The New Sudoku Players' Forum

by **rjamil** » Fri Nov 15, 2024 9:31 am

Hi P.O.,

P.O. wrote:i find the same eliminations which correspond to those mentioned above in the quote from Myth Jellies but the real power of a template procedure as a solving technique begins with the combinations of templates which consists of testing the compatibility of templates for different values and gradually eliminating all those which do not fit into any of them.
Do you intend to develop your procedure in this way?

Well, yes, of course, for sure, I do, most welcome, why not.

But the problem is that, there are no detail information available for the multi-digit POM. Myth Jellies declare POM as single digit elimination/placement technic. I was trying to start coding only for those technics that have sufficient information to understand (keeping language barrier) as well as to program the same (most preferably pattern based).

I see how powerful POM technic is but lack of interest and information due huge data set as one of the main reason. I finally able to shrink the entire data set, by importing the 46656 x 9 templates in XLSX file in rows 1 to 46656 x column A to I respectively, taken from StrmCkr's post. Then, sort whole data by A to I column in ascending order. Copy each 5184 chunk in to separate sheet and trying to shrink as much as possible, without loosing any data. My final compilation is available here. See, how all 46656 templates are carefully examined and converted in to 9 first row cell positions x 60 rest of the rows disjoint cell positions, without any loss of data.

For your information, I find that, AI does not have enough information and useless to help explain step-by-step and programming too.

Well, as an off topic, I think that, the multi-digit POM is exactly opposite to the multi-digit impossible patterns, for which, i started another topic to discuss here. But, again, lack of information on most powerful technics, like impossible pattern, unavoidable sets, deadly pattern, etc.

R. Jamil

by **P.O.** » Fri Nov 15, 2024 1:27 pm

to compute the templates is fast and they don't need that much memory, i don't organize them, at the initialization of the puzzle i retrieve from them the possible templates for each value, which does not amount to much, less than 8000 i think
for example these must be among those which have the most:

Code: Select all: ...4.6.........12.......5..2........5.1..8..........47.7..1...8.6.9.........2.... (12 7 5984 56 74 44 32 68 872) = 7149 ..34...........12........6...4.....7....61.3.8.............7..861..2.......5..... (5 135 132 30 858 5 113 47 6229) = 7554 .2...67....6...........1.4..94.............15.........5.1.4.......3..92.........7 (6 119 827 12 36 146 156 6276 57) = 7635 1....6...4...........2..5.......7.4.......9...68.1........4...8..29.....7.......1 (6 82 6698 8 720 102 92 68 60) = 7836 1...............36..82.......5...4...7....8......13....3.....1.6.......7...845... (6 660 4 64 77 62 90 4 6256) = 7223 ......7.9.5.1......8...3...23..9........7..1.....4..5......8......5....39.4...... (65 736 8 43 6 6288 54 68 4) = 7272 ..34.6.........2....9...5...4.....93....28....6.......5.......48......6....31.... (574 32 8 8 70 9 6290 68 103) = 7162

to give an overview of what i do
i have the templates in 9 lists like this:

Code: Select all: val 1 (1T0 1T1 1T2 1T3 ..... 1Tn) val 2 (2T0 2T1 2T2 2T3 ..... 2Tn) val 3 (3T0 3T1 3T2 3T3 ..... 3Tn) etc.

and i start by forming the 36 size 2 combinations:

Code: Select all: ((1 2) (1 3) (1 4) (1 5) (1 6) (1 7) (1 8) (1 9) (2 3) (2 4) (2 5) (2 6) (2 7) (2 8) (2 9) (3 4) (3 5) (3 6) (3 7) (3 8) (3 9) (4 5) (4 6) (4 7) (4 8) (4 9) (5 6) (5 7) (5 8) (5 9) (6 7) (6 8) (6 9) (7 8) (7 9) (8 9))

2 templates are compatible if their intersection is empty
i thus get a variable number of instances for each combination, the templates are indexed by their rank in their list and i preceded the instances by the combination label:

Code: Select all: ((1 2) (0 0) (0 1) (1 2) (1 3) ...) ((1 3) (0 0) (0 2) (2 2) (2 3) ...) ((2 3) (0 0) (0 3) (2 2) (2 3) ...) etc.

then i analyze the combinations, for example a template for value 1 is eliminated if it does not appear in each combinations: ((1 2) (1 3) (1 4) (1 5) (1 6) (1 7) (1 8) (1 9)

from there i only use the indexes to form the following combinations, for example a size 3 combination as (1 2 3) is made up of 3 combinations of size 2: (1 2) (1 3) (2 3) and it is from the indexes of these combinations that i form the instances of (1 2 3)

then a combination of size 4 (1 2 3 4) is made up of 4 combinations of size 3 (1 2 3) (1 2 4) (1 3 4) (2 3 4) that i use to obtain the instances of (1 2 3 4)

etc.

by **rjamil** » Fri Nov 15, 2024 2:38 pm

Hi P.O.,

In order to understand your logic, let me run your seven example puzzles with my RJSudoku.c program and a quick comparison of maximum number of valid templates is sharing below:

Code: Select all: ...4.6.........12.......5..2........5.1..8..........47.7..1...8.6.9.........2.... PO: (12 7 5984 56 74* 44 32 68 872) = 7149 RJ: (12 7 0 56 34* 44 0 0 0) ..34...........12........6...4.....7....61.3.8.............7..861..2.......5..... PO: (5 135 132 30 858 5 113 47 6229) = 7554 RJ: (5 0 0 30 0 5 0 0 0) stte .2...67....6...........1.4..94.............15.........5.1.4.......3..92.........7 PO: (6 119 827 12 36 146 156 6276 57) = 7635 RJ: (6 119 0 0 36 0 0 0 57) 1....6...4...........2..5.......7.4.......9...68.1........4...8..29.....7.......1 PO: (6 82 6698 8 720 102 92 68 60) = 7836 RJ: (6 82 0 8 0 0 92 0 60) 1...............36..82.......5...4...7....8......13....3.....1.6.......7...845... PO: (6 660 4 64 77 62* 90* 4 6256) = 7223 RJ: (6 0 4 0 0 14* 58* 4 0) ......7.9.5.1......8...3...23..9........7..1.....4..5......8......5....39.4...... PO: (65 736 8 43 6 6288 54 68* 4) = 7272 RJ: (65 0 8 43 6 0 0 20* 4) ..34.6.........2....9...5...4.....93....28....6.......5.......48......6....31.... PO: (574 32 8 8 70 9* 6290 68* 103*) = 7162 RJ: (0 32 8 8 0 3* 0 28* 13*) stte

Please note that, checking a digit beyond 5184 templates? Break your two dimensional array 46,656 x 9 templates into three dimensional 9 x 5,184 x 8 templates and then search each first row cell position to it's corresponding 5184 x 8 array.

R. Jamil

by **P.O.** » Fri Nov 15, 2024 3:51 pm

here is how i retrieve the templates for each value taking as an example the first puzzle
...4.6.........12.......5..2........5.1..8..........47.7..1...8.6.9.........2....
i initialize the puzzle with Singles
...4.6.........12.......5..2.7......5.1..8..........47.7..1...8.6.9.........2....
i retrieve the cells of each value
(16 39 59) (17 28 77) NIL (4 53) (25 37) (6 65) (30 54 56) (42 63) (67)
for each value these are subsets of templates
i retrieve these templates
64 64 46656 576 576 576 144 576 5184
and for each value i remove the templates which have cells from the other values
(12 7 5984 56 74 44 32 68 872)

by **rjamil** » Fri Nov 15, 2024 7:33 pm

Hi P.O.,

Did you not mentioned the MNVT values of each digit? Are you counting the number of templates utilized for checking each digit?

My program uses fully sorted order of 46,656 templates in highly compressed form. Let me show you how the worst case is searched:

Code: Select all: +-------+-------+-------+ +-------+-------+-------+ | X . . | . . . | . . . | | . . . | . . . | . . X | | . . . | X . . | . . . | | . . . | . . X | . . . | | . . . | . . . | X . . | | . . X | . . . | . . . | +-------+-------+-------+ +-------+-------+-------+ | . X . | . . . | . . . | | . . . | . . . | . X . | | . . . | . X . | . . . | | . . . | . X . | . . . | | . . . | . . . | . X . | | . X . | . . . | . . . | +-------+-------+-------+ +-------+-------+-------+ | . . X | . . . | . . . | | . . . | . . . | X . . | | . . . | . . X | . . . | | . . . | X . . | . . . | | . . . | . . . | . . X | | X . . | . . . | . . . | +-------+-------+-------+ +-------+-------+-------+

1) for each digit (1 to 9)
2) for each cell position in first row (0 to 8)
3) if first row cell position does not contain digit in either clue or solved or unsolved then go to step 2
4) for each template in first row cell position chunk (total 5184)
5) for i = 0 to 8 (2nd to 9th row disjoint cell position)
6) if i'th row disjoint cell location does not contain particular digit then go to 8
7) next for (row or go to step 5)
8) if i > 8 then increment count for all rows disjoint cell locations contain particular digit found else skip T[i] template
9) next for (template or go to step 4)
10) eliminate digit from disjoint cell location whose count is zero tmplate
11) place digit in disjoint cell location whose count is all templates
12) next for (digit or go to step 1)

Now keeping above steps in mind, for best and worst case, routine will always check all 9 digit x 9 times check in first row x (6 times skip for 2nd row disjoint cell check + 3 times for 3rd + 6 times for 4th + 4 times for 5th + 2 times for 6th + 3 times for 7th + 2 times for 8th and 9th = total 26 templates searched)

R. Jamil

by **rjamil** » Sat Nov 16, 2024 12:46 am

Ok.

I did search POM in blank Sudoku grid and here I am sharing my results showing MNVT count:

After T&E: 1r1c1, 2r1c2, 3r1c3, 4r1c4, 5r1c5 and 6r1c6:

Code: Select all: POM: Digit 7 in r1c789 r2c123456789 r3c123456789 r4c123456789 r5c123456789 r6c123456789 r7c123456789 r8c123456789 r9c123456789 Digit 7 not in 15552 Templates => -7 @ r2c7 r2c8 r2c9 r3c7 r3c8 r3c9 POM: Digit 8 in r1c789 r2c123456789 r3c123456789 r4c123456789 r5c123456789 r6c123456789 r7c123456789 r8c123456789 r9c123456789 Digit 8 not in 15552 Templates => -8 @ r2c7 r2c8 r2c9 r3c7 r3c8 r3c9 POM: Digit 9 in r1c789 r2c123456789 r3c123456789 r4c123456789 r5c123456789 r6c123456789 r7c123456789 r8c123456789 r9c123456789 Digit 9 not in 15552 Templates => -9 @ r2c7 r2c8 r2c9 r3c7 r3c8 r3c9

Then T&E: 7r1c7, 8r1c8
NS: 9r1c9
T&E: 4r2c1, 5r2c2 and 6r2c3

Code: Select all: POM: Digit 7 in r1c7 r2c456 r3c123456 r4c12345689 r5c12345689 r6c12345689 r7c12345689 r8c12345689 r9c12345689 Digit 7 not in 2592 Templates => -7 @ r3c4 r3c5 r3c6 POM: Digit 8 in r1c8 r2c456 r3c123456 r4c12345679 r5c12345679 r6c12345679 r7c12345679 r8c12345679 r9c12345679 Digit 8 not in 2592 Templates => -8 @ r3c4 r3c5 r3c6 POM: Digit 9 in r1c9 r2c456 r3c123456 r4c12345678 r5c12345678 r6c12345678 r7c12345678 r8c12345678 r9c12345678 Digit 9 not in 2592 Templates => -9 @ r3c4 r3c5 r3c6

R. Jamil

by **P.O.** » Sat Nov 16, 2024 6:16 am

i have the same results as you

Code: Select all: 123456........................................................................... #VT: (5184 5184 5184 5184 5184 5184 15552 15552 15552) MNVT = 15552 Cells: nil nil nil nil nil nil nil nil nil Candidates: nil nil nil nil nil nil (16 17 18 25 26 27) (16 17 18 25 26 27) (16 17 18 25 26 27) 1 2 3 4 5 6 789 789 789 456789 456789 456789 123789 123789 123789 123456 123456 123456 456789 456789 456789 123789 123789 123789 123456 123456 123456 23456789 13456789 12456789 12356789 12346789 12345789 123456789 123456789 123456789 23456789 13456789 12456789 12356789 12346789 12345789 123456789 123456789 123456789 23456789 13456789 12456789 12356789 12346789 12345789 123456789 123456789 123456789 23456789 13456789 12456789 12356789 12346789 12345789 123456789 123456789 123456789 23456789 13456789 12456789 12356789 12346789 12345789 123456789 123456789 123456789 23456789 13456789 12456789 12356789 12346789 12345789 123456789 123456789 123456789 567 candidates. 123456789456..................................................................... #VT: (5184 5184 5184 864 864 864 2592 2592 2592) MNVT = 5184 Cells: nil nil nil nil nil nil nil nil nil Candidates: nil nil nil nil nil nil (22 23 24) (22 23 24) (22 23 24) 1 2 3 4 5 6 7 8 9 4 5 6 123789 123789 123789 123 123 123 789 789 789 123 123 123 123456 123456 123456 2356789 1346789 1245789 12356789 12346789 12345789 12345689 12345679 12345678 2356789 1346789 1245789 12356789 12346789 12345789 12345689 12345679 12345678 2356789 1346789 1245789 12356789 12346789 12345789 12345689 12345679 12345678 2356789 1346789 1245789 12356789 12346789 12345789 12345689 12345679 12345678 2356789 1346789 1245789 12356789 12346789 12345789 12345689 12345679 12345678 2356789 1346789 1245789 12356789 12346789 12345789 12345689 12345679 12345678 477 candidates.

consider these puzzles and the template count calculated by Dobrichev, these are for each value the number of valid templates at the beginning even before Singles is applied, with Singles the count is slightly different for Platinum Blonde and Fata Morgana,

the MNVT, an acronym coined by Mathimagics, designates for each puzzle the highest value, for example the MNVT of Easter Monster is 148, that of Golden Nugget is 108 etc.

these are the templates i'm talking about and that i retrieve at the beginning of the template procedure after Singles in the way i indicated in the previous post

can we start from there, can you get the same number of templates at the beginning of the procedure

by **rjamil** » Sat Nov 16, 2024 2:37 pm

Hi P.O.,

My program is far inferior to solve those well known the hardest puzzles.

Sorry for unable to solve the puzzles with singleton, POM and T&E searching moves enabled. (Please note that, I can't remove singleton moves because it is highly integrated with T&E move. However, rest of the moves are independently coded and may be stop searching any or all of them.)

However, if MNVT means maximum number of VALID templates, then VT means VALID templates. Right?
Similarly, you showed some count against digit not contain elimination/placement for which I don't.

I showed my program valid templates count for which it does not match with few of your provided seven puzzles.

R. Jamil

by **P.O.** » Sat Nov 16, 2024 4:26 pm

rjamil wrote:However, if MNVT means maximum number of VALID templates, then VT means VALID templates. Right?
Similarly, you showed some count against digit not contain elimination/placement for which I don't.
I showed my program valid templates count for which it does not match with few of your provided seven puzzles.

yes VT means valid templates, but more precisely the templates that are possible, like candidates that are possible, for each value
and for these puzzles the template count has been done before Singles

these puzzles are just an example of what needs to be calculated at the beginning of a template procedure: for each value the set of possible templates, this is the first thing to do in order to be able to make the template combinations

by **rjamil** » Sun Nov 17, 2024 6:59 pm

Hi P.O.,

P.O. wrote:these puzzles are just an example of what needs to be calculated at the beginning of a template procedure: for each value the set of possible templates, this is the first thing to do in order to be able to make the template combinations

After thinking, I assume that, my program is able to search POM correctly (as I can't show VT for neither elimination nor placement POM move atm). So, assuming MNVT count is correct for single-digit POM move (without any filter use), how to check multi-digit POM?

R. Jamil

by **P.O.** » Mon Nov 18, 2024 7:58 am

let's solve this puzzle, the fifth one from the second list here, it is initialized with Singles and gives the following resolution state

Code: Select all: . 2 . . . 6 7 . . 4 . . 7 . 9 . . . . . 9 . 2 3 . 6 . . . . . . . . 5 1 . . 8 9 . . 3 . . 9 7 . . . . . . . 3 . . . 6 . 2 . 4 6 . 5 . 1 . . . 7 . . . . . . . . . .2...67..4..7.9.....9.23.6........51..89..3..97.......3...6.2.46.5.1...7......... ( n2r4c1 n2r9c3 n7r5c8 n7r7c3 n7r3c1 n9r9c5 n9r4c7 n8r8c7 n7r9c6 n7r4c5 n9r1c9 n3r6c5 ) 158 2 13 1458 458 6 7 1348 9 4 13568 136 7 58 9 15 1238 2358 7 158 9 1458 2 3 145 6 58 2 346 346 468 7 48 9 5 1 15 1456 8 9 45 1245 3 7 26 9 7 146 124568 3 12458 46 248 268 3 189 7 58 6 58 2 19 4 6 49 5 234 1 24 8 39 7 18 148 2 3458 9 7 156 13 356 130 candidates.

then for each value the templates are retrieved: #VT: (13 4 5 4 12 5 1 11 2)

Hidden Text: Show

i always start their analyses by looking for cells common to all templates and here the analysis of templates for 4 allows to put 4 in cell 30

Code: Select all: 4: ((8 10 22 30 41 52 63 69 74) (5 10 25 30 42 53 63 67 74) (5 10 25 30 42 53 63 65 76) (4 10 25 30 41 53 63 69 74))

i update the resolution state which allows 20 placements in addition to r4c3=4

Code: Select all: ( n4r4c3 n8r4c6 n5r7c6 n6r4c4 n8r7c4 n3r4c2 n5r6c4 n4r5c5 n2r8c4 n3r8c8 n3r9c4 n4r9c2 n4r8c6 n8r9c1 n9r7c8 n9r8c2 n1r7c2 n1r9c8 n3r2c9 n3r1c3 n2r2c8 ) 15 2 3 14 58 6 7 48 9 4 568 16 7 58 9 15 2 3 7 58 9 14 2 3 145 6 58 2 3 4 6 7 8 9 5 1 15 56 8 9 4 12 3 7 26 9 7 16 5 3 12 46 48 268 3 1 7 8 6 5 2 9 4 6 9 5 2 1 4 8 3 7 8 4 2 3 9 7 56 1 56 49 candidates.

i calculate the templates again: #VT: (2 2 1 2 5 3 1 2 1)

Code: Select all: 1: ((4 12 25 36 37 51 56 68 80) (1 16 22 36 42 48 56 68 80)) 2: ((2 17 23 28 45 51 61 67 75) (2 17 23 28 42 54 61 67 75)) 3: ((3 18 24 29 43 50 55 71 76)) 4: ((8 10 22 30 41 52 63 69 74) (4 10 25 30 41 53 63 69 74)) 5: ((5 16 20 35 37 49 60 66 81) (5 11 27 35 37 49 60 66 79) (5 11 25 35 37 49 60 66 81) (1 14 27 35 38 49 60 66 79) (1 14 25 35 38 49 60 66 81)) 6: ((6 12 26 31 38 54 59 64 79) (6 12 26 31 38 52 59 64 81) (6 11 26 31 45 48 59 64 79)) 7: ((7 13 19 32 44 47 57 72 78)) 8: ((8 14 20 33 39 54 58 70 73) (5 11 27 33 39 53 58 70 73)) 9: ((9 15 21 34 40 46 62 65 77)))

but their analyses give nothing, so it is with this set of templates that the size 2 combinations will be made
in conclusion in order to be able to make the template combinations you must be able to retrieve the templates
from there several strategies are possible

by **rjamil** » Mon Nov 18, 2024 2:39 pm

Hi P.O.,

I have one good news and a bad news.

Good news is that, by solving above mentioned vanilla Sudoku puzzle, after 12 singleton moves, my program detect exactly the same POM VT VALUES as yours:

Code: Select all: POM: Digit 1 in r1c1348 r2c2378 r3c247 r4c9 r5c126 r6c346 r7c28 r8c5 r9c1278 Digit 1 not in 13 Templates => -1 @ r6c4 POM: Digit 4 in r1c458 r2c1 r3c47 r4c2346 r5c256 r6c34678 r7c9 r8c246 r9c24 Digit 4 not in 4 Templates => -4 @ r4c2 r4c4 r4c6 r5c2 r6c3 r6c4 r6c6 Digit 4 in all 4 Templates => 4 @ r4c3 POM: Digit 5 in r1c145 r2c2579 r3c2479 r4c8 r5c1256 r6c46 r7c46 r8c3 r9c479 Digit 5 not in 12 Templates => -5 @ r1c4 r3c4 r5c5 r5c6 r9c4 POM: Digit 8 in r1c1458 r2c2589 r3c249 r4c46 r5c3 r6c4689 r7c246 r8c7 r9c124 Digit 8 not in 11 Templates => -8 @ r1c4 r3c4 r6c4 r6c6

However, the bad news is that, my program is unable to solve this puzzle with singleton, POM and T&E moves, but, solve with basic moves enabled:

Hidden Text: Show

Code: Select all: .78......5..2.4....1.3...........9.........733....916263.58.7......92.....27....1#http://forum.enjoysudoku.com/post282522.html#p282522 +---+---+---+ |.78|...|...| |5..|2.4|...| |.1.|3..|...| +---+---+---+ |...|...|9..| |...|...|.73| |3..|..9|162| +---+---+---+ |63.|58.|7..| |...|.92|...| |..2|7..|..1| +---+---+---+ +----------------------+-----------------------+-----------------------+ | 249 7 8 | 169 156 156 | 23456 123459 4569 | | 5 69 369 | 2 167 4 | 368 1389 6789 | | 249 1 469 | 3 567 5678 | 24568 24589 456789 | +----------------------+-----------------------+-----------------------+ | 12478 24568 14567 | 1468 1234567 135678 | 9 458 458 | | 12489 245689 14569 | 1468 12456 1568 | 458 7 3 | | 3 458 457 | 48 457 9 | 1 6 2 | +----------------------+-----------------------+-----------------------+ | 6 3 149 | 5 8 1 | 7 249 49 | | 1478 458 1457 | 146 9 2 | 34568 3458 4568 | | 489 4589 2 | 7 346 36 | 34568 34589 1 | +----------------------+-----------------------+-----------------------+ 1st) Hidden single: 9 @ r1c4 from values 169 Column 4 wise +----------------------+-----------------------+----------------------+ | 24 7 8 | 9 156 156 | 23456 12345 456 | | 5 69 369 | 2 167 4 | 368 1389 6789 | | 249 1 469 | 3 567 5678 | 24568 24589 456789 | +----------------------+-----------------------+----------------------+ | 12478 24568 14567 | 1468 1234567 135678 | 9 458 458 | | 12489 245689 14569 | 1468 12456 1568 | 458 7 3 | | 3 458 457 | 48 457 9 | 1 6 2 | +----------------------+-----------------------+----------------------+ | 6 3 149 | 5 8 1 | 7 249 49 | | 1478 458 1457 | 146 9 2 | 34568 3458 4568 | | 489 4589 2 | 7 346 36 | 34568 34589 1 | +----------------------+-----------------------+----------------------+ 2nd) Hidden single: 3 @ r2c3 from values 369 Column 3 wise +----------------------+-----------------------+----------------------+ | 24 7 8 | 9 156 156 | 23456 12345 456 | | 5 69 3 | 2 167 4 | 68 189 6789 | | 249 1 469 | 3 567 5678 | 24568 24589 456789 | +----------------------+-----------------------+----------------------+ | 12478 24568 14567 | 1468 1234567 135678 | 9 458 458 | | 12489 245689 14569 | 1468 12456 1568 | 458 7 3 | | 3 458 457 | 48 457 9 | 1 6 2 | +----------------------+-----------------------+----------------------+ | 6 3 149 | 5 8 1 | 7 249 49 | | 1478 458 1457 | 146 9 2 | 34568 3458 4568 | | 489 4589 2 | 7 346 36 | 34568 34589 1 | +----------------------+-----------------------+----------------------+ 3rd) Hidden single: 8 @ r3c6 from values 5678 Box 2 wise +----------------------+----------------------+---------------------+ | 24 7 8 | 9 156 156 | 23456 12345 456 | | 5 69 3 | 2 167 4 | 68 189 6789 | | 249 1 469 | 3 567 8 | 2456 2459 45679 | +----------------------+----------------------+---------------------+ | 12478 24568 14567 | 1468 1234567 13567 | 9 458 458 | | 12489 245689 14569 | 1468 12456 156 | 458 7 3 | | 3 458 457 | 48 457 9 | 1 6 2 | +----------------------+----------------------+---------------------+ | 6 3 149 | 5 8 1 | 7 249 49 | | 1478 458 1457 | 146 9 2 | 34568 3458 4568 | | 489 4589 2 | 7 346 36 | 34568 34589 1 | +----------------------+----------------------+---------------------+ 4th) Hidden single: 7 @ r4c6 from values 13567 Column 6 wise +----------------------+-------------------+---------------------+ | 24 7 8 | 9 156 156 | 23456 12345 456 | | 5 69 3 | 2 167 4 | 68 189 6789 | | 249 1 469 | 3 567 8 | 2456 2459 45679 | +----------------------+-------------------+---------------------+ | 1248 24568 1456 | 1468 123456 7 | 9 458 458 | | 12489 245689 14569 | 1468 12456 156 | 458 7 3 | | 3 458 457 | 48 45 9 | 1 6 2 | +----------------------+-------------------+---------------------+ | 6 3 149 | 5 8 1 | 7 249 49 | | 1478 458 1457 | 146 9 2 | 34568 3458 4568 | | 489 4589 2 | 7 346 36 | 34568 34589 1 | +----------------------+-------------------+---------------------+ 5th) Hidden single: 3 @ r4c5 from values 123456 Row 4 wise +----------------------+------------------+---------------------+ | 24 7 8 | 9 156 156 | 23456 12345 456 | | 5 69 3 | 2 167 4 | 68 189 6789 | | 249 1 469 | 3 567 8 | 2456 2459 45679 | +----------------------+------------------+---------------------+ | 1248 24568 1456 | 1468 3 7 | 9 458 458 | | 12489 245689 14569 | 1468 12456 156 | 458 7 3 | | 3 458 457 | 48 45 9 | 1 6 2 | +----------------------+------------------+---------------------+ | 6 3 149 | 5 8 1 | 7 249 49 | | 1478 458 1457 | 146 9 2 | 34568 3458 4568 | | 489 4589 2 | 7 46 36 | 34568 34589 1 | +----------------------+------------------+---------------------+ 6th) Hidden single: 2 @ r5c5 from values 12456 Column 5 wise +--------------------+----------------+---------------------+ | 24 7 8 | 9 156 156 | 23456 12345 456 | | 5 69 3 | 2 167 4 | 68 189 6789 | | 249 1 469 | 3 567 8 | 2456 2459 45679 | +--------------------+----------------+---------------------+ | 1248 24568 1456 | 1468 3 7 | 9 458 458 | | 1489 45689 14569 | 1468 2 156 | 458 7 3 | | 3 458 457 | 48 45 9 | 1 6 2 | +--------------------+----------------+---------------------+ | 6 3 149 | 5 8 1 | 7 249 49 | | 1478 458 1457 | 146 9 2 | 34568 3458 4568 | | 489 4589 2 | 7 46 36 | 34568 34589 1 | +--------------------+----------------+---------------------+ 7th) Hidden single: 2 @ r4c2 from values 24568 Column 2 wise +--------------------+----------------+---------------------+ | 24 7 8 | 9 156 156 | 23456 12345 456 | | 5 69 3 | 2 167 4 | 68 189 6789 | | 249 1 469 | 3 567 8 | 2456 2459 45679 | +--------------------+----------------+---------------------+ | 148 2 1456 | 1468 3 7 | 9 458 458 | | 1489 45689 14569 | 1468 2 156 | 458 7 3 | | 3 458 457 | 48 45 9 | 1 6 2 | +--------------------+----------------+---------------------+ | 6 3 149 | 5 8 1 | 7 249 49 | | 1478 458 1457 | 146 9 2 | 34568 3458 4568 | | 489 4589 2 | 7 46 36 | 34568 34589 1 | +--------------------+----------------+---------------------+ 8th) Hidden single: 7 @ r6c3 from values 457 Row 6 wise +--------------------+----------------+---------------------+ | 24 7 8 | 9 156 156 | 23456 12345 456 | | 5 69 3 | 2 167 4 | 68 189 6789 | | 249 1 469 | 3 567 8 | 2456 2459 45679 | +--------------------+----------------+---------------------+ | 148 2 1456 | 1468 3 7 | 9 458 458 | | 1489 45689 14569 | 1468 2 156 | 458 7 3 | | 3 458 7 | 48 45 9 | 1 6 2 | +--------------------+----------------+---------------------+ | 6 3 149 | 5 8 1 | 7 249 49 | | 1478 458 145 | 146 9 2 | 34568 3458 4568 | | 489 4589 2 | 7 46 36 | 34568 34589 1 | +--------------------+----------------+---------------------+ 9th) Hidden single: 2 @ r7c8 from values 249 Row 7 wise +--------------------+----------------+---------------------+ | 24 7 8 | 9 156 156 | 23456 1345 456 | | 5 69 3 | 2 167 4 | 68 189 6789 | | 249 1 469 | 3 567 8 | 2456 459 45679 | +--------------------+----------------+---------------------+ | 148 2 1456 | 1468 3 7 | 9 458 458 | | 1489 45689 14569 | 1468 2 156 | 458 7 3 | | 3 458 7 | 48 45 9 | 1 6 2 | +--------------------+----------------+---------------------+ | 6 3 149 | 5 8 1 | 7 2 49 | | 1478 458 145 | 146 9 2 | 34568 3458 4568 | | 489 4589 2 | 7 46 36 | 34568 34589 1 | +--------------------+----------------+---------------------+ 10th) Hidden single: 7 @ r8c1 from values 1478 Row 8 wise +--------------------+----------------+---------------------+ | 24 7 8 | 9 156 156 | 23456 1345 456 | | 5 69 3 | 2 167 4 | 68 189 6789 | | 249 1 469 | 3 567 8 | 2456 459 45679 | +--------------------+----------------+---------------------+ | 148 2 1456 | 1468 3 7 | 9 458 458 | | 1489 45689 14569 | 1468 2 156 | 458 7 3 | | 3 458 7 | 48 45 9 | 1 6 2 | +--------------------+----------------+---------------------+ | 6 3 149 | 5 8 1 | 7 2 49 | | 7 458 145 | 146 9 2 | 34568 3458 4568 | | 489 4589 2 | 7 46 36 | 34568 34589 1 | +--------------------+----------------+---------------------+ 11th) Hidden single: 3 @ r9c6 from values 36 Column 6 wise +--------------------+----------------+--------------------+ | 24 7 8 | 9 156 156 | 23456 1345 456 | | 5 69 3 | 2 167 4 | 68 189 6789 | | 249 1 469 | 3 567 8 | 2456 459 45679 | +--------------------+----------------+--------------------+ | 148 2 1456 | 1468 3 7 | 9 458 458 | | 1489 45689 14569 | 1468 2 156 | 458 7 3 | | 3 458 7 | 48 45 9 | 1 6 2 | +--------------------+----------------+--------------------+ | 6 3 149 | 5 8 1 | 7 2 49 | | 7 458 145 | 146 9 2 | 34568 3458 4568 | | 489 4589 2 | 7 46 3 | 4568 4589 1 | +--------------------+----------------+--------------------+ 12th) Naked single: 1 @ r7c6 +--------------------+---------------+--------------------+ | 24 7 8 | 9 156 56 | 23456 1345 456 | | 5 69 3 | 2 167 4 | 68 189 6789 | | 249 1 469 | 3 567 8 | 2456 459 45679 | +--------------------+---------------+--------------------+ | 148 2 1456 | 1468 3 7 | 9 458 458 | | 1489 45689 14569 | 1468 2 56 | 458 7 3 | | 3 458 7 | 48 45 9 | 1 6 2 | +--------------------+---------------+--------------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 458 145 | 46 9 2 | 34568 3458 4568 | | 489 4589 2 | 7 46 3 | 4568 4589 1 | +--------------------+---------------+--------------------+ 13th) Hidden single: 1 @ r8c3 from values 145 Row 8 wise +-------------------+---------------+--------------------+ | 24 7 8 | 9 156 56 | 23456 1345 456 | | 5 69 3 | 2 167 4 | 68 189 6789 | | 249 1 469 | 3 567 8 | 2456 459 45679 | +-------------------+---------------+--------------------+ | 148 2 456 | 1468 3 7 | 9 458 458 | | 1489 45689 4569 | 1468 2 56 | 458 7 3 | | 3 458 7 | 48 45 9 | 1 6 2 | +-------------------+---------------+--------------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 458 1 | 46 9 2 | 34568 3458 4568 | | 489 4589 2 | 7 46 3 | 4568 4589 1 | +-------------------+---------------+--------------------+ 13) POM: Digit 5 in r1c56789 r2c1 r3c5789 r4c389 r5c2367 r6c25 r7c4 r8c2789 r9c278 Digit 5 not in 10 Templates => -5 @ r1c5 r1c7 r1c8 r1c9 r3c5 r5c2 r5c6 r6c2 Digit 5 in all 10 Templates => 5 @ r1c6 r6c5 13) POM: Digit 6 in r1c579 r2c2579 r3c3579 r4c34 r5c2346 r6c8 r7c1 r8c479 r9c57 Digit 6 not in 2 Templates => -6 @ r1c7 r2c5 r2c7 r2c9 r3c3 r3c7 r4c4 r5c2 r5c3 r5c4 r8c7 r8c9 r9c5 Digit 6 in all 2 Templates => 6 @ r2c2 r4c3 r5c6 r8c4 r9c7 13) POM: Digit 9 in r1c4 r2c89 r3c1389 r4c7 r5c123 r6c6 r7c39 r8c5 r9c128 Digit 9 not in 4 Templates => -9 @ r3c8 r3c9 +-----------------+------------+------------------+ | 24 7 8 | 9 16 5 | 234 134 46 | | 5 6 3 | 2 17 4 | 8 189 789 | | 249 1 49 | 3 67 8 | 245 45 4567 | +-----------------+------------+------------------+ | 148 2 6 | 148 3 7 | 9 458 458 | | 1489 489 459 | 148 2 6 | 458 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +-----------------+------------+------------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 458 1 | 6 9 2 | 3458 3458 458 | | 489 4589 2 | 7 4 3 | 6 4589 1 | +-----------------+------------+------------------+ 14th) Naked single: 6 @ r4c3 +-----------------+------------+------------------+ | 24 7 8 | 9 16 5 | 234 134 46 | | 5 6 3 | 2 17 4 | 8 189 789 | | 249 1 49 | 3 67 8 | 245 45 4567 | +-----------------+------------+------------------+ | 148 2 6 | 148 3 7 | 9 458 458 | | 1489 489 459 | 148 2 6 | 458 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +-----------------+------------+------------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 458 1 | 6 9 2 | 3458 3458 458 | | 489 4589 2 | 7 4 3 | 6 4589 1 | +-----------------+------------+------------------+ 15th) Naked single: 5 @ r1c6 +-----------------+------------+------------------+ | 24 7 8 | 9 16 5 | 234 134 46 | | 5 6 3 | 2 17 4 | 8 189 789 | | 249 1 49 | 3 67 8 | 245 45 4567 | +-----------------+------------+------------------+ | 148 2 6 | 148 3 7 | 9 458 458 | | 1489 489 459 | 148 2 6 | 458 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +-----------------+------------+------------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 458 1 | 6 9 2 | 3458 3458 458 | | 489 4589 2 | 7 4 3 | 6 4589 1 | +-----------------+------------+------------------+ 16th) Hidden single: 5 @ r5c3 from values 459 Column 3 wise +----------------+------------+------------------+ | 24 7 8 | 9 16 5 | 234 134 46 | | 5 6 3 | 2 17 4 | 8 189 789 | | 249 1 49 | 3 67 8 | 245 45 4567 | +----------------+------------+------------------+ | 148 2 6 | 148 3 7 | 9 458 458 | | 1489 489 5 | 148 2 6 | 48 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +----------------+------------+------------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 458 1 | 6 9 2 | 3458 3458 458 | | 489 4589 2 | 7 4 3 | 6 4589 1 | +----------------+------------+------------------+ 17th) Naked single: 6 @ r5c6 +----------------+------------+------------------+ | 24 7 8 | 9 16 5 | 234 134 46 | | 5 6 3 | 2 17 4 | 8 189 789 | | 249 1 49 | 3 67 8 | 245 45 4567 | +----------------+------------+------------------+ | 148 2 6 | 148 3 7 | 9 458 458 | | 1489 489 5 | 148 2 6 | 48 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +----------------+------------+------------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 458 1 | 6 9 2 | 3458 3458 458 | | 489 4589 2 | 7 4 3 | 6 4589 1 | +----------------+------------+------------------+ 18th) Naked single: 6 @ r2c2 +----------------+------------+------------------+ | 24 7 8 | 9 16 5 | 234 134 46 | | 5 6 3 | 2 17 4 | 8 189 789 | | 249 1 49 | 3 67 8 | 245 45 4567 | +----------------+------------+------------------+ | 148 2 6 | 148 3 7 | 9 458 458 | | 1489 489 5 | 148 2 6 | 48 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +----------------+------------+------------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 458 1 | 6 9 2 | 3458 3458 458 | | 489 4589 2 | 7 4 3 | 6 4589 1 | +----------------+------------+------------------+ 19th) Naked single: 5 @ r6c5 +----------------+------------+------------------+ | 24 7 8 | 9 16 5 | 234 134 46 | | 5 6 3 | 2 17 4 | 8 189 789 | | 249 1 49 | 3 67 8 | 245 45 4567 | +----------------+------------+------------------+ | 148 2 6 | 148 3 7 | 9 458 458 | | 1489 489 5 | 148 2 6 | 48 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +----------------+------------+------------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 458 1 | 6 9 2 | 3458 3458 458 | | 489 4589 2 | 7 4 3 | 6 4589 1 | +----------------+------------+------------------+ 20th) Naked single: 6 @ r8c4 +----------------+------------+------------------+ | 24 7 8 | 9 16 5 | 234 134 46 | | 5 6 3 | 2 17 4 | 8 189 789 | | 249 1 49 | 3 67 8 | 245 45 4567 | +----------------+------------+------------------+ | 148 2 6 | 148 3 7 | 9 458 458 | | 1489 489 5 | 148 2 6 | 48 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +----------------+------------+------------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 458 1 | 6 9 2 | 3458 3458 458 | | 489 4589 2 | 7 4 3 | 6 4589 1 | +----------------+------------+------------------+ 21st) Naked single: 4 @ r9c5 +---------------+------------+------------------+ | 24 7 8 | 9 16 5 | 234 134 46 | | 5 6 3 | 2 17 4 | 8 189 789 | | 249 1 49 | 3 67 8 | 245 45 4567 | +---------------+------------+------------------+ | 148 2 6 | 148 3 7 | 9 458 458 | | 1489 489 5 | 148 2 6 | 48 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +---------------+------------+------------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 458 1 | 6 9 2 | 3458 3458 458 | | 89 589 2 | 7 4 3 | 6 589 1 | +---------------+------------+------------------+ 22nd) Naked single: 6 @ r9c7 +---------------+------------+------------------+ | 24 7 8 | 9 16 5 | 234 134 46 | | 5 6 3 | 2 17 4 | 8 189 789 | | 249 1 49 | 3 67 8 | 245 45 4567 | +---------------+------------+------------------+ | 148 2 6 | 148 3 7 | 9 458 458 | | 1489 489 5 | 148 2 6 | 48 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +---------------+------------+------------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 458 1 | 6 9 2 | 3458 3458 458 | | 89 589 2 | 7 4 3 | 6 589 1 | +---------------+------------+------------------+ 23rd) Naked single: 8 @ r2c7 +---------------+------------+-----------------+ | 24 7 8 | 9 16 5 | 234 134 46 | | 5 6 3 | 2 17 4 | 8 19 79 | | 249 1 49 | 3 67 8 | 245 45 4567 | +---------------+------------+-----------------+ | 148 2 6 | 148 3 7 | 9 458 458 | | 1489 489 5 | 148 2 6 | 4 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +---------------+------------+-----------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 458 1 | 6 9 2 | 345 3458 458 | | 89 589 2 | 7 4 3 | 6 589 1 | +---------------+------------+-----------------+ 24th) Naked single: 4 @ r5c7 +--------------+------------+----------------+ | 24 7 8 | 9 16 5 | 23 134 46 | | 5 6 3 | 2 17 4 | 8 19 79 | | 249 1 49 | 3 67 8 | 25 45 4567 | +--------------+------------+----------------+ | 148 2 6 | 148 3 7 | 9 58 58 | | 189 89 5 | 18 2 6 | 4 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +--------------+------------+----------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 458 1 | 6 9 2 | 35 3458 458 | | 89 589 2 | 7 4 3 | 6 589 1 | +--------------+------------+----------------+ 24) POM: Digit 8 in r1c3 r2c7 r3c6 r4c1489 r5c124 r6c24 r7c5 r8c289 r9c128 Digit 8 not in 7 Templates => -8 @ r4c1 r4c4 +--------------+-----------+----------------+ | 24 7 8 | 9 16 5 | 23 134 46 | | 5 6 3 | 2 17 4 | 8 19 79 | | 249 1 49 | 3 67 8 | 25 45 4567 | +--------------+-----------+----------------+ | 14 2 6 | 14 3 7 | 9 58 58 | | 189 89 5 | 18 2 6 | 4 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +--------------+-----------+----------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 458 1 | 6 9 2 | 35 3458 458 | | 89 589 2 | 7 4 3 | 6 589 1 | +--------------+-----------+----------------+ 25th) Trial & Error: 2 @ r1c7 from values 23 +--------------+-----------+----------------+ | 4 7 8 | 9 16 5 | 2 134 46 | | 5 6 3 | 2 17 4 | 8 19 79 | | 249 1 49 | 3 67 8 | 5 45 4567 | +--------------+-----------+----------------+ | 14 2 6 | 14 3 7 | 9 58 58 | | 189 89 5 | 18 2 6 | 4 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +--------------+-----------+----------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 458 1 | 6 9 2 | 35 3458 458 | | 89 589 2 | 7 4 3 | 6 589 1 | +--------------+-----------+----------------+ 26th) Hidden single: 3 @ r1c8 from values 134 Row 1 wise +--------------+-----------+---------------+ | 4 7 8 | 9 16 5 | 2 3 46 | | 5 6 3 | 2 17 4 | 8 19 79 | | 249 1 49 | 3 67 8 | 5 45 4567 | +--------------+-----------+---------------+ | 14 2 6 | 14 3 7 | 9 58 58 | | 189 89 5 | 18 2 6 | 4 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +--------------+-----------+---------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 458 1 | 6 9 2 | 35 458 458 | | 89 589 2 | 7 4 3 | 6 589 1 | +--------------+-----------+---------------+ 27th) Hidden single: 1 @ r1c5 from values 16 Row 1 wise +--------------+-----------+---------------+ | 4 7 8 | 9 1 5 | 2 3 46 | | 5 6 3 | 2 7 4 | 8 19 79 | | 249 1 49 | 3 67 8 | 5 45 4567 | +--------------+-----------+---------------+ | 14 2 6 | 14 3 7 | 9 58 58 | | 189 89 5 | 18 2 6 | 4 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +--------------+-----------+---------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 458 1 | 6 9 2 | 35 458 458 | | 89 589 2 | 7 4 3 | 6 589 1 | +--------------+-----------+---------------+ 28th) Hidden single: 6 @ r3c5 from values 67 Column 5 wise +--------------+----------+--------------+ | 4 7 8 | 9 1 5 | 2 3 46 | | 5 6 3 | 2 7 4 | 8 19 79 | | 249 1 49 | 3 6 8 | 5 45 457 | +--------------+----------+--------------+ | 14 2 6 | 14 3 7 | 9 58 58 | | 189 89 5 | 18 2 6 | 4 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +--------------+----------+--------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 458 1 | 6 9 2 | 35 458 458 | | 89 589 2 | 7 4 3 | 6 589 1 | +--------------+----------+--------------+ 29th) Hidden single: 1 @ r2c8 from values 19 Row 2 wise +--------------+----------+--------------+ | 4 7 8 | 9 1 5 | 2 3 46 | | 5 6 3 | 2 7 4 | 8 1 79 | | 249 1 49 | 3 6 8 | 5 45 457 | +--------------+----------+--------------+ | 14 2 6 | 14 3 7 | 9 58 58 | | 189 89 5 | 18 2 6 | 4 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +--------------+----------+--------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 458 1 | 6 9 2 | 35 458 458 | | 89 589 2 | 7 4 3 | 6 589 1 | +--------------+----------+--------------+ 30th) Naked single: 7 @ r2c5 +--------------+----------+--------------+ | 4 7 8 | 9 1 5 | 2 3 46 | | 5 6 3 | 2 7 4 | 8 1 9 | | 249 1 49 | 3 6 8 | 5 45 457 | +--------------+----------+--------------+ | 14 2 6 | 14 3 7 | 9 58 58 | | 189 89 5 | 18 2 6 | 4 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +--------------+----------+--------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 458 1 | 6 9 2 | 35 458 458 | | 89 589 2 | 7 4 3 | 6 589 1 | +--------------+----------+--------------+ 31st) Naked single: 9 @ r2c9 +--------------+----------+--------------+ | 4 7 8 | 9 1 5 | 2 3 46 | | 5 6 3 | 2 7 4 | 8 1 9 | | 249 1 49 | 3 6 8 | 5 45 457 | +--------------+----------+--------------+ | 14 2 6 | 14 3 7 | 9 58 58 | | 189 89 5 | 18 2 6 | 4 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +--------------+----------+--------------+ | 6 3 49 | 5 8 1 | 7 2 4 | | 7 458 1 | 6 9 2 | 35 458 458 | | 89 589 2 | 7 4 3 | 6 589 1 | +--------------+----------+--------------+ 32nd) Hidden single: 9 @ r7c3 from values 49 Row 7 wise +-------------+----------+--------------+ | 4 7 8 | 9 1 5 | 2 3 46 | | 5 6 3 | 2 7 4 | 8 1 9 | | 249 1 4 | 3 6 8 | 5 45 457 | +-------------+----------+--------------+ | 14 2 6 | 14 3 7 | 9 58 58 | | 189 89 5 | 18 2 6 | 4 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +-------------+----------+--------------+ | 6 3 9 | 5 8 1 | 7 2 4 | | 7 458 1 | 6 9 2 | 35 458 458 | | 8 58 2 | 7 4 3 | 6 589 1 | +-------------+----------+--------------+ 33rd) Naked single: 4 @ r3c3 +-------------+----------+--------------+ | 0 7 8 | 9 1 5 | 2 3 46 | | 5 6 3 | 2 7 4 | 8 1 9 | | 29 1 4 | 3 6 8 | 5 5 57 | +-------------+----------+--------------+ | 14 2 6 | 14 3 7 | 9 58 58 | | 189 89 5 | 18 2 6 | 4 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +-------------+----------+--------------+ | 6 3 9 | 5 8 1 | 7 2 4 | | 7 458 1 | 6 9 2 | 35 458 458 | | 8 58 2 | 7 4 3 | 6 589 1 | +-------------+----------+--------------+ 33) Zero state: 0 @ Row 1 25th) Undo Trial & Error: -2 @ r1c7 from values 23 +--------------+-----------+----------------+ | 24 7 8 | 9 16 5 | 3 134 46 | | 5 6 3 | 2 17 4 | 8 19 79 | | 249 1 49 | 3 67 8 | 25 45 4567 | +--------------+-----------+----------------+ | 14 2 6 | 14 3 7 | 9 58 58 | | 189 89 5 | 18 2 6 | 4 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +--------------+-----------+----------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 458 1 | 6 9 2 | 35 3458 458 | | 89 589 2 | 7 4 3 | 6 589 1 | +--------------+-----------+----------------+ 25th) Hidden single: 2 @ r3c7 from values 25 Column 7 wise +--------------+-----------+----------------+ | 24 7 8 | 9 16 5 | 3 134 46 | | 5 6 3 | 2 17 4 | 8 19 79 | | 49 1 49 | 3 67 8 | 2 45 4567 | +--------------+-----------+----------------+ | 14 2 6 | 14 3 7 | 9 58 58 | | 189 89 5 | 18 2 6 | 4 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +--------------+-----------+----------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 458 1 | 6 9 2 | 35 3458 458 | | 89 589 2 | 7 4 3 | 6 589 1 | +--------------+-----------+----------------+ 26th) Hidden single: 2 @ r1c1 from values 24 Row 1 wise +--------------+-----------+----------------+ | 2 7 8 | 9 16 5 | 3 134 46 | | 5 6 3 | 2 17 4 | 8 19 79 | | 49 1 49 | 3 67 8 | 2 45 4567 | +--------------+-----------+----------------+ | 14 2 6 | 14 3 7 | 9 58 58 | | 189 89 5 | 18 2 6 | 4 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +--------------+-----------+----------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 458 1 | 6 9 2 | 35 3458 458 | | 89 589 2 | 7 4 3 | 6 589 1 | +--------------+-----------+----------------+ 27th) Hidden single: 5 @ r8c7 from values 35 Column 7 wise +--------------+-----------+--------------+ | 2 7 8 | 9 16 5 | 3 134 46 | | 5 6 3 | 2 17 4 | 8 19 79 | | 49 1 49 | 3 67 8 | 2 45 4567 | +--------------+-----------+--------------+ | 14 2 6 | 14 3 7 | 9 58 58 | | 189 89 5 | 18 2 6 | 4 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +--------------+-----------+--------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 48 1 | 6 9 2 | 5 348 48 | | 89 589 2 | 7 4 3 | 6 89 1 | +--------------+-----------+--------------+ 28th) Naked single: 3 @ r1c7 +--------------+-----------+--------------+ | 2 7 8 | 9 16 5 | 3 14 46 | | 5 6 3 | 2 17 4 | 8 19 79 | | 49 1 49 | 3 67 8 | 2 45 4567 | +--------------+-----------+--------------+ | 14 2 6 | 14 3 7 | 9 58 58 | | 189 89 5 | 18 2 6 | 4 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +--------------+-----------+--------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 48 1 | 6 9 2 | 5 348 48 | | 89 589 2 | 7 4 3 | 6 89 1 | +--------------+-----------+--------------+ 29th) Hidden single: 3 @ r8c8 from values 348 Row 8 wise +--------------+-----------+-------------+ | 2 7 8 | 9 16 5 | 3 14 46 | | 5 6 3 | 2 17 4 | 8 19 79 | | 49 1 49 | 3 67 8 | 2 45 4567 | +--------------+-----------+-------------+ | 14 2 6 | 14 3 7 | 9 58 58 | | 189 89 5 | 18 2 6 | 4 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +--------------+-----------+-------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 48 1 | 6 9 2 | 5 3 48 | | 89 589 2 | 7 4 3 | 6 89 1 | +--------------+-----------+-------------+ 30th) Hidden single: 5 @ r9c2 from values 589 Row 9 wise +-------------+-----------+-------------+ | 2 7 8 | 9 16 5 | 3 14 46 | | 5 6 3 | 2 17 4 | 8 19 79 | | 49 1 49 | 3 67 8 | 2 45 4567 | +-------------+-----------+-------------+ | 14 2 6 | 14 3 7 | 9 58 58 | | 189 89 5 | 18 2 6 | 4 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +-------------+-----------+-------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 48 1 | 6 9 2 | 5 3 48 | | 89 5 2 | 7 4 3 | 6 89 1 | +-------------+-----------+-------------+ 31st) Hidden single: 9 @ r5c2 from values 89 Column 2 wise +------------+-----------+-------------+ | 2 7 8 | 9 16 5 | 3 14 46 | | 5 6 3 | 2 17 4 | 8 19 79 | | 49 1 49 | 3 67 8 | 2 45 4567 | +------------+-----------+-------------+ | 14 2 6 | 14 3 7 | 9 58 58 | | 18 9 5 | 18 2 6 | 4 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +------------+-----------+-------------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 48 1 | 6 9 2 | 5 3 48 | | 89 5 2 | 7 4 3 | 6 89 1 | +------------+-----------+-------------+ 31) BUG+1: 4 @ r3c9 => -567 @ r3c9 +------------+-----------+-----------+ | 2 7 8 | 9 16 5 | 3 14 46 | | 5 6 3 | 2 17 4 | 8 19 79 | | 49 1 49 | 3 67 8 | 2 45 4 | +------------+-----------+-----------+ | 14 2 6 | 14 3 7 | 9 58 58 | | 18 9 5 | 18 2 6 | 4 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +------------+-----------+-----------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 48 1 | 6 9 2 | 5 3 48 | | 89 5 2 | 7 4 3 | 6 89 1 | +------------+-----------+-----------+ 32nd) Hidden single: 6 @ r3c5 from values 67 Row 3 wise +------------+-----------+-----------+ | 2 7 8 | 9 1 5 | 3 14 46 | | 5 6 3 | 2 17 4 | 8 19 79 | | 49 1 49 | 3 6 8 | 2 45 4 | +------------+-----------+-----------+ | 14 2 6 | 14 3 7 | 9 58 58 | | 18 9 5 | 18 2 6 | 4 7 3 | | 3 48 7 | 48 5 9 | 1 6 2 | +------------+-----------+-----------+ | 6 3 49 | 5 8 1 | 7 2 49 | | 7 48 1 | 6 9 2 | 5 3 48 | | 89 5 2 | 7 4 3 | 6 89 1 | +------------+-----------+-----------+ 32) Zero state: 7 @ Row 3 Error: Unsolvable Sudoku! # U2 # C25 # P214 # N15 # H25 # G1 # D1 # 0.000000

R. Jamil

by **P.O.** » Mon Nov 18, 2024 3:32 pm

rjamil wrote:how to check multi-digit POM?

we were discussing template procedure and i was answering this question and my answer is: you have to calculate the templates for each value in order to be able to combine them.
POM as you practice it has only a weak resolving power

this puzzle is solved with 2 combinations of size 3: (5 6 7) (4 5 8)
.78......5..2.4....1.3...........9.........733....916263.58.7......92.....27....1

Hidden Text: Show

Code: Select all: #VT: (4 2 2 39 10 8 2 20 6) Cells: nil nil nil nil (6 50) nil nil nil nil SetVC: ( n5r1c6 n6r5c6 n5r6c5 n4r9c5 n6r9c7 n6r8c4 n6r4c3 n6r2c2 n8r2c7 n5r5c3 n4r5c7 ) #VT: (4 2 2 5 5 2 2 7 3) Cells: nil nil nil nil nil nil nil nil nil Candidates:nil nil nil nil nil nil nil (28 31) (26 27) 24 7 8 9 16 5 23 134 46 5 6 3 2 17 4 8 19 79 249 1 49 3 67 8 25 45 4567 14 2 6 14 3 7 9 58 58 189 89 5 18 2 6 4 7 3 3 48 7 48 5 9 1 6 2 6 3 49 5 8 1 7 2 49 7 458 1 6 9 2 35 3458 458 89 589 2 7 4 3 6 589 1 75 candidates. 1: (5 6 7) 8 instances #VT: (4 2 2 5 4 2 2 7 3) Cells: nil nil nil nil nil nil nil nil nil Candidates:nil nil nil nil (27) nil nil nil nil 24 7 8 9 16 5 23 134 46 5 6 3 2 17 4 8 19 79 249 1 49 3 67 8 25 45 467 14 2 6 14 3 7 9 58 58 189 89 5 18 2 6 4 7 3 3 48 7 48 5 9 1 6 2 6 3 49 5 8 1 7 2 49 7 458 1 6 9 2 35 3458 458 89 589 2 7 4 3 6 589 1 74 candidates. 2: (4 5 8) 11 instances #VT: (4 2 2 3 4 2 2 5 3) Cells: nil nil nil nil nil nil nil nil nil Candidates:nil nil nil (26 72) nil nil nil nil nil EraseCC: ( n5r3c8 n8r4c8 n5r4c9 n8r8c9 n9r9c8 n1r2c8 n2r3c7 n4r7c9 n3r8c8 n8r9c1 n5r9c2 n3r1c7 n4r1c8 n6r1c9 n7r2c5 n9r2c9 n6r3c5 n7r3c9 n9r7c3 n4r8c2 n5r8c7 n2r1c1 n1r1c5 n4r3c3 n8r6c2 n4r6c4 n9r3c1 n1r4c4 n1r5c1 n9r5c2 n8r5c4 n4r4c1 ) 2 7 8 9 1 5 3 4 6 5 6 3 2 7 4 8 1 9 9 1 4 3 6 8 2 5 7 4 2 6 1 3 7 9 8 5 1 9 5 8 2 6 4 7 3 3 8 7 4 5 9 1 6 2 6 3 9 5 8 1 7 2 4 7 4 1 6 9 2 5 3 8 8 5 2 7 4 3 6 9 1

by **rjamil** » Mon Nov 18, 2024 6:18 pm

Hi P.O.,

P.O. wrote:we were discussing template procedure and i was answering this question and my answer is: you have to calculate the templates for each value in order to be able to combine them.
POM as you practice it has only a weak resolving power

Well, I assume that, just like single-digit POM, we must check each combination of digits, i.e., 2-digit combination, 3-digit combination, etc., at once against templates. If single-digit templates consist of valid patterns then same may be valid for multi-digit too.

this puzzle is solved with 2 combinations of size 3: (5 6 7) (4 5 8)
.78......5..2.4....1.3...........9.........733....916263.58.7......92.....27....1

But, is your program not detecting elimination/placement of single-digit POM? My program is unable to solve with single-digit elimination/placement POM.

R. Jamil

by **P.O.** » Tue Nov 19, 2024 4:11 am

combining templates has the effect of eliminating some and examining the templates in order to find assertions or eliminations of candidates is what you call single-digit POM

let's finish the resolution of this puzzle
consider the combination (1 5) and forms its instances: only 3 are possible

Code: Select all: 1: ((4 12 25 36 37 51 56 68 80) (1 16 22 36 42 48 56 68 80)) 5: ((5 16 20 35 37 49 60 66 81) (5 11 27 35 37 49 60 66 79) (5 11 25 35 37 49 60 66 81) (1 14 27 35 38 49 60 66 79) (1 14 25 35 38 49 60 66 81)) 5 . . 1 . . . . . 1 . . . 5 . . . . 1 . . . 5 . . . . . . 1 . 5 . . . . . 5 . . . . 1 . . . 5 . . . . 1 . . . . . . . . 1 . 5 . . . 1 . . . . 5 . . . 1 . . 5 . . . . . . . . . 5 1 . . . . . . . 5 1 . . . . . . . 5 1 1 5 . . . . . . . 5 . . . . 1 . . . 5 . . . . 1 . . . . . . 5 . 1 . . . . . 1 5 . . . . . . . 1 5 . . . . . . 1 . . . 5 . . . . 1 . . . 5 . . . . 1 . . . 5 . . . . . 5 . 1 . . . . . . 5 . 1 . . . . . . 5 . 1 . . . . . . . . . . 5 1 . . . . . . . 5 1 . . . . . . . . 1 5 1: (4 12 25 36 37 51 56 68 80) 1: (1 16 22 36 42 48 56 68 80) 1: (1 16 22 36 42 48 56 68 80) 5: (1 14 27 35 38 49 60 66 79) 5: (5 11 27 35 37 49 60 66 79) 5: (5 11 25 35 37 49 60 66 81)

none of these 2 templates for 5 are compatible with a template for 1, so they are eliminated
(5 16 20 35 37 49 60 66 81) (1 14 25 35 38 49 60 66 81)
checking the remaining templates removes 5 from cells 16 and 20 and an update of the grid solves the puzzle

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Code: Select all: .2...67..4..7.9.....9.23.6........51..89..3..97.......3...6.2.46.5.1...7......... #VT: (13 4 5 4 12 5 1 11 2) Cells: nil nil nil (30) nil nil nil nil nil SetVC: ( n4r4c3 n8r4c6 n5r7c6 n6r4c4 n8r7c4 n3r4c2 n5r6c4 n4r5c5 n2r8c4 n3r8c8 n3r9c4 n4r9c2 n4r8c6 n8r9c1 n9r7c8 n9r8c2 n1r7c2 n1r9c8 n3r2c9 n3r1c3 n2r2c8 ) #VT: (2 2 1 2 5 3 1 2 1) Cells: nil nil nil nil nil nil nil nil nil Candidates:nil nil nil nil nil nil nil nil nil 15 2 3 14 58 6 7 48 9 4 568 16 7 58 9 15 2 3 7 58 9 14 2 3 145 6 58 2 3 4 6 7 8 9 5 1 15 56 8 9 4 12 3 7 26 9 7 16 5 3 12 46 48 268 3 1 7 8 6 5 2 9 4 6 9 5 2 1 4 8 3 7 8 4 2 3 9 7 56 1 56 49 candidates. 1: (1 5): 3 instances 5..1.......1.5..........1.5.......5115..........5.1....1...5.....5.1..........51. 1...5.....5....1.....1....5.......515....1.....15......1...5.....5.1..........51. 1...5.....5....1.....1..5.........515....1.....15......1...5.....5.1...........15 ...1.......1............1..........11.............1....1...........1...........1. 1..............1.....1.............1.....1.....1.......1...........1...........1. ....5.....5...............5.......5.5...........5..........5.....5............5.. ....5.....5.............5.........5.5...........5..........5.....5..............5 5............5............5.......5..5..........5..........5.....5............5.. #VT: (2 2 1 2 3 3 1 2 1) Cells: nil nil nil nil nil nil nil nil nil Candidates: nil nil nil nil (16 20) nil nil nil nil EraseCC: ( n1r2c7 n8r3c2 n5r3c9 n6r9c9 n6r2c3 n4r3c7 n2r5c9 n1r6c3 n2r6c6 n6r6c7 n8r6c9 n5r9c7 n8r1c8 n5r2c2 n8r2c5 n1r3c4 n5r5c1 n6r5c2 n1r5c6 n4r6c8 n1r1c1 n4r1c4 n5r1c5 ) 1 2 3 4 5 6 7 8 9 4 5 6 7 8 9 1 2 3 7 8 9 1 2 3 4 6 5 2 3 4 6 7 8 9 5 1 5 6 8 9 4 1 3 7 2 9 7 1 5 3 2 6 4 8 3 1 7 8 6 5 2 9 4 6 9 5 2 1 4 8 3 7 8 4 2 3 9 7 5 1 6

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