       
   7         39        6         68        4         23        1         29        5         

stte


Needless to say, this is not my preferred way of solving.

[DEFISE]

.
Needless to say, this is not my preferred way of solving.

Hi Denis,
Here is my solution:

Hidden Text: Show

Singles: 7r1c4, 1r3c8
Alignment: 8r1b3 => -8r3c7 -8r3c9
Alignment: 9r1b3 => -9r3c7 -9r3c9
Alignment: 3c8b6 => -3r4c9 -3r5c9
Hidden pair: 78r3c23 => -3r3c2 -5r3c2 -6r3c2 -3r3c3 -4r3c3 -5r3c3 -6r3c3
Alignment: 6b1r2 => -6r2c4 -6r2c5 -6r2c7 -6r2c9
Xwing: 1c16r47 => -1r4c3 -1r4c4 -1r4c5 -1r7c3 -1r7c4 -1r7c5
S2-braid[8] : b9n8{r7c9 r79c8}- b9n9{r7c9 r79c8}- {n9r1c8 NP: n45r1c18}- r1c6{n4 n6}- {n6r6c6 NP: n23r6c16}- r6c8{n2 .}
=> -7L7C9
Single: 7r7c5
Alignment: 8c5b5 => -8r4c4 -8r5c4 -8r6c4
Naked pair: 12c5r28 => -2r3c5 -2r4c5 -1r5c5 -2r5c5
S3-braid[10] : r3n9{c5 c4}- {n9r6c4 NT : n236r6c146}- {n6r1c6 NP : n45r1c16}-{n4r1c8 NP : n89r16c8}- b9n8{r7c8 r7c9}- b9n9{r7c9 .} => -6r3c5
Singles and one alignment to the solution.

[Cenoman]

Code: Select all

 +---------------------------+----------------------------+------------------------------+
 | A45     1        2        |  7         3      A456     |  689-45   A4589     689-4    |
 |  9      356      3456     |  1245      12      8       |  245       7        234      |
 |  345    78       78       |  24569     269     245-6   |  2456      1        2346     |
 +---------------------------+----------------------------+------------------------------+
 |  1235   235789   135789   |  1234689   12689   1234-6  |  2456789   2345-89  246789   |
 |  6      23589    13589    |  123489    1289    7       |  24589     2345-89  2489     |
 | B23     4        789-3    |  689-23    5      B236     |  6789-2   B2389     1        |
 +---------------------------+----------------------------+------------------------------+
 |  1245   2569     14569    |  12568     12678   125-6   |  3        C2489     89-247   |
 |  8      235      1345     |  1235      12-7    9       | C247       6       C247      |
 |  7      2369     369      |  2368      4       23-6    |  1        C289      5        |
 +---------------------------+----------------------------+------------------------------+

Consider AALS A (45689)r1c168, AALS B (23689)r6c168 and ALS C (24789)b9p2468.
6 is a restricted common between A,B; 8 and 9 are restricted commons between A,B,C with a contraint degree of 2.

Code: Select all

They form the following net:
   A(2)--6--B(2)
    \       /
     \     /
      \   /
       8 9(2 each)
        |
        |
        C(1)
Freedom degree: 2+2+1-2*2-1 = 0

=> 19 eliminations: -4r1c79, -5r1c8, -2r6c47, -3r6c34, -6r3479c6, -89r56c8, -247r7c9, -7r8c5; lclste

Can be presented as a MSLS: same 10 cell truths; 10 links: 45r1, 23r6, 6c6, 89r8, 247b9; same eliminations.

Edit: corrected typo and eliminations in PM

[mith]

Cenoman's is the primary one I was looking at (though there are of course others). This is the SET* presentation of it:

Consider the following partitions:

Red: r2345789c168 (21 cells)
Blue: r16c234579 (12 cells)
Red = Blue + 1 set of 1-9 (Red + r16c168 is 3 columns; Blue + r16c168 is 2 rows)

Blue contains 6 low (12345) digits as givens, so Red must contain (at least) 11 low digits. Red contains 9 high digits as givens.

However, in b9, at most one of r79c8 can be low - if both were low, r8c79 would both be 7. (This is ALS C given by Cenoman.) So we have a 10th high digit in Red, and Red must therefore be exactly 11 low and 10 high (and Blue must be exactly 6 low and 6 high).

Deductions:

Red: -6r3479c6, -89r56c8
Blue: -4r1c79, -5r1c7, -3r6c34, -2r6c47
ALS C: -247r7c9, -7r8c5

The logic for the one corresponding to Leren's 14 cell is more akin to what I used in Garam Masalas (one of the b7 cells has to be low simply by counting the number of high digits available to place - there's already an 8 in the box), and is also more powerful in this case (more eliminations, and reduces to singles), but I thought the box 9 ALS was pretty slick.

*SET is a term that has caught on elsewhere for this sort of partition argument. It's a nested acronym: SET Equivalence Theory. It's just another way of looking at MSLS deductions, replacing cell truths with equivalent partitions (modulo complete sets of 1-9) and links with digit counts within those partitions. I personally find it much easier to visualize than MSLS, for whatever that's worth.

[denis_berthier]

.
Needless to say, this is not my preferred way of solving.

Hi Denis,
Here is my solution:

Hidden Text: Show

Singles: 7r1c4, 1r3c8
Alignment: 8r1b3 => -8r3c7 -8r3c9
Alignment: 9r1b3 => -9r3c7 -9r3c9
Alignment: 3c8b6 => -3r4c9 -3r5c9
Hidden pair: 78r3c23 => -3r3c2 -5r3c2 -6r3c2 -3r3c3 -4r3c3 -5r3c3 -6r3c3
Alignment: 6b1r2 => -6r2c4 -6r2c5 -6r2c7 -6r2c9
Xwing: 1c16r47 => -1r4c3 -1r4c4 -1r4c5 -1r7c3 -1r7c4 -1r7c5
S2-braid[8] : b9n8{r7c9 r79c8}- b9n9{r7c9 r79c8}- {n9r1c8 NP: n45r1c18}- r1c6{n4 n6}- {n6r6c6 NP: n23r6c16}- r6c8{n2 .}
=> -7L7C9
Single: 7r7c5
Alignment: 8c5b5 => -8r4c4 -8r5c4 -8r6c4
Naked pair: 12c5r28 => -2r3c5 -2r4c5 -1r5c5 -2r5c5
S3-braid[10] : r3n9{c5 c4}- {n9r6c4 NT : n236r6c146}- {n6r1c6 NP : n45r1c16}-{n4r1c8 NP : n89r16c8}- b9n8{r7c8 r7c9}- b9n9{r7c9 .} => -6r3c5
Singles and one alignment to the solution.

Hi François,
I wonder why you don't find the simpler whip[8] before the S2-braid[8]:

Code: Select all

whip[8]: c7n8{r6 r1} - c9n8{r1 r7} - r7n7{c9 c5} - c5n8{r7 r4} - c5n6{r4 r3} - r1n6{c6 c9} - r1n9{c9 c8} - b9n9{r7c8 .} ==> r5c8 ≠ 8

Anyway, If you have an implementation of S-braids, that's very interesting. I never tried to implement them due to combinatorial explosion whenever Subsets are involved in chains.
Do you have all the types of Subsets (Naked, Hidden, Super-Hidden)? (Here, I see only Naked ones).
Do you have S-whips?

[DEFISE]

I wonder why you don't find the simpler whip[8] before the S2-braid[8]:

Code: Select all

whip[8]: c7n8{r6 r1} - c9n8{r1 r7} - r7n7{c9 c5} - c5n8{r7 r4} - c5n6{r4 r3} - r1n6{c6 c9} - r1n9{c9 c8} - b9n9{r7c8 .} ==> r5c8 ≠ 8

Anyway, If you have an implementation of S-braids, that's very interesting. I never tried to implement them due to combinatorial explosion whenever Subsets are involved in chains.
Do you have all the types of Subsets (Naked, Hidden, Super-Hidden)? (Here, I see only Naked ones).
Do you have S-whips?

Hi Denis,
The 8r5c8 is not an interesting target for reducing the number of steps because its deletion does not lead to any basic technique.
I only implemented s2-whips and s2-braids (naked & hidden pairs). I was lucky, for this puzzle there is no combinatorial explosion.
First of all my program “Few Step” with option s2-braids found a solution with 3 eliminations:
7r7c9, 7r6c7, 6r3c5.
Then, with other tests, I saw that a naked triplet allowed to directly conclude with the 6r3c5 target.
N.B: there are solutions with s2-whips but of greater length.

[denis_berthier]

I wonder why you don't find the simpler whip[8] before the S2-braid[8]:

Code: Select all

whip[8]: c7n8{r6 r1} - c9n8{r1 r7} - r7n7{c9 c5} - c5n8{r7 r4} - c5n6{r4 r3} - r1n6{c6 c9} - r1n9{c9 c8} - b9n9{r7c8 .} ==> r5c8 ≠ 8

Anyway, If you have an implementation of S-braids, that's very interesting. I never tried to implement them due to combinatorial explosion whenever Subsets are involved in chains.
Do you have all the types of Subsets (Naked, Hidden, Super-Hidden)? (Here, I see only Naked ones).
Do you have S-whips?

Hi Denis,
The 8r5c8 is not an interesting target for reducing the number of steps because its deletion does not lead to any basic technique.
I only implemented s2-whips and s2-braids (naked & hidden pairs). I was lucky, for this puzzle there is no combinatorial explosion.
First of all my program “Few Step” with option s2-braids found a solution with 3 eliminations:
7r7c9, 7r6c7, 6r3c5.
Then, with other tests, I saw that a naked triplet allowed to directly conclude with the 6r3c5 target.
N.B: there are solutions with s2-whips but of greater length.

OK, I didn't think you applied step-optimisation.
Excellent solution.