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 » Sun Oct 23, 2022 6:24 am

.
1) T&E-depth is a universal classification (not rating) of all the instances of all the finite binary CSPs.
Each level of T&E requires specific but universal resolution rules (chain rules) and allows precise sub-classifications and/or ratings;
- T&E(1) sub-classified and rated by braids[n];
- T&E(2) sub-classified by Bn-B (computable) and rated by B-braids[n] (impractical);
- ...
In the most general context, it is much more useful to use pairs (T&E-depth, sub-classification within this T&E level) than a single rating for all.

2) 9x9 Sudoku puzzles in T&E(2) are a very small fraction of all the 9x9 Sudoku puzzles (1 in a few millions);
9x9 Sudoku puzzles in T&E(3) are probably also a very small fraction of all the 9x9 Sudoku puzzles in T&E(≥2).
Why would one care to define a unique general rating based on such exceptional cases?

3) Exotic patterns (such as sk-loops, J-Exocets, anti-tridagons ...) are very brittle: if some of the defining candidates disappears, what remains is a degenerated form of the pattern. [The situation is similar to some uniqueness rules.]
The degenerated form may be totally impossible to find (for a manual solver and for a computer as well). This is true of the above 3 examples.
And for the anti-tridagon case, this is true in spite of the non-degenerated form being in T&E(3) and the degenerated one in T&E(2).

4) CSP-Rules deals with this degeneracy problem by making the anti-tridagon ORk-relations ultra-persistent, thus allowing to "keep in mind" the fact that an anti-tridagon pattern has been identified and that it may loose some of its defining candidates and/or some of its guardians during the resolution process.
This works well for puzzles in mith's collection because all the known 9x9 T&E(3) puzzles have a full anti-tridagon pattern at the start.

5) At this point, nobody knows if this is true of all the T&E(3) 9x9 puzzles and nobody knows if there are puzzles in T&E(2) with a degenerated form of the anti-tridagon pattern that would not derive from any non-degenerated form.
The fact is, T&E(2) is a very ill-known land. The "hardest collection" is strongly biased towards puzzles with specific patterns (sk-loops...), but we have no idea of any set of patterns that would allow to simplify all the puzzles in T&E(2) in the same way as anti-tridagons may simplify puzzles in T&E(3).
I'm not aware of any systematic search for patterns that would make a puzzle potentially be in T&E(2). Such search would be much more interesting to me than talks about defining a unique rating.

6) For puzzles in T&E(3) (with the still limited knowledge we have of this level), for any k, Wn+ORkWn is a good rating system. You can change Wn to gWn or to Bn or to gBn; you'll get slightly different ratings. Or you can change Wn to FWn. Or you can use Wn+ORkWn+ORkFWn. See the tables I've given in the "tridagon rule" thread (http://forum.enjoysudoku.com/the-tridagon-rule-t39859.html). They clearly show what can be hoped for each value of k and n.

7) About defining the complexity of the tridagon rule.
My first approach was to say: it is based on 12 CSP-Variables, so that it must be 12. However, mith's additional criterion (3 digits + a block such that the 3 digits are not decided in any other block) drastically simplifies the search. Once such a block has been found, the rest is mere checking of the other conditions.
[The absence of such an additional criterion is also what makes the degenerated forms impossible to find.]
As a result, I've now granted the anti-tridagon pattern complexity 3: it will be found immediately after the Subsets[3]. If you want to revert to the first view (complexity 12), SudoRules has a control variable to allow this (but the default is now 3) - see the forthcoming release.
There's another reason for detecting the anti-tridagon pattern as soon as possible: prevent it from degenerating before being found.

8) Complexity verus priority
Because the default strategy in CSP-Rules is simplest-first, the priority of a rule is normally defined inversely to its complexity. But this can easily be changed and I could have granted the anti-tridagon detection rule a high priority while keeping its complexity 12.
The problem with this would be, most puzzles with this pattern would be granted the same complexity 12. But in reality, the hard part of solving is in the Wn+ORkWn.
denis_berthier
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Location: Paris

Re: The hardest sudokus (new thread)

Postby eleven » Mon Oct 24, 2022 1:56 pm

If we talk about random puzzles, we only need a small amount of solution techniques, the hardest being chains (however they are defined in detail), to get a rating, which has high correlation to what, let's say, is common sense about hardness for manual solving.
Other techniques, which only are applicable in (relatively) very rare puzzles, for the ones are salt and fun in solving sudokus, for the others uninteresting exceptional cases.

But when it goes to answer the popular question "what are the hardest sudokus ?", we talk about exceptional cases by definition.
champagne had carefully called his collection a list of potentially hardest sudokus, and back in 2012 he made some efforts to determine, how many of them could be solved easier with so-called exotic patterns, and therefore can be argued to be removed from the list. Because the patterns are rather complex, naturally there are different opinions, which of them are accepted as manually applicable, or what rating (position in hierarchy) they deserve.

I say this to point out, that much serious work had been done in order to answer that popular question - and to show, what difficulties arise, if you want to do that correctly. It is definitely a question, which cannot be answered in a clear or obvious way, or by a static rating system.

Coming to TH (tridagons) the things are much simpler. Since both spotting and using them is easy (at least in the 1 extra candidate case), a big part of the now high rated puzzles can be solved also by non experienced manual solvers (as soon as you have shown them the trick). So it's no question for me, that this pattern has to be integrated in any rating system, which claims to be able to rate also very hard puzzles.
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Joined: 10 February 2008

Re: The hardest sudokus (new thread)

Postby champagne » Mon Oct 24, 2022 2:43 pm

I am fully in line with eleven remarks. When I worked in this field, I followed carefully what experts in manual solving could do. This followed the claim of one telling he had the "hardest sudoku".

One evidence to me, for a manual solver, the so-called "hardest step" is not a rating criteria. Some of them have the nose to apply very very hard steps not defined in any of the common sets of rules. "ttt" "abi" and many other surprised me. they have often been key actors to improve the "exotic patterns" knowledge.

I have heavy doubts that any agreement can be made in the area of potential hardest, BYW a tiny percentage of the sudokus as said above.
champagne
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Re: The hardest sudokus (new thread)

Postby P.O. » Mon Oct 24, 2022 3:10 pm

puzzles have the complexity of finding their solutions through a process without guessing so no need to bother finding a general rating system it already exists, it is a template based solver.

all puzzles have the same formula: nine subsets, one and the same from each of the nine templates its solutions are made of.

from the puzzle to its solutions the solver uses the logical and hierarchical process of combining its possible templates, this process eliminates templates that can't be in some solutions,

when no more templates can be eliminated you get the solutions: that's the complexity of finding the solutions, it can be graded by the highest combination that has made some eliminations

finding Loki solution has complexity 4
Hidden Text: Show
Code: Select all
57....9..........8.1.........168..4......28.9..2.9416.....2.....6.9.82.4...41.6..

#VT: (4 6 410 14 68 7 73 10 14)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2
#VT: (4 6 389 14 65 7 70 10 14)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2 3
#VT: (3 6 200 14 35 7 38 10 14)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:(62) nil (62 80) nil (62 80) nil (62 80) nil nil
2 3
#VT: (3 6 127 14 27 7 30 10 14)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil (56) nil (55) nil nil
2 3
#VT: (3 6 127 14 27 7 28 10 14)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2 3
#VT: (3 6 117 14 25 7 28 10 14)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2 3
#VT: (3 6 117 14 23 7 26 10 14)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2 3 4
#VT: (1 6 2 14 5 7 6 10 13)
Cells: (9 15 55 71) nil (6 34 41 47 58 64 81) nil nil nil nil nil nil
SetVC: ( n3r1c6   n1r1c9   n1r2c6   n3r4c7   n3r5c5   n3r6c2
         n1r7c1   n3r7c4   n3r8c1   n1r8c8   n3r9c9   n2r1c8
         n8r1c4   n6r3c9   n8r6c1   n9r3c6   n8r3c3   n3r3c8
         n3r2c3 )

#VT: (1 3 1 14 5 2 6 2 4)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil (57) nil (57) nil (56 73 74)
2
#VT: (1 3 1 6 3 2 4 2 4)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil (11) (13 39 74) nil (13) nil nil
EraseCC: ( n2r2c4   n9r2c2   n5r4c2   n7r4c6   n4r5c2   n5r6c4
           n7r6c9   n8r7c2   n9r7c8   n5r7c9   n2r9c2   n5r9c6
           n8r9c8   n7r3c4   n9r4c1   n5r5c8   n4r7c3   n7r7c7
           n7r8c5   n7r9c1   n9r9c3   n6r1c3   n4r1c5   n4r2c1
           n5r2c7   n7r2c8   n2r3c1   n5r3c5   n4r3c7   n6r5c1
           n7r5c3   n5r8c3   n6r2c5 )
5 7 6   8 4 3   9 2 1
4 9 3   2 6 1   5 7 8
2 1 8   7 5 9   4 3 6
9 5 1   6 8 7   3 4 2
6 4 7   1 3 2   8 5 9
8 3 2   5 9 4   1 6 7
1 8 4   3 2 6   7 9 5
3 6 5   9 7 8   2 1 4
7 2 9   4 1 5   6 8 3


finding the solutions of the puzzle obtained by removing n5r1c1 from Loki has complexity 6
Hidden Text: Show
Code: Select all
.7....9..........8.1.........168..4......28.9..2.9416.....2.....6.9.82.4...41.6..

#VT: (4 8 478 18 478 8 73 12 14)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2
#VT: (4 8 454 18 454 8 70 12 14)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2 3
#VT: (3 8 268 18 268 8 38 12 14)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:(62) nil (62 80) nil (62 80) nil (62 80) nil nil
2 3
#VT: (3 8 202 18 202 8 32 12 14)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil (55) nil nil
2 3 4
#VT: (1 8 53 18 53 8 10 12 14)
Cells: (9 15 55 71) nil nil nil nil nil nil nil nil
SetVC: ( n1r1c9   n1r2c6   n1r7c1   n1r8c8   n6r3c9   n9r3c6 )

#VT: (1 8 53 18 53 4 10 12 7)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil (56 57) nil (56 57) nil (34 57 64) nil nil
2
#VT: (1 8 51 13 51 4 10 10 7)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2 3 4
#VT: (1 8 46 13 46 4 10 10 7)
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: (1 8 38 10 38 4 10 10 7)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil (19 21) nil nil nil nil nil
2 3 4 5 6
#VT: (1 8 37 10 37 4 10 10 7)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2 3 4 5 6 7 8 9

Left in pool: (260)
#VT: (1 8 37 10 37 4 10 10 7)

234568  7       34568   2358    3456    35      9       235     1               
234569  23459   34569   2357    34567   1       3457    2357    8               
2358    1       358     23578   3457    9       3457    2357    6               
3579    359     1       6       8       357     35      4       2               
34567   345     34567   1       357     2       8       357     9               
3578    358     2       357     9       4       1       6       357             
1       489     489     357     2       6       357     89      357             
35      6       357     9       357     8       2       1       4               
235789  23589   35789   4       1       357     6       89      357         

its 260 solutions
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Code: Select all
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P.O.
 
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Re: The hardest sudokus (new thread)

Postby eleven » Mon Oct 24, 2022 5:11 pm

One more time:
we are not interested here in global classification systems, but in the hardness of puzzles for manual solvers. I don't know even one, who solves puzzles manually using templates.
And i am very sure, that any chaining rating system correlates better with the rating, that manual solvers would give a puzzle, than a template rating.
eleven
 
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Re: The hardest sudokus (new thread)

Postby P.O. » Mon Oct 24, 2022 6:37 pm

the hardest puzzles are not manually solved, they get this qualification by being solved by a computer
i agree that templates are not for human solvers but neither are the rules of SE that rate the hardest puzzles

you shouldn't refer to yourself in the plural or set the subject of a thread you didn't author.
P.O.
 
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Re: The hardest sudokus (new thread)

Postby champagne » Mon Oct 24, 2022 7:11 pm

P.O. wrote:the hardest puzzles are not manually solved, they get this qualification by being solved by a computer
i agree that templates are not for human solvers but neither are the rules of SE that rate the hardest puzzles

you shouldn't refer to yourself in the plural or set the subject of a thread you didn't author.


Sorry, but this is just the kind of post that many of the old members of this forum would dislike (and sorry for the wording, English is not my mother language)
champagne
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Re: The hardest sudokus (new thread)

Postby P.O. » Mon Oct 24, 2022 7:30 pm

is it a characteristic of an old member to speak in place of the others? are you their representative?
P.O.
 
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Re: The hardest sudokus (new thread)

Postby champagne » Mon Oct 24, 2022 7:45 pm

worst and worst
just an historical view of the background

https://www.mirror.co.uk/news/weird-news/worlds-hardest-sudoku-puzzle-ever-942299

and even if I did not go in details in your "template" approach, you can search in this forum the "multi floors" analysis.
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Re: The hardest sudokus (new thread)

Postby P.O. » Mon Oct 24, 2022 8:31 pm

are opinions such as: I dislike your post or I am not interested in global classification systems? so difficult to express that they require a group effect?

the puzzle by Arto Inkala is in 4-template, not near the hardest from a computer point of view
Hidden Text: Show
Code: Select all
8..........36......7..9.2...5...7.......457.....1...3...1....68..85...1..9....4..

#VT: (22 327 68 100 14 57 17 21 48)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2
#VT: (22 288 66 95 14 54 17 19 45)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2 3
#VT: (21 273 47 84 14 46 17 16 44)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2 3
#VT: (21 270 47 84 14 46 17 16 44)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2 3 4
#VT: (4 213 6 69 14 7 17 4 42)
Cells: (16) nil nil nil nil nil nil (26 34) nil
SetVC: ( n1r2c7   n8r3c8   n8r4c7 )

#VT: (4 213 6 69 14 7 17 4 42)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil (77) (4 9 22 38 55 58 60 64 69 72 73 77) nil nil (3 9 28 30 37 46 48 54 64 65 77 78) nil nil nil
EraseCC: ( n4r3c4 )

#VT: (4 213 6 28 14 7 17 4 42)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2
#VT: (4 202 6 20 10 7 17 4 35)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil (10) nil nil nil nil
2 3
#VT: (4 123 6 13 8 7 14 4 24)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil (36) nil (36) (3 7 27) nil (77) nil (7 36)
2
#VT: (4 123 6 13 8 7 14 4 23)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2 3
#VT: (2 95 6 13 8 6 11 4 18)
Cells: (2 24 77) nil nil nil nil (7) (48) nil nil
SetVC: ( n1r1c2   n6r1c7   n1r3c6   n3r3c9   n7r6c3   n1r9c5 )

#VT: (2 95 21 13 8 18 13 4 18)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil (75) nil nil nil nil nil nil nil
2
#VT: (2 77 18 7 8 15 13 3 17)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil (9 55) nil nil nil nil nil
2
#VT: (2 72 16 7 8 14 12 3 17)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil (38) nil nil nil nil nil nil nil
2 3
#VT: (2 36 5 7 8 6 8 3 12)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil (37 40 47 78) (59 68) nil nil (39) (8 76) nil (37)
2 3
#VT: (2 24 5 5 6 6 8 2 8)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil (14 28 45 76) nil (17 28 47) nil nil nil (51) (17 28 45)
2
#VT: (2 24 5 5 6 6 7 2 8)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil (55 68) nil nil
2 3
#VT: (2 16 5 5 6 5 5 2 8)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil (50 69) nil nil nil nil (18) nil nil
2 3
#VT: (2 16 5 5 6 5 5 2 7)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2 3
#VT: (2 16 5 4 6 5 5 2 7)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2 3
#VT: (2 11 5 4 4 4 5 2 6)
Cells: nil nil nil nil (21 52 55) nil nil nil nil
SetVC: ( n5r3c3   n5r6c7   n5r7c1   n6r9c3   n6r3c1 )

#VT: (2 11 5 4 4 4 5 2 6)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil (30) nil nil nil nil nil nil nil
2
#VT: (2 10 5 4 4 3 5 2 3)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil (60) nil nil nil nil nil nil (8 30 40 54 72)
EraseCC: ( n4r4c3   n4r6c9   n4r1c8 )

#VT: (2 10 5 3 4 3 5 2 3)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil nil nil nil nil nil nil nil nil
2
#VT: (2 9 5 3 3 3 5 2 3)
Cells: nil nil nil nil nil nil nil nil nil
Candidates:nil (6 32 80) nil nil nil nil nil nil nil
EraseCC: ( n3r1c6   n8r9c6   n2r2c6   n3r9c4   n7r1c4   n5r1c5
           n9r1c9   n4r2c2   n8r2c5   n5r2c9   n8r5c4   n6r6c5
           n9r6c6   n4r7c6   n2r8c5   n6r8c6   n7r8c9   n5r9c8
           n2r9c9   n2r1c3   n9r2c1   n7r2c8   n2r4c4   n3r4c5
           n9r4c8   n6r5c2   n9r5c3   n2r5c8   n1r5c9   n2r6c1
           n8r6c2   n9r7c4   n7r7c5   n3r7c7   n4r8c1   n3r8c2
           n9r8c7   n7r9c1   n1r4c1   n6r4c9   n3r5c1   n2r7c2 )
8 1 2   7 5 3   6 4 9
9 4 3   6 8 2   1 7 5
6 7 5   4 9 1   2 8 3
1 5 4   2 3 7   8 9 6
3 6 9   8 4 5   7 2 1
2 8 7   1 6 9   5 3 4
5 2 1   9 7 4   3 6 8
4 3 8   5 2 6   9 1 7
7 9 6   3 1 8   4 5 2


i read these posts and many more, my interest in templates did not come from nowhere, writing a template-based solver is within the reach of any programmer.
P.O.
 
Posts: 591
Joined: 07 June 2021

Re: The hardest sudokus (new thread)

Postby eleven » Mon Oct 24, 2022 9:59 pm

P.O.,

this thread is the successor of Ravel's The hardest sudokus, started in 2006. From the beginning the intention was to find the hardest sudokus for manual solving. It is trivial, that the candidates for the hardest list had to be found by computer programs.
Some months later Ravel wrote:

"Sudoku Explainer is the only program i know so far, that can solve the hardest puzzles in human readable (and graphical) form. It is also great for "normal" puzzles to look what techniques it uses to solve them.
..
[Edit:] With version 1.2 now also the current hardest puzzles can be solved."

This clearly shows the intention of these threads. It's about manual solving and possible solutions of very hard puzzles, that can be represented in a form, that manual solvers can understand (and - if not too hard - could find themselves). As far as i can judge it, almost all of the contributors (and readers) had this intention in mind.

So please, if you want to discuss the worth of templates for (hardest) sudoku rating, start another thread.
eleven
 
Posts: 2938
Joined: 10 February 2008

Re: The hardest sudokus (new thread)

Postby P.O. » Tue Oct 25, 2022 6:26 am

hardest for manual solver is not hardest at all, the first post of the 'The hardest sudokus (new thread)' recalls the relationship between hardest puzzles and rating programs: "The following is a "Top 5" list of hardest sudokus according to some popular sudoku rating programs"

some of the recently found puzzles rated by SE in the range 10. 11. are by a logical and hierarchical process classified in the same category as puzzles rated by SE in the range 5. 7. this could lead to a revision of the current rating procedures

i doubt that the graphic of a chain with 100 nodes can have the least educational value on anyone unlike a chain with 10 nodes and it is the same with templates, 100 possible templates for a value is impracticable but 10 possible templates can be used by a manual solver to find eliminations, but so far no one has written a graphic interface to visualize them and encourage their use by manual solvers

but they can be made more accessible with some explication
http://forum.enjoysudoku.com/post326863.html#p326863
P.O.
 
Posts: 591
Joined: 07 June 2021

Re: The hardest sudokus (new thread)

Postby eleven » Tue Oct 25, 2022 9:33 am

I suggest, that you move your posts to this thread from 2012.
eleven
 
Posts: 2938
Joined: 10 February 2008

Re: The hardest sudokus (new thread)

Postby P.O. » Tue Oct 25, 2022 9:50 am

the discussion is about hardest puzzles and their rating and not the templates as such so this thread is the most appropriate.
P.O.
 
Posts: 591
Joined: 07 June 2021

Re: The hardest sudokus (new thread)

Postby eleven » Tue Oct 25, 2022 10:47 am

If you read the last page, you can see, that 10 years ago i had checked, that the hardest list contained less than 2000 puzzles, which needed 6 digit templates for solving.
Now guess, why neither i nor anyone else had your glorious idea to use that as a rating for the hardest puzzles thread.
eleven
 
Posts: 2938
Joined: 10 February 2008

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