The hardest sudokus (new thread)

Everything about Sudoku that doesn't fit in one of the other sections

Re: The hardest sudokus (new thread)

Postby denis_berthier » Wed Oct 05, 2022 5:11 am

.
I've long ago given up with the idea of defining a unique rating that would be able to rate all the puzzles and would encompass all the known patterns. I've also realised that such a requirement has no rational support and would have very little practical use.
As I said long ago, a rating system is always defined with some purpose in mind. SER has satisfied most solvers purposes for a long time (in spite of all its shortcomings). IMO, it still does.
When a new pattern is discovered, the first question is not "how to include it into some predefined rating system?", but "how does it simplify a puzzle wrt some rating?"
What might be interesting is to have a version of SER able to rate the Sukakus we obtain after applying anti-tridagon rules (and that still works on any platform). But this is only part of the question, because anti-tridagon rules and other rules may have to be intertwined to get a solution: see all my examples from mith database in the "Puzzles" forum.

As for the anti-tridagon patterns:
- any degenerated form (i.e. some of the 3 candidates missing in some cell) is in T&E(2), as I've shown in "the tridagon rule" thread; but it is also much harder to spot than the full form in T&E(3). I've tried to code the pattern, but it leads to combinatorial explosion;
- all the known puzzles in T&E(3) have the full non-degenerated form; we have an example where it degenerates during the resolution process; but we have no example of a puzzle with a degenerated form at the start (or at least, before applying any T&E(3) rule);
- as my classification results for the Wn+ORkWn rating show (also in "the Tridagon rule" thread), there is a huge scale of complexity for puzzles with the anti-tridagon pattern; so that the complexity of finding it is generally a neglectable part of the complexity of using it.

Notice that:
- for any k, W+ORkW IS a rating system and it is stable under Sudoku isomorphisms;
- but it cannot rate the puzzles in T&E(2) with a degenerated anti-tridagon pattern.
.
denis_berthier
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Re: The hardest sudokus (new thread)

Postby P.O. » Wed Oct 05, 2022 4:01 pm

templates is what sudokus are made of so as an indication of the difficulty of a puzzle, i would suggest the classification that solving by templates does; it is universal, all puzzles with unique solution are solvable by templates and therefore can be classified (for puzzles with multiple solutions it gives all the solutions).
it relies only on checking that a template for a value is compatible with at least one template for each of the other values, it is therefore independent of all existing or future rules;
the checking can be done progressively, first all the combinations of two templates, then all the combinations of three templates etc. and in parallel with the elimination of non-compatible templates the grid evolves by a very simple logic: for each value candidates that are not in any templates are eliminated and candidates that are in all the templates are set;
when following this procedure a puzzle is solved it has the classification of the combination of templates under consideration: solved with 2-templates 3-templates etc.

this puzzle "....5.7.....9.1.6.........52....8.16.......2..3....4.7.7..4......82.9...9.68....." by coloin with SER 11.7 would be classified as 4-templates, by curiosity i checked the combination (3 4 5 7), after it is considered the puzzle is solved with basic:
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initialization:
#VT: (68 22 249 70 87 30 17 21 35)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

1: (3 4 5 7)
#VT: (68 22 22 26 11 30 2 21 35)
Cells: nil nil (34) nil (53) nil (22 32 71 78) nil nil
SetVC: ( n7r3c4   n7r4c5   n3r4c7   n5r6c8   n7r8c8   n7r9c6
         n8r6c1 )

#VT: (68 22 46 26 14 30 2 11 35)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil (2 9 20 29 38) (55 57 61) nil nil nil nil

1346   12689  12349  346    5      2346   7      3489   12389           
3457   2458   23457  9      238    1      28     6      2348           
1346   12689  12349  7      2368   2346   1289   3489   5               
2      59     459    45     7      8      3      1      6               
14567  1569   14579  13456  1369   3456   89     2      89             
8      3      19     16     1269   26     4      5      7               
13     7      123    1356   4      356    12689  389    12389           
1345   145    8      2      136    9      156    7      134             
9      1245   6      8      13     7      125    34     1234           

intersections:
((((9 0) (5 7 6) (8 9)) ((9 0) (5 9 6) (8 9)))
 (((9 0) (4 2 4) (5 9)) ((9 0) (4 3 4) (4 5 9)))
 (((6 0) (6 4 5) (1 6)) ((6 0) (6 5 5) (1 2 6 9)) ((6 0) (6 6 5) (2 6)))
 ( n9r6c5   n2r6c6   n6r6c4   n1r6c3 ))

PAIR COL: ((5 5 5) (1 3)) ((9 5 8) (1 3)) 
(((2 5 2) (2 3 8)) ((3 5 2) (2 3 6 8)) ((8 5 8) (1 3 6)))

( n6r8c5   n6r7c7 )

PAIR ROW: ((2 5 2) (2 8)) ((2 7 3) (2 8)) 
(((2 2 1) (2 4 5 8)) ((2 3 1) (2 3 4 5 7)) ((2 9 3) (2 3 4 8)))

TRIPLET ROW: ((7 1 7) (1 3)) ((7 4 8) (1 3 5)) ((7 6 8) (3 5))
(((7 3 7) (2 3)) ((7 8 9) (3 8 9)) ((7 9 9) (1 2 3 8 9)))

( n2r7c3 )

intersection:
((((3 0) (7 1 7) (1 3)) ((3 0) (8 1 7) (1 3 4 5))))

PAIR COL: ((5 9 6) (8 9)) ((7 9 9) (8 9)) 
(((1 9 3) (1 2 3 8 9)))

TRIPLET COL: ((2 2 1) (4 5)) ((8 2 7) (1 4 5)) ((9 2 7) (1 4 5))
(((1 2 1) (1 2 6 8 9)) ((3 2 1) (1 2 6 8 9)) ((4 2 4) (5 9)) ((5 2 4) (5 6)))

( n9r4c2   n6r5c2 )

intersections:
((((1 0) (8 2 7) (1 4 5)) ((1 0) (9 2 7) (1 4 5)))
 ( n2r3c5   n8r3c2   n8r2c5   n2r1c2   n8r7c9   n9r7c8   n9r5c9
   n8r5c7   n3r3c8   n9r3c7   n4r3c3   n2r2c7   n8r1c8   n3r1c4
   n9r1c3   n5r5c4   n7r5c3   n4r4c4   n5r4c3   n6r3c6   n1r3c1
   n4r1c6   n6r1c1   n3r5c6   n4r5c1   n1r8c7   n5r8c1   n5r9c7
   n4r8c2   n3r2c3   n7r2c1   n1r1c9   n2r9c9   n1r9c2   n5r2c2
   n4r2c9   n3r8c9   n1r5c5   n4r9c8   n3r9c5   n1r7c4   n5r7c6
   n3r7c1 ))

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

what is the usefulness of this classification, at least it makes you ask questions like this: does it make sense for SE to give the next two puzzles the same rating of 11.9s when the first is solved with 3-templates and the second with 5-templates?
the first of the following list and eleven hardest #4
http://forum.enjoysudoku.com/post325781.html#p325781
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27...........8.......75.2..43.....177.9...32..12...9.49.7.314..14........23..91..

#VT: (6 5 14 18 68 432 4 62 10)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2
#VT: (6 5 14 18 66 391 4 60 10)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2 3
#VT: (6 4 14 18 42 240 3 39 10)
Cells: nil (63) nil nil nil nil nil nil nil
SetVC: ( n2r7c9 )

#VT: (6 4 14 18 42 240 3 39 10)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil (49 71 72) (71 72) (71) (49 71 72) nil
2
#VT: (6 4 14 18 33 196 3 31 10)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2 3
#VT: (6 4 14 18 15 56 3 14 10)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil (8 17) (8 17 26 31) nil (8 26 31) nil
2 3
#VT: (6 4 14 18 10 23 3 9 10)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil (3 9 12 21 40 68) nil (45) nil
2
#VT: (6 4 14 18 10 19 3 9 10)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2 3
#VT: (6 4 13 18 6 4 3 7 10)
Cells: nil nil nil nil nil (16 20 45) nil nil nil
SetVC: ( n6r2c7   n6r3c2   n6r5c9   n7r8c7   n7r2c8   n9r2c2
         n2r8c5   n7r9c5   n7r6c6   n6r6c5   n9r4c5   n3r6c4
         n4r9c4   n6r4c3   n6r9c1   n9r1c4   n6r7c8   n6r1c6
         n6r8c4 )

#VT: (6 2 4 4 6 1 1 7 2)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil (18 26 27) nil (31) nil nil nil nil
EraseCC: ( n2r4c4   n1r2c4   n5r2c9   n8r9c9   n4r1c5   n8r1c7
           n3r1c8   n1r1c9   n3r2c1   n4r2c3   n2r2c6   n8r3c1
           n1r3c3   n3r3c6   n9r3c9   n5r4c7   n1r5c5   n5r6c1
           n8r6c8   n9r8c8   n3r8c9   n5r9c8   n5r1c3   n4r3c8
           n8r4c6   n8r5c2   n5r5c4   n4r5c6   n5r7c2   n8r7c4
           n8r8c3   n5r8c6 )
2 7 5   9 4 6   8 3 1
3 9 4   1 8 2   6 7 5
8 6 1   7 5 3   2 4 9
4 3 6   2 9 8   5 1 7
7 8 9   5 1 4   3 2 6
5 1 2   3 6 7   9 8 4
9 5 7   8 3 1   4 6 2
1 4 8   6 2 5   7 9 3
6 2 3   4 7 9   1 5 8

(2 2 3 2 2 3 2 3 2 2 3)


Hidden Text: Show
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eleven #4
..3..6.8....1..2......7...4..9..8.6..3..4...1.7.2.....3....5.....5...6..98.....5.

#VT: (51 31 44 37 26 8 38 20 133)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2 3
#VT: (51 31 40 37 26 8 38 20 123)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2 3 4
#VT: (37 31 40 34 26 8 37 20 118)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2 3 4 5
#VT: (14 8 36 12 24 7 12 20 92)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:(20) (20 56) nil (48 62 76) (1) nil (72) nil nil
2
#VT: (14 8 29 12 24 7 12 20 77)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil (53) nil nil nil nil nil nil
2 3
#VT: (14 8 28 12 24 7 12 20 77)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2 3 4
#VT: (14 8 26 12 22 7 12 20 53)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2 3 4
#VT: (14 8 25 12 22 7 12 20 53)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2 3 4 5
#VT: (13 8 14 12 21 7 11 17 14)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil (78) nil (2) nil nil nil (4 24 62)
2
#VT: (13 8 14 12 21 7 11 17 13)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2 3
#VT: (13 8 14 12 19 7 11 17 13)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2 3 4
#VT: (13 7 13 9 12 7 11 16 13)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil (57 81) (36) (52 56) (43) nil nil nil nil
2 3 4
#VT: (13 7 11 9 12 6 10 14 13)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil (50) nil nil (58) (40) nil nil
2 3 4
#VT: (13 7 11 9 10 6 10 14 13)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil (19 52) nil nil nil nil
2 3 4 5
#VT: (9 6 6 7 6 4 5 10 3)
Cells: nil nil nil nil nil (56) nil nil (9 20)
SetVC: ( n9r1c9   n9r3c2   n6r7c2 )

#VT: (9 6 6 7 6 4 5 10 14)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:(19 64) (37 71) nil (11) (7 14 28) nil (1 17 43 58 61 67 71 79) nil nil
EraseCC: ( n5r2c2   n3r2c8   n1r3c8   n7r1c7   n5r3c7 )

#VT: (12 6 18 7 5 4 11 10 14)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:(59 77) nil nil nil nil nil nil nil nil
2
#VT: (12 6 17 7 5 4 11 10 13)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2 3
#VT: (8 6 9 7 3 4 7 8 11)
Cells: nil nil nil nil (54) nil nil nil nil
SetVC: ( n5r6c9   n5r5c1 )

#VT: (8 6 9 7 2 4 7 8 11)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil (79) nil nil nil nil nil (44)
2
#VT: (8 6 9 7 2 3 7 6 11)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil (61 68) nil
2
#VT: (8 4 7 7 2 3 7 6 11)
Cells: nil (44) nil nil nil nil nil nil nil
SetVC: ( n2r5c8   n7r7c8   n3r9c9   n7r4c9   n7r5c6   n7r9c4
         n6r9c5   n6r5c4   n7r8c1   n7r2c3   n9r5c7   n8r5c3
         n4r6c8   n9r8c8   n3r4c7   n8r6c7   n5r4c4   n1r4c5
         n9r6c5   n3r6c6   n4r1c4   n8r2c5   n9r2c6   n3r3c4
         n2r3c6   n2r7c5   n8r7c9   n8r8c4   n3r8c5   n2r8c9
         n5r1c5   n4r2c1   n6r3c3   n2r4c1   n4r4c2   n1r6c3
         n4r7c3   n9r7c4   n1r7c7   n1r8c2   n4r8c6   n2r9c3
         n1r9c6   n4r9c7   n1r1c1   n2r1c2   n8r3c1   n6r6c1 )
1 2 3   4 5 6   7 8 9
4 5 7   1 8 9   2 3 6
8 9 6   3 7 2   5 1 4
2 4 9   5 1 8   3 6 7
5 3 8   6 4 7   9 2 1
6 7 1   2 9 3   8 4 5
3 6 4   9 2 5   1 7 8
7 1 5   8 3 4   6 9 2
9 8 2   7 6 1   4 5 3

(2 3 2 3 4 2 3 4 5 2 2 3 2 3 4 2 3 4 2 3 4 5 2 2 3 2 3 4 2 3 4 2 3 4 2 3 4 5 2
 2 3 2 2)
P.O.
 
Posts: 591
Joined: 07 June 2021

Re: The hardest sudokus (new thread)

Postby mith » Mon Oct 17, 2022 7:30 pm

Finally updated the te3 database with hendrik's new puzzles. Of the 11.9s, six are depth 3, and of those three have expanded forms which also rate 11.9 (the others are 11.8). Additionally, one of the 11.8's expanded forms rates 11.9. Presumably the differences are due to the same issue as with Loki and friends, but could be uniqueness related as well I guess.

(The 11.9 expanded forms are from the 1st, 3rd, 8th, and 16th minimals on hendrik's list.)

Running my other scripts with the new puzzles until they stabilize again.
mith
 
Posts: 889
Joined: 14 July 2020

Re: The hardest sudokus (new thread)

Postby mith » Mon Oct 17, 2022 7:36 pm

P.O. wrote:templates is what sudokus are made of so as an indication of the difficulty of a puzzle, i would suggest the classification that solving by templates does; it is universal, all puzzles with unique solution are solvable by templates and therefore can be classified (for puzzles with multiple solutions it gives all the solutions).
[snip]
what is the usefulness of this classification, at least it makes you ask questions like this: does it make sense for SE to give the next two puzzles the same rating of 11.9s when the first is solved with 3-templates and the second with 5-templates?


I don't know why anyone would expect the SE rating to correlate with a template based rating; if the question is whether the two puzzles should be considered the same difficulty, probably not (though I personally wouldn't base a rating on templates). But both are of course an 11.9 by the rules SE uses (absent any bugs that might be affecting it, which we know exist). It's like asking if we should call two things blue if one is a cat and the other is a dog. The different categorizations don't have anything to do with each other.

There are of course also other objective ratings one could use - the T&E depth is one we have been focused on lately - and it's unlikely all such systems would agree on a "hardest" puzzle anyway.
mith
 
Posts: 889
Joined: 14 July 2020

Re: The hardest sudokus (new thread)

Postby P.O. » Tue Oct 18, 2022 10:00 am

hi Mith thank you for your reply.

what sense does it make to use SE to rate puzzles that are solved by techniques that are not in SE? absolutely none. no wonder they seem so difficult. it is like using a hammer for the work of a saw.
rating systems are a collection of solving techniques developed over time and must be updated as new techniques are found or they will become obsolete, and the hierarchy of rules they employ must accordingly be revised.

solving with templates consists in directly manipulating what sudokus are: 9 compatible templates because the solutions of puzzles are sought by verifying what makes them valid sudokus: compatibility of templates and the resolution process is logically hierarchical,

it starts by calculating the possible templates for each value, and that can be enough for solving some puzzles, then from them the combinations of 2 templates are formed, compatible meaning dijoint the intersection operator is used, and this is the only time it is used because after that all the combinations are formed by combining the previous hence an objective and invariable hierarchy of difficulties as it is a characteristic of the puzzle not succeptible to vary over time.

with the rating systems it is the rules used that make the difficulty of the puzzles, adding or removing them, changing their hierarchy modifies this difficulty, with templates it is an inherent property of the puzzles.
P.O.
 
Posts: 591
Joined: 07 June 2021

Re: The hardest sudokus (new thread)

Postby mith » Wed Oct 19, 2022 2:19 pm

There's nothing "wrong" with rating puzzles with templates. I just don't know that it's all that useful of a measure for human difficulty.

SE, for all its flaws, does a reasonable job of separating levels of difficulty for a human solver, with the (big) caveat that some puzzles have techniques available which shortcut that difficulty. There are of course other rating systems which do much the same thing, just as there are other objective metrics one can use. T&E depth we've mentioned of course, as that is the driver for the current search (though obviously T&E(singles) only has 4 possible ratings, even less than using templates).

Another is the "fractional dimension", which I haven't talked about here but is something Mitchell Lee (of the "Miracle Sudoku") studied. The idea is that one can generalize sudoku to allow for fractions of digits per cell (say r1c1 contains 1/2 of a 4, 1/3 of an 8, and 1/6 of a 3) such that each cell contains a total of 1 digit (possibly fractions of multiple digits) and each house contains a total of 1 of each digit (but possibly spread across multiple cells). For any uniquely solvable classic sudoku, the same puzzle treated as a fractional sudoku either also has one solution (in which case the fractional dimension is 0 - this corresponds to being solvable by rank 0 techniques, I think?) or infinitely many, and one can talk about the dimension of the polytope formed by the fractional solutions. (Mitchell was working on a video or paper on this, but I don't think he ever finished. He had written some code to calculate the fractional dimension and I ran it on some of the ph puzzles last year.)

And of course there are other measures that can be defined. My point is that there is not one "right" or "inherent" way to define difficulty, and rating systems have different goals. SE is used because it has been used for a long time and is (relatively) quick and convenient for getting a rough measure of difficulty. Obviously that doesn't mean all 11.9 puzzles are the "same" difficulty in any other context - but it's also not the case that all 3-template puzzles are the "same" in any other context. Ratings mean what they are defined to mean.
mith
 
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Joined: 14 July 2020

Re: The hardest sudokus (new thread)

Postby mith » Wed Oct 19, 2022 2:25 pm

mith wrote:Finally updated the te3 database with hendrik's new puzzles. Of the 11.9s, six are depth 3, and of those three have expanded forms which also rate 11.9 (the others are 11.8). Additionally, one of the 11.8's expanded forms rates 11.9. Presumably the differences are due to the same issue as with Loki and friends, but could be uniqueness related as well I guess.

(The 11.9 expanded forms are from the 1st, 3rd, 8th, and 16th minimals on hendrik's list.)

Running my other scripts with the new puzzles until they stabilize again.


Following up on this, the scripts are still running with the minimizer script lagging behind at the moment (about 4k expanded forms not minimized yet). Up to 172 expanded forms with SER 11.8+ (32 more than before adding hendrik's puzzles, I think the majority are directly from expanding those but a few new ones). In total, 770k expanded forms and 3.48M minimals.
mith
 
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Re: The hardest sudokus (new thread)

Postby P.O. » Thu Oct 20, 2022 5:33 pm

rating systems show with which combination of their rules you can solve a puzzle, and they prioritize their rules on an estimate of the difficulty in applying them, but i dont think that past a certain level they are useful for human solver if not to tell them that this level of difficulty is well beyond their ability.
the interesting point they make is that any puzzle can be solved without guessing.

a SER 11.8 rating says: to solve this puzzle you need to apply at least once this rule: Nested Forcing Chains (+ Dynamic Forcing Chains), ED=10.5 + 1.3 (257-384 nodes), which i dont think is within the reach of human solver,
but if a technique like the TH, which a human solver can mastered, allow to manually solve this puzzle then it shows than SE, as it is, is no longer the right tool to rate it.

T&E(n) is not a measure of difficulty, the following 2 puzzles, not in T&E(2), which from the point of view of templates are identical except for the permutation of 6 and 8: same configuration at the beginning, same templates reduction, equivalent eliminations, are in 4-template which doesn't put them among the hardest, far from it.
solved directly with combinations of 4, only those that make eliminations are listed:
http://forum.enjoysudoku.com/post317678.html#p317678
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........1.....2.......3..45..6.......71.8....23..67..8.827..1..6...23...7.381.6..

#VT: (4 10 14 68 73 6 14 7 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

5: (1 2 3 6)
#VT: (3 10 14 68 73 6 14 7 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:(31) nil nil nil nil nil nil nil nil

16: (1 4 6 8)
#VT: (3 10 14 65 73 6 14 7 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

17: (1 2 4 6)
#VT: (3 10 14 63 73 6 14 7 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

19: (1 5 6 8)
#VT: (3 10 14 63 69 6 14 7 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

22: (1 3 4 6)
#VT: (3 10 14 57 69 6 14 7 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

31: (1 2 3 4)
#VT: (3 10 14 37 69 6 14 7 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil (31 40) nil nil nil nil nil

33: (1 3 5 6)
#VT: (3 10 14 37 63 6 14 7 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

36: (1 5 7 8)
#VT: (3 10 14 37 59 6 14 7 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

44: (1 2 3 5)
#VT: (3 10 14 37 37 6 14 7 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil (31 40) nil nil nil nil

59: (1 2 4 5)
#VT: (3 10 14 36 36 6 14 7 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

60: (2 4 5 6)
#VT: (3 10 14 35 35 6 14 7 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

62: (1 4 5 7)
#VT: (3 10 14 35 34 6 14 7 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

66: (4 5 7 8)
#VT: (3 10 14 33 33 6 14 7 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

67: (3 4 5 8)
#VT: (3 10 14 32 32 6 14 7 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

71: (1 6 8 9)
#VT: (3 10 14 32 32 6 14 7 380)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

72: (1 2 6 9)
#VT: (3 10 14 32 32 6 14 7 342)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

73: (1 2 8 9)
#VT: (3 10 14 32 32 6 14 7 341)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

76: (1 3 6 9)
#VT: (3 10 14 32 32 6 14 7 327)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

77: (1 7 8 9)
#VT: (3 10 14 32 32 6 14 7 291)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

82: (1 2 3 9)
#VT: (3 10 14 32 32 6 14 7 208)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil (31 40)

83: (2 6 7 9)
#VT: (3 10 14 32 32 6 14 7 207)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

85: (2 7 8 9)
#VT: (3 10 14 32 32 6 14 7 204)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

91: (1 4 6 9)
#VT: (3 10 14 32 32 6 14 7 198)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

94: (1 2 4 9)
#VT: (3 10 14 32 32 6 14 7 197)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

95: (1 5 6 9)
#VT: (3 10 14 32 32 6 14 7 194)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

96: (2 4 6 9)
#VT: (3 10 14 32 32 6 14 7 189)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

97: (1 5 8 9)
#VT: (3 10 14 32 32 6 14 7 187)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

100: (1 4 7 9)
#VT: (3 10 14 32 32 6 14 7 178)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

102: (1 2 5 9)
#VT: (3 10 14 32 31 6 14 7 178)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

105: (4 7 8 9)
#VT: (3 10 14 32 31 6 14 7 173)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

107: (2 5 6 9)
#VT: (3 10 14 32 31 6 14 7 172)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

109: (1 5 7 9)
#VT: (3 10 14 32 31 6 14 7 170)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

115: (5 7 8 9)
#VT: (3 10 14 32 31 6 14 7 168)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

117: (3 4 7 9)
#VT: (3 10 14 32 31 6 14 7 167)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

121: (1 4 5 9)
#VT: (1 10 14 5 6 6 14 7 2)
Cells: (10 24 35 49) nil nil nil nil nil nil nil (19 29 42 53 59 66 81)
SetVC: ( n1r2c1   n9r3c1   n1r3c6   n9r4c2   n1r4c8   n9r5c6
         n1r6c4   n9r6c8   n9r7c5   n9r8c3   n9r9c9   n6r3c4
         n2r3c2   n2r9c8   n3r1c1   n8r1c6   n2r1c7   n9r2c7
         n9r1c4 )

#VT: (1 2 4 5 6 3 14 2 1)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil (36 44 45) (34 36) (34) nil nil nil nil

122: (4 5 6 9)
#VT: (1 2 4 3 4 1 14 2 1)
Cells: nil nil nil nil nil (8 11 45) nil nil nil
SetVC: ( n6r1c8   n6r2c2   n5r5c8   n6r5c9   n4r6c7   n3r7c8
         n4r7c9   n7r8c9   n3r2c9   n2r4c9   n4r5c1   n3r5c7
         n5r6c3   n5r7c1   n8r8c8   n4r9c2   n5r9c6   n5r1c2
         n7r2c8   n8r3c7   n3r4c4   n4r4c6   n7r4c7   n2r5c4
         n4r8c4   n5r8c7   n5r2c4   n4r2c5   n7r3c3   n5r4c5
         n4r1c3   n7r1c5   n8r2c3 )
3 5 4   9 7 8   2 6 1
1 6 8   5 4 2   9 7 3
9 2 7   6 3 1   8 4 5
8 9 6   3 5 4   7 1 2
4 7 1   2 8 9   3 5 6
2 3 5   1 6 7   4 9 8
5 8 2   7 9 6   1 3 4
6 1 9   4 2 3   5 8 7
7 4 3   8 1 5   6 2 9



........1.....2.......3..45..1.23....267.81..73.61.8...17.6.....8.......2.3.87..6

#VT: (4 10 14 68 73 7 14 6 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

5: (1 2 3 8)
#VT: (3 10 14 68 73 7 14 6 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:(67) nil nil nil nil nil nil nil nil

16: (1 4 6 8)
#VT: (3 10 14 65 73 7 14 6 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

17: (1 2 4 8)
#VT: (3 10 14 63 73 7 14 6 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

19: (1 5 6 8)
#VT: (3 10 14 63 69 7 14 6 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

22: (1 3 4 8)
#VT: (3 10 14 57 69 7 14 6 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

31: (1 2 3 4)
#VT: (3 10 14 37 69 7 14 6 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil (58 67) nil nil nil nil nil

33: (1 3 5 8)
#VT: (3 10 14 37 63 7 14 6 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

36: (1 5 6 7)
#VT: (3 10 14 37 59 7 14 6 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

44: (1 2 3 5)
#VT: (3 10 14 37 37 7 14 6 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil (58 67) nil nil nil nil

59: (1 2 4 5)
#VT: (3 10 14 36 36 7 14 6 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

60: (2 4 5 8)
#VT: (3 10 14 35 35 7 14 6 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

62: (1 4 5 7)
#VT: (3 10 14 35 34 7 14 6 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

66: (4 5 6 7)
#VT: (3 10 14 33 33 7 14 6 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

67: (3 4 5 6)
#VT: (3 10 14 32 32 7 14 6 410)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

71: (1 6 8 9)
#VT: (3 10 14 32 32 7 14 6 380)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

72: (1 2 8 9)
#VT: (3 10 14 32 32 7 14 6 342)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

73: (1 2 6 9)
#VT: (3 10 14 32 32 7 14 6 341)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

76: (1 3 8 9)
#VT: (3 10 14 32 32 7 14 6 327)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

77: (1 6 7 9)
#VT: (3 10 14 32 32 7 14 6 291)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

82: (1 2 3 9)
#VT: (3 10 14 32 32 7 14 6 208)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil (58 67)

83: (2 7 8 9)
#VT: (3 10 14 32 32 7 14 6 207)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

85: (2 6 7 9)
#VT: (3 10 14 32 32 7 14 6 204)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

91: (1 4 8 9)
#VT: (3 10 14 32 32 7 14 6 198)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

94: (1 2 4 9)
#VT: (3 10 14 32 32 7 14 6 197)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

95: (1 5 8 9)
#VT: (3 10 14 32 32 7 14 6 194)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

96: (2 4 8 9)
#VT: (3 10 14 32 32 7 14 6 189)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

97: (1 5 6 9)
#VT: (3 10 14 32 32 7 14 6 187)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

100: (1 4 7 9)
#VT: (3 10 14 32 32 7 14 6 178)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

102: (1 2 5 9)
#VT: (3 10 14 32 31 7 14 6 178)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

105: (4 6 7 9)
#VT: (3 10 14 32 31 7 14 6 173)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

107: (2 5 8 9)
#VT: (3 10 14 32 31 7 14 6 172)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

109: (1 5 7 9)
#VT: (3 10 14 32 31 7 14 6 170)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

115: (5 6 7 9)
#VT: (3 10 14 32 31 7 14 6 168)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

117: (3 4 7 9)
#VT: (3 10 14 32 31 7 14 6 167)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil

121: (1 4 5 9)
#VT: (1 10 14 5 6 7 14 6 2)
Cells: (10 24 71 76) nil nil nil nil nil nil nil (19 29 41 54 60 66 80)
SetVC: ( n1r2c1   n9r3c1   n1r3c6   n9r4c2   n9r5c5   n9r6c9
         n9r7c6   n9r8c3   n1r8c8   n1r9c4   n9r9c8   n8r3c4
         n2r3c3   n2r6c8   n3r1c1   n6r1c6   n2r1c7   n9r2c7
         n9r1c4 )

#VT: (1 2 4 5 6 2 14 3 1)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil (62 63 72) (70 72) (70) nil nil nil nil

122: (4 5 8 9)
#VT: (1 2 4 3 4 2 14 1 1)
Cells: nil nil nil nil nil nil nil (8 12 63) nil
SetVC: ( n8r1c8   n8r2c3   n5r7c8   n8r7c9   n4r9c7   n3r5c8
         n4r5c9   n4r7c1   n3r7c7   n7r8c7   n2r8c9   n5r9c2
         n6r3c7   n5r4c7   n7r4c9   n5r5c1   n4r6c3   n5r6c6
         n2r7c4   n3r8c4   n4r8c6   n5r1c3   n7r2c8   n3r2c9
         n7r3c2   n4r4c4   n6r4c8   n5r8c5   n4r1c2   n7r1c5
         n6r2c2   n5r2c4   n4r2c5 )
3 4 5   9 7 6   2 8 1
1 6 8   5 4 2   9 7 3
9 7 2   8 3 1   6 4 5
8 9 1   4 2 3   5 6 7
5 2 6   7 9 8   1 3 4
7 3 4   6 1 5   8 2 9
4 1 7   2 6 9   3 5 8
6 8 9   3 5 4   7 1 2
2 5 3   1 8 7   4 9 6

P.O.
 
Posts: 591
Joined: 07 June 2021

Re: The hardest sudokus (new thread)

Postby eleven » Fri Oct 21, 2022 10:52 am

Remember this manual solution of a puzzle not in gT&E(2) a half year ago. Since then it was clear, that both SE and T&E(n) at least have to be expanded to be taken serious as (hardest) sudoku rating systems.
Note that this problem did not arise before the discovery of the TH pattern. Only a few puzzles in the hardest list could be simplified/solved with (much more) complex techniques (like SK loop, Exocets, MSLS), which are manually applicable.

I agree with mith, that templates can be a valuable classification for puzzles, but have little to do with the hardness for manual solving.
eleven
 
Posts: 2938
Joined: 10 February 2008

Re: The hardest sudokus (new thread)

Postby denis_berthier » Fri Oct 21, 2022 11:19 am

.
But then consider also this puzzle: http://forum.enjoysudoku.com/19828-in-mith-s-t-e-3-min-expands-t40425.html
Even after applying rules in W9+OR5W9, it remains in T&E(2).

In and of itself, the anti-tridagon pattern doesn't define a rating level.

Even if you consider all the possible Trdagon-related rules, they appear almost never in unbiased puzzles. They can' be the basis of any rating. For sk-loops and for tradigons alike, the best one can say is: they bring classification XX from level xx1 down to level xx2 - whatever you choose for XX.
denis_berthier
2010 Supporter
 
Posts: 3536
Joined: 19 June 2007
Location: Paris

Re: The hardest sudokus (new thread)

Postby eleven » Fri Oct 21, 2022 5:14 pm

Yes, TH (tridagon) is a pattern (like oddagons/UR's/MUG's etc.), which can be used to simplify a puzzle, in some cases very much, in others very little. It does not define a certain rating of a puzzle, but you can define a low rating for the pattern itself, because it is very easy to spot.
eleven
 
Posts: 2938
Joined: 10 February 2008

Re: The hardest sudokus (new thread)

Postby eleven » Fri Oct 21, 2022 8:53 pm

One more note: In my eyes the really hardest puzzles are the most annoying and boring stuff for manual solvers.
eleven
 
Posts: 2938
Joined: 10 February 2008

Re: The hardest sudokus (new thread)

Postby P.O. » Sat Oct 22, 2022 3:51 pm

http://forum.enjoysudoku.com/post318378.html#p318378
rules like the tridagon should not be considered any different from any other rule, they are of the form condition => action, they appear in the resolution path, they are assigned a level of complexity which places them in the hierarchy of rules, they are no auxiliary techniques used to reduce the difficulty of a puzzle

if the tridagon rule is the highest rated rule used to solve a puzzle, then the puzzle should be assigned that difficulty level, it doesn't shortcut or lower a difficulty the puzzle would have without applying it.

Loki (SER 11.9) is in 4-template
#19828 in mith's T&E(3) min-expands (SER 11.1) is in 5-template
P.O.
 
Posts: 591
Joined: 07 June 2021

Re: The hardest sudokus (new thread)

Postby eleven » Sat Oct 22, 2022 10:12 pm

I don't understand you. A tridagon is not a rule, but an impossible pattern like an oddagon. If it has a single extra candidate in a grid, it must be true, so you get a number, which simplifies the puzzle (solving) more or less. If it has more than one extra candidates, at least one of them must be true, which can be helpful either.
How high tridagon moves are placed in the hierarchy does not depend on, how hard it is to prove it or to resolve it with other techniques, but only how hard it is to spot it. Personally i would rate the step to place a single extra candidate of a tridagon not harder than ER 7.0.
eleven
 
Posts: 2938
Joined: 10 February 2008

Re: The hardest sudokus (new thread)

Postby P.O. » Sun Oct 23, 2022 5:56 am

my point is that continuing to use SE without incorporating newly developed techniques falsifies the perception of what a difficult puzzle is.
P.O.
 
Posts: 591
Joined: 07 June 2021

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