The New Sudoku Players' Forum

[P.O.]

Code: Select all

.  3  .  .  1  .  .  8  .
.  .  4  9  .  .  1  .  .
2  .  .  .  .  7  .  .  5
.  5  .  .  .  .  .  2  .
6  .  .  .  .  .  3  .  .
.  .  7  .  .  .  .  .  4
7  .  .  .  8  .  .  3  .
.  .  6  .  .  2  .  .  8
.  4  .  1  .  .  5  .  .

.3..1..8...49..1..2....7..5.5.....2.6.....3....7.....47...8..3...6..2..8.4.1..5..


basics:

Hidden Text: Show

Code: Select all

( n3r2c9   n4r4c1   n7r2c2   n6r2c8   n2r2c5   n6r3c2   n1r3c3   n8r3c4   n8r2c1
  n5r2c6   n3r3c5   n8r9c3   n2r9c9   n3r4c3   n2r1c7   n7r1c9 )

intersection:
((((6 0) (7 7 9) (4 6 9)) ((6 0) (7 9 9) (1 6 9))))


Code: Select all

59     3      59     46     1      46     2      8      7               
8      7      4      9      2      5      1      6      3               
2      6      1      8      3      7      49     49     5               
4      5      3      67     679    1689   6789   2      169             
6      1289   29     2457   4579   1489   3      1579   19             
19     1289   7      2356   569    13689  689    159    4               
7      129    259    45     8      49     469    3      169             
1359   19     6      3457   4579   2      479    1479   8               
39     4      8      1      679    369    5      79     2     


[jco]

After basics

Code: Select all

,--------------------------------------------------------------------,
| 59     3      59     |a46     1      46     | 2      8      7      |
| 8      7      4      | 9      2      5      | 1      6      3      |
| 2      6      1      | 8      3      7      | 49    d49     5      |
|----------------------+----------------------+----------------------|
| 4      5      3      |a67     679    1689   |b6789   2      169    |
| 6      1289   29     | 2457   4579   1489   | 3      1579   19     |
| 19     1289   7      | 2356   569    13689  | 689    159    4      |
|----------------------+----------------------+----------------------|
| 7      129    259    | 5-4    8     e49     |e469    3     e169    |
| 1359   19     6      | 3457   4579   2      |c479   d1479   8      |
| 39     4      8      | 1      679    369    | 5     d79     2      |
'--------------------------------------------------------------------'


(4=67)r14c4 - r4c7 = r8c7 - (7=941)r389c8 - (1=694) => -4 r7c4; lclste

[Leren]

Code: Select all

*------------------------------------------------------*
| 59   3    59  | 46    1     46    | 2      8     7   |
| 8    7    4   | 9     2     5     | 1      6     3   |
| 2    6    1   | 8     3     7     | 49     49    5   |
|---------------+-------------------+------------------|
| 4    5    3   |a67    679   1689  | 689-7  2    b169 |
| 6    1289 29  | 245-7 459-7 1489  | 3     b1579 b19  |
| 19   1289 7   | 2356  569   13689 | 689   b159   4   |
|---------------+-------------------+------------------|
| 7    129  259 | 45    8     49    | 469    3     169 |
| 1359 19   6   | 3457  4579  2     | 479    1479  8   |
| 39   4    8   | 1     679   369   | 5      79    2   |
*------------------------------------------------------*

ALS XZ Rule: X = 6, Z = 7: (7=6) r4c4 - (6=7) r4c9, r5c89, r6c8 => - 7 r3c7, r4c45; stte

Leren

[Cenoman]

Code: Select all

 +----------------------+------------------------+----------------------+
 |  59     3      59    |  46     1      46      |  2      8      7     |
 |  8      7      4     |  9      2      5       |  1      6      3     |
 |  2      6      1     |  8      3      7       |  49     49     5     |
 +----------------------+------------------------+----------------------+
 |  4      5      3     |  67     679    1689    |  6789   2      69-1  |
 |  6      128-9 a29    |  2457   457-9  148-9   |  3      157-9 a19    |
 |  19     1289   7     |  2356   569    13689   |  689    159    4     |
 +----------------------+------------------------+----------------------+
 |  7     c12-9  b259   |  45     8      49      |  469    3     d169   |
 |  1359   19     6     |  3457   4579   2       |  479    1479   8     |
 |  39     4      8     |  1      679    369     |  5      79     2     |
 +----------------------+------------------------+----------------------+

ALS M-Ring: (1=92)r5c39 - r7c3 = (2-1)r7c2 = (1)r7c9 => -1 r4c9, -9 r5c2568, -9 r7c2; ste
(Note: -1 r4c9 is the effective elimination)

[Ngisa]

Code: Select all

+-------------------+---------------------+-------------------+
| 59     3      59  | 46     1      46    | 2     8       7   |
| 8      7      4   | 9      2      5     | 1      6      3   |
| 2      6      1   | 8      3      7     | 49     49     5   |
+-------------------+---------------------+-------------------+
| 4      5      3   | 7-6   d679    1689  | 6789   2     a169 |
| 6      1289   29  | 2457   4579   1489  | 3     a1579  a19  |
| 19     1289   7   | 2356  d569    13689 | 689   a159    4   |
+-------------------+---------------------+-------------------+
| 7      129    259 | 45     8      49    | 469    3      169 |
| 1359   19     6   | 3457   4579   2     | 479    1479   8   |
| 39     4      8   | 1     c679    369   | 5     b79     2   |
+-------------------+---------------------+-------------------+


(6=1|957)b6p3568 - (7)r9c8 = (7-6)r9c5 = (6)r46c5 => - 6r4c4; ste

Clement

[pjb]

Looking for something different:

Code: Select all

 59      3       59     | 46     1      46     | 2      8      7     
 8       7       4      | 9      2      5      | 1      6      3     
 2       6       1      | 8      3      7      | 49     49     5     
------------------------+----------------------+---------------------
 4       5       3      | 67     679   d1689   |c6789   2     e169   
 6       1289    29     | 2457   4579   1489   | 3     b579-1   19     
 19      1289    7      | 2356   569    13689  | 689   b59-1   4     
------------------------+----------------------+---------------------
 7       129     259    | 45     8      49     | 469    3      69--1   
 1359    19      6      | 3457   4579   2      | 479   a1479   8     
 39      4       8      | 1      679    369    | 5      79     2     


(1)r8c8 = (HP 15 at r56c8) - (7)r5c8 = (7-8)r4c7 = (8-1)r4c6 = r4c9 => -1 r56c8, r7c9; stte

Phil

[P.O.]

Code: Select all

7r4c7 => r4c4 <> 6,7
 r4c7=7 - c7n8{r4 r6} - b6n6{r6c7 r4c9}
 
=> r4c7 <> 7
ste.


[rjamil]

Well, understand that POM technique is powerful, but, considered to be as long and boring. Here is my solution path:

Code: Select all

 +-----------------+-------------------+-----------------+
 | 59    3     59  | 46    1     46    | 2     8     7   |
 | 8     7     4   | 9     2     5     | 1     6     3   |
 | 2     6     1   | 8     3     7     | 49    49    5   |
 +-----------------+-------------------+-----------------+
 | 4     5     3   | 67    679   1689  | 6789  2     169 |
 | 6     1289  29  | 2457  4579  1489  | 3     1579  19  |
 | 19    1289  7   | 2356  569   13689 | 689   159   4   |
 +-----------------+-------------------+-----------------+
 | 7     129   259 | 456   8     469   | 469   3     169 |
 | 1359  19    6   | 3457  4579  2     | 479   1479  8   |
 | 39    4     8   | 1     679   369   | 5     79    2   |
 +-----------------+-------------------+-----------------+

Single-digit POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c2689 r6c1268 r7c29 r8c128 r9c4
Digit 1 not in 5 Templates => -1 @ r6c6
Single-digit POM: 6 @ r1c46 r2c8 r3c2 r4c45679 r5c1 r6c4567 r7c4679 r8c3 r9c56
Digit 6 not in 9 Templates => -6 @ r7c4 r7c6
Single-digit POM: 7 @ r1c9 r2c2 r3c6 r4c457 r5c458 r6c3 r7c1 r8c4578 r9c58
Digit 7 not in 4 Templates => -7 @ r4c5

Code: Select all

 +-----------------+------------------+-----------------+
 | 59    3     59  | 46    1     46   | 2     8     7   |
 | 8     7     4   | 9     2     5    | 1     6     3   |
 | 2     6     1   | 8     3     7    | 49    49    5   |
 +-----------------+------------------+-----------------+
 | 4     5     3   | 67    69    1689 | 6789  2     169 |
 | 6     1289  29  | 2457  4579  1489 | 3     1579  19  |
 | 19    1289  7   | 2356  569   3689 | 689   159   4   |
 +-----------------+------------------+-----------------+
 | 7     129   259 | 45    8     49   | 469   3     169 |
 | 1359  19    6   | 3457  4579  2    | 479   1479  8   |
 | 39    4     8   | 1     679   369  | 5     79    2   |
 +-----------------+------------------+-----------------+

Double-digit POM: 5 @ r1c13 r2c6 r3c9 r4c2 r5c458 r6c458 r7c34 r8c145 r9c7
and POM: 2 @ r1c7 r2c5 r3c1 r4c8 r5c234 r6c24 r7c23 r8c6 r9c9
Digit 5 not in 5 Templates => -5 @ r6c4

Code: Select all

 +-----------------+------------------+-----------------+
 | 59    3     59  | 46    1     46   | 2     8     7   |
 | 8     7     4   | 9     2     5    | 1     6     3   |
 | 2     6     1   | 8     3     7    | 49    49    5   |
 +-----------------+------------------+-----------------+
 | 4     5     3   | 67    69    1689 | 6789  2     169 |
 | 6     1289  29  | 2457  4579  1489 | 3     1579  19  |
 | 19    1289  7   | 236   569   3689 | 689   159   4   |
 +-----------------+------------------+-----------------+
 | 7     129   259 | 45    8     49   | 469   3     169 |
 | 1359  19    6   | 3457  4579  2    | 479   1479  8   |
 | 39    4     8   | 1     679   369  | 5     79    2   |
 +-----------------+------------------+-----------------+

Triple-digit POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c2689 r6c128 r7c29 r8c128 r9c4
and POM: 3 @ r1c2 r2c9 r3c5 r4c3 r5c7 r6c46 r7c8 r8c14 r9c16
and POM: 5 @ r1c13 r2c6 r3c9 r4c2 r5c458 r6c58 r7c34 r8c145 r9c7
Digit 1 not in 4 Templates => -1 @ r5c2 r6c8
Triple-digit POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c689 r6c12 r7c29 r8c128 r9c4
and POM: 3 @ r1c2 r2c9 r3c5 r4c3 r5c7 r6c46 r7c8 r8c14 r9c16
and POM: 8 @ r1c8 r2c1 r3c4 r4c67 r5c26 r6c267 r7c5 r8c9 r9c3
Digit 1 not in 3 Templates => -1 @ r6c2 r8c1
Digit 1 in all 3 Templates => 1 @ r6c1
Triple-digit POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c689 r6c1 r7c29 r8c28 r9c4
and POM: 7 @ r1c9 r2c2 r3c6 r4c47 r5c458 r6c3 r7c1 r8c4578 r9c58
and POM: 8 @ r1c8 r2c1 r3c4 r4c67 r5c26 r6c267 r7c5 r8c9 r9c3
Digit 1 not in 2 Templates => -1 @ r5c8 r7c9 r8c2
Digit 1 in all 2 Templates => 1 @ r7c2 r8c8
Triple-digit POM: 2 @ r1c7 r2c5 r3c1 r4c8 r5c234 r6c24 r7c3 r8c6 r9c9
and POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c69 r6c1 r7c2 r8c8 r9c4
and POM: 3 @ r1c2 r2c9 r3c5 r4c3 r5c7 r6c46 r7c8 r8c14 r9c16
Digit 2 not in 2 Templates => -2 @ r5c3
Digit 2 in all 2 Templates => 2 @ r7c3
Triple-digit POM: 3 @ r1c2 r2c9 r3c5 r4c3 r5c7 r6c46 r7c8 r8c14 r9c16
and POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c69 r6c1 r7c2 r8c8 r9c4
and POM: 5 @ r1c13 r2c6 r3c9 r4c2 r5c458 r6c58 r7c4 r8c145 r9c7
Digit 3 not in 1 Template => -3 @ r6c4 r8c1 r9c6
Digit 3 in all 1 Template => 3 @ r6c6 r8c4 r9c1
Triple-digit POM: 4 @ r1c46 r2c3 r3c78 r4c1 r5c456 r6c9 r7c467 r8c57 r9c2
and POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c69 r6c1 r7c2 r8c8 r9c4
and POM: 2 @ r1c7 r2c5 r3c1 r4c8 r5c24 r6c24 r7c3 r8c6 r9c9
Digit 4 not in 4 Templates => -4 @ r3c7
Digit 4 in all 4 Templates => 4 @ r3c8
Triple-digit POM: 4 @ r1c46 r2c3 r3c8 r4c1 r5c456 r6c9 r7c467 r8c57 r9c2
and POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c69 r6c1 r7c2 r8c8 r9c4
and POM: 5 @ r1c13 r2c6 r3c9 r4c2 r5c458 r6c58 r7c4 r8c15 r9c7
Digit 4 not in 3 Templates => -4 @ r7c4
Triple-digit POM: 4 @ r1c46 r2c3 r3c8 r4c1 r5c456 r6c9 r7c67 r8c57 r9c2
and POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c69 r6c1 r7c2 r8c8 r9c4
and POM: 8 @ r1c8 r2c1 r3c4 r4c67 r5c26 r6c27 r7c5 r8c9 r9c3
Digit 4 not in 2 Templates => -4 @ r5c6
Triple-digit POM: 5 @ r1c13 r2c6 r3c9 r4c2 r5c458 r6c58 r7c4 r8c15 r9c7
and POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c69 r6c1 r7c2 r8c8 r9c4
and POM: 2 @ r1c7 r2c5 r3c1 r4c8 r5c24 r6c24 r7c3 r8c6 r9c9
Digit 5 not in 2 Templates => -5 @ r1c1 r5c4 r8c5
Digit 5 in all 2 Templates => 5 @ r1c3 r8c1
Triple-digit POM: 6 @ r1c46 r2c8 r3c2 r4c45679 r5c1 r6c457 r7c79 r8c3 r9c56
and POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c69 r6c1 r7c2 r8c8 r9c4
and POM: 8 @ r1c8 r2c1 r3c4 r4c67 r5c26 r6c27 r7c5 r8c9 r9c3
Digit 6 not in 6 Templates => -6 @ r4c6
Triple-digit POM: 6 @ r1c46 r2c8 r3c2 r4c4579 r5c1 r6c457 r7c79 r8c3 r9c56
and POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c69 r6c1 r7c2 r8c8 r9c4
and POM: 9 @ r1c1 r2c4 r3c7 r4c5679 r5c235689 r6c2578 r7c679 r8c257 r9c568
Digit 6 not in 4 Templates => -6 @ r6c5
Triple-digit POM: 6 @ r1c46 r2c8 r3c2 r4c4579 r5c1 r6c47 r7c79 r8c3 r9c56
and POM: 2 @ r1c7 r2c5 r3c1 r4c8 r5c24 r6c24 r7c3 r8c6 r9c9
and POM: 7 @ r1c9 r2c2 r3c6 r4c47 r5c458 r6c3 r7c1 r8c57 r9c58
Digit 6 not in 2 Templates => -6 @ r4c7 r4c9 r6c4 r7c7
Digit 6 in all 2 Templates => 6 @ r6c7 r7c9
Triple-digit POM: 6 @ r1c46 r2c8 r3c2 r4c45 r5c1 r6c7 r7c9 r8c3 r9c56
and POM: 4 @ r1c46 r2c3 r3c8 r4c1 r5c45 r6c9 r7c67 r8c57 r9c2
and POM: 9 @ r1c1 r2c4 r3c7 r4c5679 r5c235689 r6c258 r7c67 r8c257 r9c568
Digit 6 not in 1 Template => -6 @ r1c6 r4c4 r9c5
Digit 6 in all 1 Template => 6 @ r1c4 r4c5 r9c6
Triple-digit POM: 7 @ r1c9 r2c2 r3c6 r4c47 r5c458 r6c3 r7c1 r8c57 r9c58
and POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c69 r6c1 r7c2 r8c8 r9c4
and POM: 2 @ r1c7 r2c5 r3c1 r4c8 r5c24 r6c24 r7c3 r8c6 r9c9
Digit 7 not in 2 Templates => -7 @ r5c5
Triple-digit POM: 7 @ r1c9 r2c2 r3c6 r4c47 r5c48 r6c3 r7c1 r8c57 r9c58
and POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c69 r6c1 r7c2 r8c8 r9c4
and POM: 4 @ r1c6 r2c3 r3c8 r4c1 r5c45 r6c9 r7c67 r8c57 r9c2
Digit 7 not in 1 Template => -7 @ r4c7 r5c4 r8c5 r9c8
Digit 7 in all 1 Template => 7 @ r5c8 r8c7 r9c5
Triple-digit POM: 8 @ r1c8 r2c1 r3c4 r4c67 r5c26 r6c2 r7c5 r8c9 r9c3
and POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c69 r6c1 r7c2 r8c8 r9c4
and POM: 2 @ r1c7 r2c5 r3c1 r4c8 r5c24 r6c24 r7c3 r8c6 r9c9
Digit 8 not in 1 Template => -8 @ r4c6 r5c2
Digit 8 in all 1 Template => 8 @ r4c7 r5c6 r6c2
Triple-digit POM: 9 @ r1c1 r2c4 r3c7 r4c69 r5c2359 r6c58 r7c67 r8c25 r9c8
and POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c9 r6c1 r7c2 r8c8 r9c4
and POM: 2 @ r1c7 r2c5 r3c1 r4c8 r5c24 r6c4 r7c3 r8c6 r9c9
Digit 9 not in 1 Template => -9 @ r4c6 r5c2 r5c5 r5c9 r6c8 r7c7 r8c5
Digit 9 in all 1 Template => 9 @ r4c9 r6c5 r7c6; stte

R. Jamil

[P.O.]

Well, understand that POM technique is powerful, but, considered to be as long and boring. Here is my solution path:

a resolution process using usual techniques can also be long and boring, while resolutions using templates can be short and interesting

without any validity checks, this puzzle can be solved with a single combination of size 3 (6 7 8)

Hidden Text: Show

Code: Select all

.3..1..8...49..1..2....7..5.5.....2.6.....3....7.....47...8..3...6..2..8.4.1..5..
#VT: (5 3 2 7 6 9 4 3 82)
Cells: nil nil nil nil nil nil nil nil nil
Candidates: (51) nil nil nil nil (58 60) (32) nil nil

59    3     59    46    1     46    2     8     7             
8     7     4     9     2     5     1     6     3             
2     6     1     8     3     7     49    49    5             
4     5     3     67    69    1689  6789  2     169           
6     1289  29    2457  4579  1489  3     1579  19             
19    1289  7     2356  569   3689  689   159   4             
7     129   259   45    8     49    469   3     169           
1359  19    6     3457  4579  2     479   1479  8             
39    4     8     1     679   369   5     79    2             
121 candidates. 40 values.

#VT: (5 3 2 7 6 9 4 3 82)
Cells: nil nil nil nil nil nil nil nil nil
Candidates: nil nil nil nil nil nil nil nil nil

1: (6 7 8)   13 instances

#VT: (5 3 2 7 6 5 4 3 82)
Cells: nil nil nil nil nil nil nil nil nil
Candidates: nil nil nil nil nil (31 33) nil nil nil
EraseCC: ( n7r4c4   n7r8c7   n7r5c8   n9r9c8   n4r3c8   n1r8c8   n3r9c1   n6r9c6   n4r1c6   n9r3c7
           n5r6c8   n9r7c6   n6r7c9   n9r8c2   n7r9c5   n6r1c4   n4r7c7   n5r8c1   n4r8c5   n9r1c1
           n5r1c3   n1r6c1   n2r7c3   n5r7c4   n3r8c4   n9r5c3   n5r5c5   n1r5c9   n2r6c4   n1r7c2
           n9r4c9   n4r5c4   n8r5c6   n8r6c2   n3r6c6   n6r6c7   n6r4c5   n1r4c6   n8r4c7   n2r5c2
           n9r6c5 )
9 3 5   6 1 4   2 8 7
8 7 4   9 2 5   1 6 3
2 6 1   8 3 7   9 4 5
4 5 3   7 6 1   8 2 9
6 2 9   4 5 8   3 7 1
1 8 7   2 9 3   6 5 4
7 1 2   5 8 9   4 3 6
5 9 6   3 4 2   7 1 8
3 4 8   1 7 6   5 9 2


with the validity check, a single combination of size two is sufficient (1 7)

Hidden Text: Show

Code: Select all

.3..1..8...49..1..2....7..5.5.....2.6.....3....7.....47...8..3...6..2..8.4.1..5..
init
(5 3 2 7 6 9 4 3 82)
(5 3 2 7 5 7 4 3 61)

#VT: (5 3 2 7 5 7 4 3 61)
Cells: nil nil nil nil nil nil nil nil nil
Candidates: (51) nil nil nil (49) (58 60) (32) nil nil

59    3     59    46    1     46    2     8     7             
8     7     4     9     2     5     1     6     3             
2     6     1     8     3     7     49    49    5             
4     5     3     67    69    1689  6789  2     169           
6     1289  29    2457  4579  1489  3     1579  19             
19    1289  7     236   569   3689  689   159   4             
7     129   259   45    8     49    469   3     169           
1359  19    6     3457  4579  2     479   1479  8             
39    4     8     1     679   369   5     79    2             
120 candidates. 40 values.

#VT: (5 3 2 7 5 7 4 3 61)
Cells: nil nil nil nil nil nil nil nil nil
Candidates: nil nil nil nil nil nil nil nil nil

1: (1 7)   3 instances

#VT: (2 2 1 3 2 4 2 2 2)
Cells: (46 56 71) (57) (51 67 73) (26) (3 58 64) nil nil nil (1 25 39 65)
SetVC: ( n9r1c1   n5r1c3   n9r3c7   n4r3c8   n9r5c3   n1r5c9   n1r6c1   n3r6c6   n1r7c2   n2r7c3
         n5r7c4   n5r8c1   n9r8c2   n3r8c4   n1r8c8   n3r9c1   n1r4c6   n8r5c6   n8r4c7   n2r5c2
         n8r6c2   n6r6c7   n4r7c7   n7r8c7   n9r9c8   n9r4c9   n2r6c4   n5r6c8   n9r7c6   n6r7c9
         n4r8c5   n6r9c6   n4r1c6   n6r4c5   n7r5c8   n9r6c5   n7r9c5   n6r1c4   n7r4c4   n4r5c4
         n5r5c5 )
9 3 5   6 1 4   2 8 7
8 7 4   9 2 5   1 6 3
2 6 1   8 3 7   9 4 5
4 5 3   7 6 1   8 2 9
6 2 9   4 5 8   3 7 1
1 8 7   2 9 3   6 5 4
7 1 2   5 8 9   4 3 6
5 9 6   3 4 2   7 1 8
3 4 8   1 7 6   5 9 2


[rjamil]

Hi P.O.,

Well, I see your two OTP solution using POM. Where as, I generate all possible combinations for single, double and triple digit POM moves.

My technique for double-digit POM move is to check all possible/valid templates of first digit against one of to possible/valid template of another digit each.

Similarly, for triple-digit POM move, to check all possible/valid templates of first digit against one of the both other digits' possible/valid template each.

My program neither detected triple-digit (6 7 8), nor double-digit (1 7) combinations as mentioned by your good self.

BTW, I did not understand what do you mean about 'with the validity' and 'without any validity' terms.

R. Jamil

[P.O.]

BTW, I did not understand what do you mean about 'with the validity' and 'without any validity' terms.

yet we have already discussed it, starting here

My program neither detected triple-digit (6 7 8), nor double-digit (1 7) combinations

it is normal that you do not find that the combination (1 7) solves the puzzle because to find it the validity check must have been carried out, on the other hand you should be able to find the combination (6 7 8)
after initializing the puzzle, which corresponds for you to the first use of Single-digit POM, we have the same resolution state, if at this point you retrieve the templates for 6, 7 and 8, make their combinations and analyze the result you will see that 6 is eliminated from r4c4 and r4c6 which has the effect of solving the puzzle

[rjamil]

Hi P.O.,

I am sharing herewith my same solution path with each individual template validation steps:

Hidden Text: Show

Code: Select all

.3..1..8...49..1..2....7..5.5.....2.6.....3....7.....47...8..3...6..2..8.4.1..5..#Puzzle 282
+---+---+---+
|.3.|.1.|.8.|
|..4|9..|1..|
|2..|..7|..5|
+---+---+---+
|.5.|...|.2.|
|6..|...|3..|
|..7|...|..4|
+---+---+---+
|7..|.8.|.3.|
|..6|..2|..8|
|.4.|1..|5..|
+---+---+---+
 +-------------------+----------------------+-------------------+
 | 59     3     59   | 2456   1      456    | 24679  8     2679 |
 | 58     678   4    | 9      2356   3568   | 1      67    2367 |
 | 2      1689  189  | 3468   346    7      | 469    469   5    |
 +-------------------+----------------------+-------------------+
 | 13489  5     1389 | 34678  34679  134689 | 6789   2     1679 |
 | 6      1289  1289 | 24578  24579  14589  | 3      1579  179  |
 | 1389   1289  7    | 23568  23569  135689 | 689    1569  4    |
 +-------------------+----------------------+-------------------+
 | 7      129   1259 | 456    8      4569   | 2469   3     1269 |
 | 1359   19    6    | 3457   34579  2      | 479    1479  8    |
 | 389    4     2389 | 1      3679   369    | 5      679   2679 |
 +-------------------+----------------------+-------------------+
7 @ r2c2, 3 @ r2c9, 2 @ r2c5, 6 @ r3c2, 1 @ r3c3, 8 @ r2c1, 5 @ r2c6, 6 @ r2c8,
8 @ r3c4, 3 @ r3c5, 4 @ r4c1, 8 @ r9c3, 3 @ r4c3, 2 @ r9c9, 2 @ r1c7, 7 @ r1c9;
 +-----------------+-------------------+-----------------+
 | 59    3     59  | 46    1     46    | 2     8     7   |
 | 8     7     4   | 9     2     5     | 1     6     3   |
 | 2     6     1   | 8     3     7     | 49    49    5   |
 +-----------------+-------------------+-----------------+
 | 4     5     3   | 67    679   1689  | 6789  2     169 |
 | 6     1289  29  | 2457  4579  1489  | 3     1579  19  |
 | 19    1289  7   | 2356  569   13689 | 689   159   4   |
 +-----------------+-------------------+-----------------+
 | 7     129   259 | 456   8     469   | 469   3     169 |
 | 1359  19    6   | 3457  4579  2     | 479   1479  8   |
 | 39    4     8   | 1     679   369   | 5     79    2   |
 +-----------------+-------------------+-----------------+
VT (5 3 2 7 6 9 4 3 82)
1-d POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c2689 r6c1268 r7c29 r8c128 r9c4
Digit 1 not in 5 Templates => -1 @ r6c6
1-d POM: 6 @ r1c46 r2c8 r3c2 r4c45679 r5c1 r6c4567 r7c4679 r8c3 r9c56
Digit 6 not in 9 Templates => -6 @ r7c4 r7c6
1-d POM: 7 @ r1c9 r2c2 r3c6 r4c457 r5c458 r6c3 r7c1 r8c4578 r9c58
Digit 7 not in 4 Templates => -7 @ r4c5
 +-----------------+------------------+-----------------+
 | 59    3     59  | 46    1     46   | 2     8     7   |
 | 8     7     4   | 9     2     5    | 1     6     3   |
 | 2     6     1   | 8     3     7    | 49    49    5   |
 +-----------------+------------------+-----------------+
 | 4     5     3   | 67    69    1689 | 6789  2     169 |
 | 6     1289  29  | 2457  4579  1489 | 3     1579  19  |
 | 19    1289  7   | 2356  569   3689 | 689   159   4   |
 +-----------------+------------------+-----------------+
 | 7     129   259 | 45    8     49   | 469   3     169 |
 | 1359  19    6   | 3457  4579  2    | 479   1479  8   |
 | 39    4     8   | 1     679   369  | 5     79    2   |
 +-----------------+------------------+-----------------+
VT (5 3 2 7 6 9 4 3 82)
5 @ r1c1 r2c6 r3c9 r4c2 r5c4 r6c8 r7c3 r8c5 r9c7 2 @ r1c7 r2c5 r3c1 r4c8 r5c3 r6c4 r7c2 r8c6 r9c9
5 @ r1c1 r2c6 r3c9 r4c2 r5c5 r6c8 r7c3 r8c4 r9c7 2 @ r1c7 r2c5 r3c1 r4c8 r5c3 r6c4 r7c2 r8c6 r9c9
5 @ r1c1 r2c6 r3c9 r4c2 r5c8 r6c4 r7c3 r8c5 r9c7
5 @ r1c1 r2c6 r3c9 r4c2 r5c8 r6c5 r7c3 r8c4 r9c7 2 @ r1c7 r2c5 r3c1 r4c8 r5c3 r6c4 r7c2 r8c6 r9c9
5 @ r1c3 r2c6 r3c9 r4c2 r5c5 r6c8 r7c4 r8c1 r9c7 2 @ r1c7 r2c5 r3c1 r4c8 r5c2 r6c4 r7c3 r8c6 r9c9
5 @ r1c3 r2c6 r3c9 r4c2 r5c8 r6c5 r7c4 r8c1 r9c7 2 @ r1c7 r2c5 r3c1 r4c8 r5c2 r6c4 r7c3 r8c6 r9c9
2-d POM: 5 @ r1c13 r2c6 r3c9 r4c2 r5c458 r6c458 r7c34 r8c145 r9c7
and POM: 2 @ r1c7 r2c5 r3c1 r4c8 r5c234 r6c24 r7c23 r8c6 r9c9
Digit 5 not in 5 Templates => -5 @ r6c4
 +-----------------+------------------+-----------------+
 | 59    3     59  | 46    1     46   | 2     8     7   |
 | 8     7     4   | 9     2     5    | 1     6     3   |
 | 2     6     1   | 8     3     7    | 49    49    5   |
 +-----------------+------------------+-----------------+
 | 4     5     3   | 67    69    1689 | 6789  2     169 |
 | 6     1289  29  | 2457  4579  1489 | 3     1579  19  |
 | 19    1289  7   | 236   569   3689 | 689   159   4   |
 +-----------------+------------------+-----------------+
 | 7     129   259 | 45    8     49   | 469   3     169 |
 | 1359  19    6   | 3457  4579  2    | 479   1479  8   |
 | 39    4     8   | 1     679   369  | 5     79    2   |
 +-----------------+------------------+-----------------+
VT (5 3 2 7 5 9 4 3 82)
1 @ r1c5 r2c7 r3c3 r4c6 r5c2 r6c8 r7c9 r8c1 r9c4
1 @ r1c5 r2c7 r3c3 r4c6 r5c8 r6c1 r7c9 r8c2 r9c4 3 @ r1c2 r2c9 r3c5 r4c3 r5c7 r6c4 r7c8 r8c1 r9c6 5 @ r1c1 r2c6 r3c9 r4c2 r5c4 r6c8 r7c3 r8c5 r9c7
1 @ r1c5 r2c7 r3c3 r4c6 r5c8 r6c2 r7c9 r8c1 r9c4 3 @ r1c2 r2c9 r3c5 r4c3 r5c7 r6c6 r7c8 r8c4 r9c1 5 @ r1c1 r2c6 r3c9 r4c2 r5c4 r6c8 r7c3 r8c5 r9c7
1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 3 @ r1c2 r2c9 r3c5 r4c3 r5c7 r6c4 r7c8 r8c1 r9c6 5 @ r1c1 r2c6 r3c9 r4c2 r5c4 r6c8 r7c3 r8c5 r9c7
1 @ r1c5 r2c7 r3c3 r4c9 r5c6 r6c1 r7c2 r8c8 r9c4 3 @ r1c2 r2c9 r3c5 r4c3 r5c7 r6c4 r7c8 r8c1 r9c6 5 @ r1c1 r2c6 r3c9 r4c2 r5c4 r6c8 r7c3 r8c5 r9c7
3-d POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c2689 r6c128 r7c29 r8c128 r9c4
and POM: 3 @ r1c2 r2c9 r3c5 r4c3 r5c7 r6c46 r7c8 r8c14 r9c16
and POM: 5 @ r1c13 r2c6 r3c9 r4c2 r5c458 r6c58 r7c34 r8c145 r9c7
Digit 1 not in 4 Templates => -1 @ r5c2 r6c8

1 @ r1c5 r2c7 r3c3 r4c6 r5c8 r6c1 r7c9 r8c2 r9c4 3 @ r1c2 r2c9 r3c5 r4c3 r5c7 r6c4 r7c8 r8c1 r9c6 8 @ r1c8 r2c1 r3c4 r4c7 r5c2 r6c6 r7c5 r8c9 r9c3
1 @ r1c5 r2c7 r3c3 r4c6 r5c8 r6c2 r7c9 r8c1 r9c4
1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 3 @ r1c2 r2c9 r3c5 r4c3 r5c7 r6c4 r7c8 r8c1 r9c6 8 @ r1c8 r2c1 r3c4 r4c7 r5c2 r6c6 r7c5 r8c9 r9c3
1 @ r1c5 r2c7 r3c3 r4c9 r5c6 r6c1 r7c2 r8c8 r9c4 3 @ r1c2 r2c9 r3c5 r4c3 r5c7 r6c4 r7c8 r8c1 r9c6 8 @ r1c8 r2c1 r3c4 r4c6 r5c2 r6c7 r7c5 r8c9 r9c3
3-d POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c689 r6c12 r7c29 r8c128 r9c4
and POM: 3 @ r1c2 r2c9 r3c5 r4c3 r5c7 r6c46 r7c8 r8c14 r9c16
and POM: 8 @ r1c8 r2c1 r3c4 r4c67 r5c26 r6c267 r7c5 r8c9 r9c3
Digit 1 not in 3 Templates => -1 @ r6c2 r8c1
Digit 1 in all 3 Templates => 1 @ r6c1

1 @ r1c5 r2c7 r3c3 r4c6 r5c8 r6c1 r7c9 r8c2 r9c4
1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 7 @ r1c9 r2c2 r3c6 r4c4 r5c8 r6c3 r7c1 r8c7 r9c5 8 @ r1c8 r2c1 r3c4 r4c7 r5c2 r6c6 r7c5 r8c9 r9c3
1 @ r1c5 r2c7 r3c3 r4c9 r5c6 r6c1 r7c2 r8c8 r9c4 7 @ r1c9 r2c2 r3c6 r4c4 r5c8 r6c3 r7c1 r8c7 r9c5 8 @ r1c8 r2c1 r3c4 r4c6 r5c2 r6c7 r7c5 r8c9 r9c3
3-d POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c689 r6c1 r7c29 r8c28 r9c4
and POM: 7 @ r1c9 r2c2 r3c6 r4c47 r5c458 r6c3 r7c1 r8c4578 r9c58
and POM: 8 @ r1c8 r2c1 r3c4 r4c67 r5c26 r6c267 r7c5 r8c9 r9c3
Digit 1 not in 2 Templates => -1 @ r5c8 r7c9 r8c2
Digit 1 in all 2 Templates => 1 @ r7c2 r8c8

2 @ r1c7 r2c5 r3c1 r4c8 r5c2 r6c4 r7c3 r8c6 r9c9 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 3 @ r1c2 r2c9 r3c5 r4c3 r5c7 r6c6 r7c8 r8c4 r9c1
2 @ r1c7 r2c5 r3c1 r4c8 r5c3 r6c4 r7c2 r8c6 r9c9
2 @ r1c7 r2c5 r3c1 r4c8 r5c4 r6c2 r7c3 r8c6 r9c9 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 3 @ r1c2 r2c9 r3c5 r4c3 r5c7 r6c4 r7c8 r8c1 r9c6
3-d POM: 2 @ r1c7 r2c5 r3c1 r4c8 r5c234 r6c24 r7c3 r8c6 r9c9
and POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c69 r6c1 r7c2 r8c8 r9c4
and POM: 3 @ r1c2 r2c9 r3c5 r4c3 r5c7 r6c46 r7c8 r8c14 r9c16
Digit 2 not in 2 Templates => -2 @ r5c3
Digit 2 in all 2 Templates => 2 @ r7c3

3 @ r1c2 r2c9 r3c5 r4c3 r5c7 r6c4 r7c8 r8c1 r9c6
3 @ r1c2 r2c9 r3c5 r4c3 r5c7 r6c6 r7c8 r8c4 r9c1 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 5 @ r1c3 r2c6 r3c9 r4c2 r5c5 r6c8 r7c4 r8c1 r9c7
3-d POM: 3 @ r1c2 r2c9 r3c5 r4c3 r5c7 r6c46 r7c8 r8c14 r9c16
and POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c69 r6c1 r7c2 r8c8 r9c4
and POM: 5 @ r1c13 r2c6 r3c9 r4c2 r5c458 r6c58 r7c4 r8c145 r9c7
Digit 3 not in 1 Template => -3 @ r6c4 r8c1 r9c6
Digit 3 in all 1 Template => 3 @ r6c6 r8c4 r9c1

4 @ r1c4 r2c3 r3c7 r4c1 r5c5 r6c9 r7c6 r8c8 r9c2
4 @ r1c4 r2c3 r3c8 r4c1 r5c5 r6c9 r7c6 r8c7 r9c2 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 2 @ r1c7 r2c5 r3c1 r4c8 r5c2 r6c4 r7c3 r8c6 r9c9
4 @ r1c4 r2c3 r3c8 r4c1 r5c6 r6c9 r7c7 r8c5 r9c2 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 2 @ r1c7 r2c5 r3c1 r4c8 r5c2 r6c4 r7c3 r8c6 r9c9
4 @ r1c6 r2c3 r3c7 r4c1 r5c5 r6c9 r7c4 r8c8 r9c2
4 @ r1c6 r2c3 r3c8 r4c1 r5c4 r6c9 r7c7 r8c5 r9c2 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 2 @ r1c7 r2c5 r3c1 r4c8 r5c2 r6c4 r7c3 r8c6 r9c9
4 @ r1c6 r2c3 r3c8 r4c1 r5c5 r6c9 r7c4 r8c7 r9c2 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 2 @ r1c7 r2c5 r3c1 r4c8 r5c2 r6c4 r7c3 r8c6 r9c9
4 @ r1c6 r2c3 r3c8 r4c1 r5c5 r6c9 r7c7 r8c4 r9c2
16) Triple-digit POM: 4 @ r1c46 r2c3 r3c78 r4c1 r5c456 r6c9 r7c467 r8c57 r9c2
and POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c69 r6c1 r7c2 r8c8 r9c4
and POM: 2 @ r1c7 r2c5 r3c1 r4c8 r5c24 r6c24 r7c3 r8c6 r9c9
Digit 4 not in 4 Templates => -4 @ r3c7
Digit 4 in all 4 Templates => 4 @ r3c8

4 @ r1c4 r2c3 r3c8 r4c1 r5c5 r6c9 r7c6 r8c7 r9c2 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 5 @ r1c3 r2c6 r3c9 r4c2 r5c8 r6c5 r7c4 r8c1 r9c7
4 @ r1c4 r2c3 r3c8 r4c1 r5c6 r6c9 r7c7 r8c5 r9c2 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 5 @ r1c3 r2c6 r3c9 r4c2 r5c5 r6c8 r7c4 r8c1 r9c7
4 @ r1c6 r2c3 r3c8 r4c1 r5c4 r6c9 r7c7 r8c5 r9c2 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 5 @ r1c3 r2c6 r3c9 r4c2 r5c5 r6c8 r7c4 r8c1 r9c7
4 @ r1c6 r2c3 r3c8 r4c1 r5c5 r6c9 r7c4 r8c7 r9c2
3-d POM: 4 @ r1c46 r2c3 r3c8 r4c1 r5c456 r6c9 r7c467 r8c57 r9c2
and POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c69 r6c1 r7c2 r8c8 r9c4
and POM: 5 @ r1c13 r2c6 r3c9 r4c2 r5c458 r6c58 r7c4 r8c15 r9c7
Digit 4 not in 3 Templates => -4 @ r7c4

4 @ r1c4 r2c3 r3c8 r4c1 r5c5 r6c9 r7c6 r8c7 r9c2 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 8 @ r1c8 r2c1 r3c4 r4c7 r5c6 r6c2 r7c5 r8c9 r9c3
4 @ r1c4 r2c3 r3c8 r4c1 r5c6 r6c9 r7c7 r8c5 r9c2
4 @ r1c6 r2c3 r3c8 r4c1 r5c4 r6c9 r7c7 r8c5 r9c2 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 8 @ r1c8 r2c1 r3c4 r4c7 r5c6 r6c2 r7c5 r8c9 r9c3
3-d POM: 4 @ r1c46 r2c3 r3c8 r4c1 r5c456 r6c9 r7c67 r8c57 r9c2
and POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c69 r6c1 r7c2 r8c8 r9c4
and POM: 8 @ r1c8 r2c1 r3c4 r4c67 r5c26 r6c27 r7c5 r8c9 r9c3
Digit 4 not in 2 Templates => -4 @ r5c6

5 @ r1c1 r2c6 r3c9 r4c2 r5c4 r6c8 r7c3 r8c5 r9c7
5 @ r1c1 r2c6 r3c9 r4c2 r5c5 r6c8 r7c3 r8c4 r9c7
5 @ r1c1 r2c6 r3c9 r4c2 r5c8 r6c5 r7c3 r8c4 r9c7
5 @ r1c3 r2c6 r3c9 r4c2 r5c5 r6c8 r7c4 r8c1 r9c7 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 2 @ r1c7 r2c5 r3c1 r4c8 r5c2 r6c4 r7c3 r8c6 r9c9
5 @ r1c3 r2c6 r3c9 r4c2 r5c8 r6c5 r7c4 r8c1 r9c7 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 2 @ r1c7 r2c5 r3c1 r4c8 r5c2 r6c4 r7c3 r8c6 r9c9
3-d POM: 5 @ r1c13 r2c6 r3c9 r4c2 r5c458 r6c58 r7c4 r8c15 r9c7
and POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c69 r6c1 r7c2 r8c8 r9c4
and POM: 2 @ r1c7 r2c5 r3c1 r4c8 r5c24 r6c24 r7c3 r8c6 r9c9
Digit 5 not in 2 Templates => -5 @ r1c1 r5c4 r8c5
Digit 5 in all 2 Templates => 5 @ r1c3 r8c1

6 @ r1c4 r2c8 r3c2 r4c5 r5c1 r6c7 r7c9 r8c3 r9c6 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 8 @ r1c8 r2c1 r3c4 r4c7 r5c6 r6c2 r7c5 r8c9 r9c3
6 @ r1c4 r2c8 r3c2 r4c6 r5c1 r6c7 r7c9 r8c3 r9c5
6 @ r1c4 r2c8 r3c2 r4c7 r5c1 r6c5 r7c9 r8c3 r9c6 1 @ r1c5 r2c7 r3c3 r4c9 r5c6 r6c1 r7c2 r8c8 r9c4 8 @ r1c8 r2c1 r3c4 r4c6 r5c2 r6c7 r7c5 r8c9 r9c3
6 @ r1c4 r2c8 r3c2 r4c7 r5c1 r6c6 r7c9 r8c3 r9c5
6 @ r1c4 r2c8 r3c2 r4c9 r5c1 r6c5 r7c7 r8c3 r9c6 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 8 @ r1c8 r2c1 r3c4 r4c7 r5c6 r6c2 r7c5 r8c9 r9c3
6 @ r1c4 r2c8 r3c2 r4c9 r5c1 r6c6 r7c7 r8c3 r9c5
6 @ r1c6 r2c8 r3c2 r4c4 r5c1 r6c7 r7c9 r8c3 r9c5 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 8 @ r1c8 r2c1 r3c4 r4c7 r5c6 r6c2 r7c5 r8c9 r9c3
6 @ r1c6 r2c8 r3c2 r4c7 r5c1 r6c4 r7c9 r8c3 r9c5 1 @ r1c5 r2c7 r3c3 r4c9 r5c6 r6c1 r7c2 r8c8 r9c4 8 @ r1c8 r2c1 r3c4 r4c6 r5c2 r6c7 r7c5 r8c9 r9c3
6 @ r1c6 r2c8 r3c2 r4c9 r5c1 r6c4 r7c7 r8c3 r9c5 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 8 @ r1c8 r2c1 r3c4 r4c7 r5c6 r6c2 r7c5 r8c9 r9c3
3-d POM: 6 @ r1c46 r2c8 r3c2 r4c45679 r5c1 r6c457 r7c79 r8c3 r9c56
and POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c69 r6c1 r7c2 r8c8 r9c4
and POM: 8 @ r1c8 r2c1 r3c4 r4c67 r5c26 r6c27 r7c5 r8c9 r9c3
Digit 6 not in 6 Templates => -6 @ r4c6

6 @ r1c4 r2c8 r3c2 r4c5 r5c1 r6c7 r7c9 r8c3 r9c6 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 9 @ r1c1 r2c4 r3c7 r4c9 r5c3 r6c5 r7c6 r8c2 r9c8
6 @ r1c4 r2c8 r3c2 r4c7 r5c1 r6c5 r7c9 r8c3 r9c6
6 @ r1c4 r2c8 r3c2 r4c9 r5c1 r6c5 r7c7 r8c3 r9c6
6 @ r1c6 r2c8 r3c2 r4c4 r5c1 r6c7 r7c9 r8c3 r9c5 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 9 @ r1c1 r2c4 r3c7 r4c9 r5c3 r6c5 r7c6 r8c2 r9c8
6 @ r1c6 r2c8 r3c2 r4c7 r5c1 r6c4 r7c9 r8c3 r9c5 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 9 @ r1c1 r2c4 r3c7 r4c9 r5c3 r6c5 r7c6 r8c2 r9c8
6 @ r1c6 r2c8 r3c2 r4c9 r5c1 r6c4 r7c7 r8c3 r9c5 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 9 @ r1c1 r2c4 r3c7 r4c5 r5c3 r6c8 r7c9 r8c2 r9c6
3-d POM: 6 @ r1c46 r2c8 r3c2 r4c4579 r5c1 r6c457 r7c79 r8c3 r9c56
and POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c69 r6c1 r7c2 r8c8 r9c4
and POM: 9 @ r1c1 r2c4 r3c7 r4c5679 r5c235689 r6c2578 r7c679 r8c257 r9c568
Digit 6 not in 4 Templates => -6 @ r6c5

6 @ r1c4 r2c8 r3c2 r4c5 r5c1 r6c7 r7c9 r8c3 r9c6 2 @ r1c7 r2c5 r3c1 r4c8 r5c2 r6c4 r7c3 r8c6 r9c9 7 @ r1c9 r2c2 r3c6 r4c4 r5c8 r6c3 r7c1 r8c7 r9c5
6 @ r1c6 r2c8 r3c2 r4c4 r5c1 r6c7 r7c9 r8c3 r9c5 2 @ r1c7 r2c5 r3c1 r4c8 r5c2 r6c4 r7c3 r8c6 r9c9 7 @ r1c9 r2c2 r3c6 r4c7 r5c4 r6c3 r7c1 r8c5 r9c8
6 @ r1c6 r2c8 r3c2 r4c7 r5c1 r6c4 r7c9 r8c3 r9c5
6 @ r1c6 r2c8 r3c2 r4c9 r5c1 r6c4 r7c7 r8c3 r9c5
3-d POM: 6 @ r1c46 r2c8 r3c2 r4c4579 r5c1 r6c47 r7c79 r8c3 r9c56
and POM: 2 @ r1c7 r2c5 r3c1 r4c8 r5c24 r6c24 r7c3 r8c6 r9c9
and POM: 7 @ r1c9 r2c2 r3c6 r4c47 r5c458 r6c3 r7c1 r8c57 r9c58
Digit 6 not in 2 Templates => -6 @ r4c7 r4c9 r6c4 r7c7
Digit 6 in all 2 Templates => 6 @ r6c7 r7c9

6 @ r1c4 r2c8 r3c2 r4c5 r5c1 r6c7 r7c9 r8c3 r9c6 4 @ r1c6 r2c3 r3c8 r4c1 r5c4 r6c9 r7c7 r8c5 r9c2 9 @ r1c1 r2c4 r3c7 r4c9 r5c3 r6c5 r7c6 r8c2 r9c8
6 @ r1c6 r2c8 r3c2 r4c4 r5c1 r6c7 r7c9 r8c3 r9c5
3-d POM: 6 @ r1c46 r2c8 r3c2 r4c45 r5c1 r6c7 r7c9 r8c3 r9c56
and POM: 4 @ r1c46 r2c3 r3c8 r4c1 r5c45 r6c9 r7c67 r8c57 r9c2
and POM: 9 @ r1c1 r2c4 r3c7 r4c5679 r5c235689 r6c258 r7c67 r8c257 r9c568
Digit 6 not in 1 Template => -6 @ r1c6 r4c4 r9c5
Digit 6 in all 1 Template => 6 @ r1c4 r4c5 r9c6

7 @ r1c9 r2c2 r3c6 r4c4 r5c8 r6c3 r7c1 r8c7 r9c5 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 2 @ r1c7 r2c5 r3c1 r4c8 r5c2 r6c4 r7c3 r8c6 r9c9
7 @ r1c9 r2c2 r3c6 r4c7 r5c4 r6c3 r7c1 r8c5 r9c8 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 2 @ r1c7 r2c5 r3c1 r4c8 r5c2 r6c4 r7c3 r8c6 r9c9
7 @ r1c9 r2c2 r3c6 r4c7 r5c4 r6c3 r7c1 r8c8 r9c5
7 @ r1c9 r2c2 r3c6 r4c7 r5c5 r6c3 r7c1 r8c4 r9c8
3-d POM: 7 @ r1c9 r2c2 r3c6 r4c47 r5c458 r6c3 r7c1 r8c57 r9c58
and POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c69 r6c1 r7c2 r8c8 r9c4
and POM: 2 @ r1c7 r2c5 r3c1 r4c8 r5c24 r6c24 r7c3 r8c6 r9c9
Digit 7 not in 2 Templates => -7 @ r5c5

7 @ r1c9 r2c2 r3c6 r4c4 r5c8 r6c3 r7c1 r8c7 r9c5 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 4 @ r1c6 r2c3 r3c8 r4c1 r5c4 r6c9 r7c7 r8c5 r9c2
7 @ r1c9 r2c2 r3c6 r4c7 r5c4 r6c3 r7c1 r8c5 r9c8
3-d POM: 7 @ r1c9 r2c2 r3c6 r4c47 r5c48 r6c3 r7c1 r8c57 r9c58
and POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c69 r6c1 r7c2 r8c8 r9c4
and POM: 4 @ r1c6 r2c3 r3c8 r4c1 r5c45 r6c9 r7c67 r8c57 r9c2
Digit 7 not in 1 Template => -7 @ r4c7 r5c4 r8c5 r9c8
Digit 7 in all 1 Template => 7 @ r5c8 r8c7 r9c5

8 @ r1c8 r2c1 r3c4 r4c6 r5c2 r6c7 r7c5 r8c9 r9c3
8 @ r1c8 r2c1 r3c4 r4c7 r5c2 r6c6 r7c5 r8c9 r9c3
8 @ r1c8 r2c1 r3c4 r4c7 r5c6 r6c2 r7c5 r8c9 r9c3 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 2 @ r1c7 r2c5 r3c1 r4c8 r5c2 r6c4 r7c3 r8c6 r9c9
3-d POM: 8 @ r1c8 r2c1 r3c4 r4c67 r5c26 r6c2 r7c5 r8c9 r9c3
and POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c69 r6c1 r7c2 r8c8 r9c4
and POM: 2 @ r1c7 r2c5 r3c1 r4c8 r5c24 r6c24 r7c3 r8c6 r9c9
Digit 8 not in 1 Template => -8 @ r4c6 r5c2
Digit 8 in all 1 Template => 8 @ r4c7 r5c6 r6c2

9 @ r1c1 r2c4 r3c7 r4c5 r5c3 r6c8 r7c9 r8c2 r9c6
9 @ r1c1 r2c4 r3c7 r4c5 r5c9 r6c2 r7c3 r8c8 r9c6
9 @ r1c1 r2c4 r3c7 r4c6 r5c3 r6c8 r7c9 r8c2 r9c5
9 @ r1c1 r2c4 r3c7 r4c6 r5c9 r6c2 r7c3 r8c5 r9c8
9 @ r1c1 r2c4 r3c7 r4c6 r5c9 r6c2 r7c3 r8c8 r9c5
9 @ r1c1 r2c4 r3c7 r4c9 r5c2 r6c5 r7c3 r8c8 r9c6
9 @ r1c1 r2c4 r3c7 r4c9 r5c2 r6c6 r7c3 r8c5 r9c8
9 @ r1c1 r2c4 r3c7 r4c9 r5c2 r6c6 r7c3 r8c8 r9c5
9 @ r1c1 r2c4 r3c7 r4c9 r5c3 r6c5 r7c2 r8c8 r9c6
9 @ r1c1 r2c4 r3c7 r4c9 r5c3 r6c5 r7c6 r8c2 r9c8
9 @ r1c1 r2c4 r3c7 r4c9 r5c3 r6c6 r7c2 r8c5 r9c8
9 @ r1c1 r2c4 r3c7 r4c9 r5c3 r6c6 r7c2 r8c8 r9c5
9 @ r1c1 r2c4 r3c7 r4c9 r5c5 r6c2 r7c3 r8c8 r9c6
9 @ r1c1 r2c4 r3c7 r4c9 r5c6 r6c2 r7c3 r8c5 r9c8
9 @ r1c1 r2c4 r3c7 r4c9 r5c6 r6c2 r7c3 r8c8 r9c5
9 @ r1c1 r2c4 r3c8 r4c5 r5c3 r6c7 r7c9 r8c2 r9c6
9 @ r1c1 r2c4 r3c8 r4c5 r5c9 r6c2 r7c3 r8c7 r9c6
9 @ r1c1 r2c4 r3c8 r4c6 r5c3 r6c7 r7c9 r8c2 r9c5
9 @ r1c1 r2c4 r3c8 r4c6 r5c9 r6c2 r7c3 r8c7 r9c5
9 @ r1c1 r2c4 r3c8 r4c7 r5c3 r6c5 r7c9 r8c2 r9c6
9 @ r1c1 r2c4 r3c8 r4c7 r5c3 r6c6 r7c9 r8c2 r9c5
9 @ r1c1 r2c4 r3c8 r4c9 r5c2 r6c5 r7c3 r8c7 r9c6
9 @ r1c1 r2c4 r3c8 r4c9 r5c2 r6c6 r7c3 r8c7 r9c5
9 @ r1c1 r2c4 r3c8 r4c9 r5c3 r6c5 r7c2 r8c7 r9c6
9 @ r1c1 r2c4 r3c8 r4c9 r5c3 r6c5 r7c7 r8c2 r9c6
9 @ r1c1 r2c4 r3c8 r4c9 r5c3 r6c6 r7c2 r8c7 r9c5
9 @ r1c1 r2c4 r3c8 r4c9 r5c3 r6c6 r7c7 r8c2 r9c5
9 @ r1c1 r2c4 r3c8 r4c9 r5c5 r6c2 r7c3 r8c7 r9c6
9 @ r1c1 r2c4 r3c8 r4c9 r5c6 r6c2 r7c3 r8c7 r9c5
9 @ r1c3 r2c4 r3c7 r4c5 r5c2 r6c8 r7c9 r8c1 r9c6
9 @ r1c3 r2c4 r3c7 r4c5 r5c8 r6c1 r7c9 r8c2 r9c6
9 @ r1c3 r2c4 r3c7 r4c5 r5c8 r6c2 r7c9 r8c1 r9c6
9 @ r1c3 r2c4 r3c7 r4c5 r5c9 r6c1 r7c2 r8c8 r9c6
9 @ r1c3 r2c4 r3c7 r4c5 r5c9 r6c1 r7c6 r8c2 r9c8
9 @ r1c3 r2c4 r3c7 r4c5 r5c9 r6c2 r7c6 r8c1 r9c8
9 @ r1c3 r2c4 r3c7 r4c5 r5c9 r6c2 r7c6 r8c8 r9c1
9 @ r1c3 r2c4 r3c7 r4c6 r5c2 r6c8 r7c9 r8c1 r9c5
9 @ r1c3 r2c4 r3c7 r4c6 r5c2 r6c8 r7c9 r8c5 r9c1
9 @ r1c3 r2c4 r3c7 r4c6 r5c8 r6c1 r7c9 r8c2 r9c5
9 @ r1c3 r2c4 r3c7 r4c6 r5c8 r6c2 r7c9 r8c1 r9c5
9 @ r1c3 r2c4 r3c7 r4c6 r5c8 r6c2 r7c9 r8c5 r9c1
9 @ r1c3 r2c4 r3c7 r4c6 r5c9 r6c1 r7c2 r8c5 r9c8
9 @ r1c3 r2c4 r3c7 r4c6 r5c9 r6c1 r7c2 r8c8 r9c5
9 @ r1c1 r2c4 r3c7 r4c9 r5c3 r6c5 r7c6 r8c2 r9c8 1 @ r1c5 r2c7 r3c3 r4c6 r5c9 r6c1 r7c2 r8c8 r9c4 2 @ r1c7 r2c5 r3c1 r4c8 r5c2 r6c4 r7c3 r8c6 r9c9
9 @ r1c3 r2c4 r3c7 r4c9 r5c2 r6c5 r7c6 r8c1 r9c8
9 @ r1c3 r2c4 r3c7 r4c9 r5c2 r6c5 r7c6 r8c8 r9c1
9 @ r1c3 r2c4 r3c7 r4c9 r5c5 r6c1 r7c2 r8c8 r9c6
9 @ r1c3 r2c4 r3c7 r4c9 r5c5 r6c1 r7c6 r8c2 r9c8
9 @ r1c3 r2c4 r3c7 r4c9 r5c5 r6c2 r7c6 r8c1 r9c8
9 @ r1c3 r2c4 r3c7 r4c9 r5c5 r6c2 r7c6 r8c8 r9c1
9 @ r1c3 r2c4 r3c7 r4c9 r5c6 r6c1 r7c2 r8c5 r9c8
9 @ r1c3 r2c4 r3c7 r4c9 r5c6 r6c1 r7c2 r8c8 r9c5
9 @ r1c3 r2c4 r3c8 r4c5 r5c2 r6c7 r7c9 r8c1 r9c6
9 @ r1c3 r2c4 r3c8 r4c5 r5c9 r6c1 r7c2 r8c7 r9c6
9 @ r1c3 r2c4 r3c8 r4c5 r5c9 r6c1 r7c7 r8c2 r9c6
9 @ r1c3 r2c4 r3c8 r4c5 r5c9 r6c2 r7c6 r8c7 r9c1
9 @ r1c3 r2c4 r3c8 r4c5 r5c9 r6c2 r7c7 r8c1 r9c6
9 @ r1c3 r2c4 r3c8 r4c6 r5c2 r6c7 r7c9 r8c1 r9c5
9 @ r1c3 r2c4 r3c8 r4c6 r5c2 r6c7 r7c9 r8c5 r9c1
9 @ r1c3 r2c4 r3c8 r4c6 r5c9 r6c1 r7c2 r8c7 r9c5
9 @ r1c3 r2c4 r3c8 r4c6 r5c9 r6c1 r7c7 r8c2 r9c5
9 @ r1c3 r2c4 r3c8 r4c6 r5c9 r6c2 r7c7 r8c1 r9c5
9 @ r1c3 r2c4 r3c8 r4c6 r5c9 r6c2 r7c7 r8c5 r9c1
9 @ r1c3 r2c4 r3c8 r4c7 r5c2 r6c5 r7c9 r8c1 r9c6
9 @ r1c3 r2c4 r3c8 r4c7 r5c2 r6c6 r7c9 r8c1 r9c5
9 @ r1c3 r2c4 r3c8 r4c7 r5c2 r6c6 r7c9 r8c5 r9c1
9 @ r1c3 r2c4 r3c8 r4c7 r5c5 r6c1 r7c9 r8c2 r9c6
9 @ r1c3 r2c4 r3c8 r4c7 r5c5 r6c2 r7c9 r8c1 r9c6
9 @ r1c3 r2c4 r3c8 r4c7 r5c6 r6c1 r7c9 r8c2 r9c5
9 @ r1c3 r2c4 r3c8 r4c7 r5c6 r6c2 r7c9 r8c1 r9c5
9 @ r1c3 r2c4 r3c8 r4c7 r5c6 r6c2 r7c9 r8c5 r9c1
9 @ r1c3 r2c4 r3c8 r4c9 r5c2 r6c5 r7c6 r8c7 r9c1
9 @ r1c3 r2c4 r3c8 r4c9 r5c2 r6c5 r7c7 r8c1 r9c6
9 @ r1c3 r2c4 r3c8 r4c9 r5c2 r6c6 r7c7 r8c1 r9c5
9 @ r1c3 r2c4 r3c8 r4c9 r5c2 r6c6 r7c7 r8c5 r9c1
9 @ r1c3 r2c4 r3c8 r4c9 r5c5 r6c1 r7c2 r8c7 r9c6
9 @ r1c3 r2c4 r3c8 r4c9 r5c5 r6c1 r7c7 r8c2 r9c6
9 @ r1c3 r2c4 r3c8 r4c9 r5c5 r6c2 r7c6 r8c7 r9c1
9 @ r1c3 r2c4 r3c8 r4c9 r5c5 r6c2 r7c7 r8c1 r9c6
9 @ r1c3 r2c4 r3c8 r4c9 r5c6 r6c1 r7c2 r8c7 r9c5
9 @ r1c3 r2c4 r3c8 r4c9 r5c6 r6c1 r7c7 r8c2 r9c5
9 @ r1c3 r2c4 r3c8 r4c9 r5c6 r6c2 r7c7 r8c1 r9c5
9 @ r1c3 r2c4 r3c8 r4c9 r5c6 r6c2 r7c7 r8c5 r9c1
3-d POM: 9 @ r1c1 r2c4 r3c7 r4c69 r5c2359 r6c58 r7c67 r8c25 r9c8
and POM: 1 @ r1c5 r2c7 r3c3 r4c69 r5c9 r6c1 r7c2 r8c8 r9c4
and POM: 2 @ r1c7 r2c5 r3c1 r4c8 r5c24 r6c4 r7c3 r8c6 r9c9
Digit 9 not in 1 Template => -9 @ r4c6 r5c2 r5c5 r5c9 r6c8 r7c7 r8c5
Digit 9 in all 1 Template => 9 @ r4c9 r6c5 r7c6
 +---------+-----------+---------+
 | 9  3  5 | 6   1   4 | 2  8  7 |
 | 8  7  4 | 9   2   5 | 1  6  3 |
 | 2  6  1 | 8   3   7 | 9  4  5 |
 +---------+-----------+---------+
 | 4  5  3 | 7   6   1 | 8  2  9 |
 | 6  2  9 | 24  45  8 | 3  7  1 |
 | 1  8  7 | 2   9   3 | 6  5  4 |
 +---------+-----------+---------+
 | 7  1  2 | 5   8   9 | 4  3  6 |
 | 5  9  6 | 3   4   2 | 7  1  8 |
 | 3  4  8 | 1   7   6 | 5  9  2 |
 +---------+-----------+---------+
stte

R. Jamil