## The hardest sudokus (new thread)

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

### Re: The hardest sudokus (new thread)

I made a first exploration in my new â€œpotential hardestâ€ data base to extract true â€œpotential hardestâ€.

I processed the first 9000 entries.

The first step has been to filter the properties described earlier

_ Symmetry of given
_ SK loop pattern
_ EXOCET pattern
_ And the 3 rank 0 or nearly rank 0 multi fish pattern recognized by my solver
Sets made exclusively of rows or of columns (usually a pure rank 0 pattern)
Sets made of row and columns forming a square (often not a pure rank 0 pattern)

There were only 960 puzzles left out of the 9000 after having applied that filter.

I then eliminated puzzles easily solved by my solver.
Here the cut off to apply is not quite clear, but
_ clearing obviously easy suppressed about 130 puzzles
_ a reasonable cut off left 200 puzzles.

I took a high cut off to produce the following short list.

(puzzle, sequence number, source, ER, EP, ED, difficulty index )

Hidden Text: Show
Code: Select all
`1....6.......8.2....97....5.3.9....4..5....9.....2.1....4.....7.9.3...4.8.....6..;1582;elev;11.3;1.2 ;1.2 ;9..34..7...5...9.2.....1....23.....5...1.6.3..8..........46..........8.9....17.6..;5416;elev;11.1;1.2 ;1.2 ;8.2.4.......7.8...6.....3.5...9.6...1.....23.....5...4...1...8..6...1...797.......;243 ;elev;11.4;11.4;11.3;8..34......5...9...7...2...62...7..1..9...5......3....8..1.6...78......21......4..;4024;elev;11.1;11.1;3.4 ;8.......8....1.9..66...2...4.7...8.9.5..........4.3...73.2.....5..5.6.3...1.....7.;3218;elev;11.2;1.2 ;1.2 ;898.7.....7...8.6....5..4...6..3..9...9.....2...4..1..6.3.8..7.......2.1.........5;227 ;GP  ;11.5;1.2 ;1.2 ;898.7.....6.....9....5....7..4..3..2...85..4.......4..1..69..5......2...3.....1.4.;35  ;GP  ;11.7;11.7;11.3;8...45.........9.3.6...375...4....1....8.....29...6..7.3....5.9...2...8...1..7....;620 ;elev;11.3;11.3;10.6;8..3..6.8.4.....2......3.5....89...3.5...7...4.....1....6.8...9.7...2......1..3...;2780;elev;11.2;1.2 ;1.2 ;71.......9.5.1...3...8..34...1.5.......9..8..2....6..7.3....4..8..2.......8..7..6.;3411;elev;11.1;11.1;10.8;7....5...94..1..2.......7...2.8.1.6....62......7......3.3..9..5.8..........28..4..;4971;elev;11.1;1.2 ;1.2 ;7.....6.8.4..7....3.9..3.....1..4...73..1.......8..52...3.9....4......5....2....6.;4891;elev;11.1;1.2 ;1.2 ;7.2.4....9..7......8....2.1...1..5.7..6..9...5...2..........8.5....36...2.3..2.6..;4993;elev;11.1;1.2 ;1.2 ;7.......89.5.1.....6....31...7...16......9..2...8.....4..4.2....7..5..3...6...7...;3329;elev;11.2;1.2 ;1.2 ;7.2.4.......7.8...69......5...8...6.15....1.9.....7...3........7.4.2.......1.6.3..;1263;elev;11.3;1.2 ;1.2 ;6........9..71...6.....3.1.42..6...1...5.......8...43..5......2.6.17......9...8...;1690;elev;11.3;1.2 ;1.2 ;61....6....5.........82....4..98..3...6...5.7.........27....2.1...2.4.9.3...3.....;1102;elev;11.3;11.3;2.6 ;61.......9..67...2..8....5.......83.....2...64..7.4..1...462....5..8......9...3...;889 ;elev;11.3;11.3;9.5 ;6`

All the puzzles in that list are new and have a relatively low ER rating. Most of them have been generated recently by eleven.

To-day, the best indicator I have to forecast if something simple has a chance to come is to use Allan Barker process applied to a multi floors pattern.

I am very reluctant to consider as of any use complex SLG produced by XSUDO model, unless one can clearly establish in a logical way where are the â€œrank 0â€ pieces of logic.
However, it is very simple to find the potential of a multi floors pattern and to have an idea of the chances to find a new pattern easy to explain.

Consequently, I made an exploration of that short list with the Allan Barker model an got the following table where

Seq is the sequence number as above
Floors is the list of digits leading to a potential action
Perm is the number of valid patterns in the multi floors
Elims is the number of candidates than are never set to 1
Cells is the number of cells where extra candidates exist and are never set to 1

Note : if a multi floor of â€œnâ€ digits is active, all multi floors with the same digits plus another one are ignored;

Code: Select all
` Seq  floors perm  elims   cells1582   1268  94   7  11      34579 161  14   6    5416   2579  79  13   5        13467 190   8   8 243   23459  99   6   1 4024                   empty3218   1789  49   12  8227    1245  79   12  6       36789 188   10  8   35     1234  43    6  4620   12348 178    7  1      12468 203    6  4  2780  12457 257    8      13689  77    9      23457 152    6  2  3411                   empty4971   3579  40   14  8      12468 712   11  8   4891   2568 128    6 11      13479  89   17  6  4993   1578  52   11  7      23469 412   11  8      34689 347   16  3   3329   2489  52    6  9      13567 328   13 10  1263   2459  62    5  7       13678 221   10  31690   3489 107   10  4      12567 190    6  8  1102  12567  35   13  6      23489 147   13  9 889                    empty`

From that table, one can predict that entries with no potential will remain extremely difficult.
The best chance to find a new SLG â€œeasy to justifyâ€ using XSUDO is where the number of permutations is low

Eg: number 1102 35 perms with a 5 floors pattern

One can also notice that often the complementary floors of a 4 floors with potential have also some potential.

champagne
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### Re: The hardest sudokus (new thread)

I added a new lot of puzzles in my database, but most of them are in the lowest part of the ER ratings.

The file contains now 11205 puzzles and this should not change over the next 2 weeks. (out of external feed)
Next lot should be 23 clues puzzles, the area I am now investigating

This is the link to the zip file
here

The file contains sub files with the list of puzzles having the key properties to crack high rating puzzles.

champagne
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### Re: The hardest sudokus (new thread)

Hi,

Here the second highest SE rating known so far

As most of these very high ratings, that puzzle has an EXOCET pattern and is not so hard to solve.

98.7.....6.....87...7.....5.4..3.5....65...9......2..1..86...5.....1.3.......4..2;11.9;11.9;11.6

It will enter my data base at the end of the current search cycle.

champagne
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### Re: The hardest sudokus (new thread)

Congrats, champagne,

this one i did not find in my set of 7.5 mio step1 puzzles with 23 clues.
Good luck for your further search.
eleven

Posts: 1864
Joined: 10 February 2008

### Re: The hardest sudokus (new thread)

eleven wrote:Congrats, champagne,

this one i did not find in my set of 7.5 mio step1 puzzles with 23 clues.
Good luck for your further search.

It's not too hard to find new 11 and more. I have about 800 of them waiting for the next entry.

Reversely, it seems that you covered most of the highest part. In that lot of 800, I have only 4 puzzles rating 11.6 and more.

I am just processing the next 2000 entries of my database to detect special properties. It seems that again 90% of the puzzles have at least one among (SK loop, EXOCET, symmetry of given, multi fish).

champagne
champagne
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### Re: The hardest sudokus (new thread)

champagne wrote:Here the second highest SE rating known so far
As most of these very high ratings, that puzzle has an EXOCET pattern and is not so hard to solve.

98.7.....6.....87...7.....5.4..3.5....65...9......2..1..86...5.....1.3.......4..2;11.9;11.9;11.6
It will enter my data base at the end of the current search cycle.

Congrats...!

Wow...! Perhaps, this is SE's hardest puzzle - GN: 11.9;11.9;11.3

ttt
ttt

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### Re: The hardest sudokus (new thread)

ttt wrote:
champagne wrote:Here the second highest SE rating known so far
As most of these very high ratings, that puzzle has an EXOCET pattern and is not so hard to solve.

98.7.....6.....87...7.....5.4..3.5....65...9......2..1..86...5.....1.3.......4..2;11.9;11.9;11.6
It will enter my data base at the end of the current search cycle.

Congrats...!

Wow...! Perhaps, this is SE's hardest puzzle - GN: 11.9;11.9;11.3

ttt

Hi ttt,

No SE's highest known rating is that one

98.7.....7.....6....6.5.....4...5.3...79..5......2...1..85..9......1...4.....3.2. champagne dry 11.9 11.9 11.8

may be we will see one day a ER 12.0 or more

but it would be interesting to see what a player can do with the "hardest" described 5 posts above

Code: Select all
`1....6.......8.2....97....5.3.9....4..5....9.....2.1....4.....7.9.3...4.8.....6..;1582;elev;11.3;1.2 ;1.2 ;9..34..7...5...9.2.....1....23.....5...1.6.3..8..........46..........8.9....17.6..;5416;elev;11.1;1.2 ;1.2 ;8.2.4.......7.8...6.....3.5...9.6...1.....23.....5...4...1...8..6...1...797.......;243 ;elev;11.4;11.4;11.3;8..34......5...9...7...2...62...7..1..9...5......3....8..1.6...78......21......4..;4024;elev;11.1;11.1;3.4 ;8.......8....1.9..66...2...4.7...8.9.5..........4.3...73.2.....5..5.6.3...1.....7.;3218;elev;11.2;1.2 ;1.2 ;898.7.....7...8.6....5..4...6..3..9...9.....2...4..1..6.3.8..7.......2.1.........5;227 ;GP  ;11.5;1.2 ;1.2 ;898.7.....6.....9....5....7..4..3..2...85..4.......4..1..69..5......2...3.....1.4.;35  ;GP  ;11.7;11.7;11.3;8...45.........9.3.6...375...4....1....8.....29...6..7.3....5.9...2...8...1..7....;620 ;elev;11.3;11.3;10.6;8..3..6.8.4.....2......3.5....89...3.5...7...4.....1....6.8...9.7...2......1..3...;2780;elev;11.2;1.2 ;1.2 ;71.......9.5.1...3...8..34...1.5.......9..8..2....6..7.3....4..8..2.......8..7..6.;3411;elev;11.1;11.1;10.8;7....5...94..1..2.......7...2.8.1.6....62......7......3.3..9..5.8..........28..4..;4971;elev;11.1;1.2 ;1.2 ;7.....6.8.4..7....3.9..3.....1..4...73..1.......8..52...3.9....4......5....2....6.;4891;elev;11.1;1.2 ;1.2 ;7.2.4....9..7......8....2.1...1..5.7..6..9...5...2..........8.5....36...2.3..2.6..;4993;elev;11.1;1.2 ;1.2 ;7.......89.5.1.....6....31...7...16......9..2...8.....4..4.2....7..5..3...6...7...;3329;elev;11.2;1.2 ;1.2 ;7.2.4.......7.8...69......5...8...6.15....1.9.....7...3........7.4.2.......1.6.3..;1263;elev;11.3;1.2 ;1.2 ;6........9..71...6.....3.1.42..6...1...5.......8...43..5......2.6.17......9...8...;1690;elev;11.3;1.2 ;1.2 ;61....6....5.........82....4..98..3...6...5.7.........27....2.1...2.4.9.3...3.....;1102;elev;11.3;11.3;2.6 ;61.......9..67...2..8....5.......83.....2...64..7.4..1...462....5..8......9...3...;889 ;elev;11.3;11.3;9.5 ;6`

If you have time, try them.

champagne
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### Re: The hardest sudokus (new thread)

I've been making sudokus based on what I think would make a hard puzzle.

1) There are a maximum of 3 clues in a 3x3 box none of which share a row or column with any other clue in that box.
2) There are no repeated clues of the same clone in any vertical or horizontal band of three 3x3 boxes.

These conditions do not guarantee a hard puzzle, but I think they would make one easier to create. Diagonal puzzles would satisfy requirement 1, but usually violate 2. Having a maximum of 27 clues and a maximum of 3 of a clone , which these rules imply and nothing lining up to simpleton hint, except for a pair, swordfish, or colors.

Is there a mathematical reason why these criteria could not be used to make a world class sudoku. I recently made a puzzle that rated 10.2 on the Explainer, but I'm currently possessive of it. Here's one:

8..5....7....9.1....4....6...62..4......1..8..7...3..5...7..3..5...4..9...1..8..2

rated 9.0 by the Explainer. I am in awe of the six 11.9's made and they didn't look anything like mine. I wonder what your approach is.
LordRot

Posts: 2
Joined: 05 September 2011

### Re: The hardest sudokus (new thread)

LordRot wrote:I've been making sudokus based on what I think would make a hard puzzle.

1) There are a maximum of 3 clues in a 3x3 box none of which share a row or column with any other clue in that box.
2) There are no repeated clues of the same clone in any vertical or horizontal band of three 3x3 boxes.

These conditions do not guarantee a hard puzzle, but I think they would make one easier to create. Diagonal puzzles would satisfy requirement 1, but usually violate 2. Having a maximum of 27 clues and a maximum of 3 of a clone , which these rules imply and nothing lining up to simpleton hint, except for a pair, swordfish, or colors.

Is there a mathematical reason why these criteria could not be used to make a world class sudoku. I recently made a puzzle that rated 10.2 on the Explainer, but I'm currently possessive of it. Here's one:

8..5....7....9.1....4....6...62..4......1..8..7...3..5...7..3..5...4..9...1..8..2

rated 9.0 by the Explainer. I am in awe of the six 11.9's made and they didn't look anything like mine. I wonder what your approach is.

Hi LordRot,

Welcome in that forum.

First of all, defining a "very hard" puzzle is not so easy.

Common sense would say that a "hardest" puzzle is one that nobody could solve using some "nice logic". IMO are excluded from the "Nice logic" for example ;
. Too long chain including many derivations
. XSUDO complex structures where one can not explain where is the rank 0 logic
. most of the nested chains in Sudoku Explainer
. ....

For sure, with such a definition, the family of "hardest" is not stable overtime, but one target is precisely to find new pieces of "nice logic" to crack the resisting puzzles.

A limited number of "nice pieces of logic" have been found that help to crack some puzzles nobody solved without using T&E or one of these tools I mentioned earlier.

The primary collection of"potential hardest puzzles" is generally made ignoring the "nice pieces of logic". It gives a sound bas to study that area.

Depending on the set of rules applied by the program used to filter the puzzles, the ranking will differ, so that primary rank is not a key issue.
Eleven and I are using Sudoku Explainer rating just because it's a widely spread rating technique. The raw rating is not so important. It's a kind of side game to publish the highest SE rating. (SE is the referee and the base for the pattern game in that forum)

I try now to answer to your question on how to find hardest puzzles and on the value of your 2 rules.

It's commonly agreed that "diagonal patterns" have a high potential to produce hard puzzles.

Such pattern have been deeply explored and many examples can be seen in the pattern game. The current gamer is exploring a 27 clues pattern meeting your rule 1.

with so many clues, it could not produce so far puzzles of the family of hardest

Your second rule sounds well, but I am afraid that it"s just limiting the search to specific families.
It could be that applying such a rule you find the same thing as in the 17 clues family;

Common sense would let think that the less you have clues the tougher will be the puzzles.
Experience has shown that this is false. The constraints induced to produce a 17 clues puzzles led to the fact that none by far would enter the family of hardest.

The search of "hardest" has been made in several ways. Some, as in the pattern game, have explored specific patterns (ocean ...).

To day, intensive search is made mainly using some vicinity techniques plus some filters.

The underlying concept is that the reason why a puzzle is hard are still valid for some puzzles nearby, whatever is the reason.

This led to the collection of puzzles I published the zip file
here

I am convinced that if anybody finds some specific characteristics leading to hardest puzzles, It could be a better approach to generation.
The EXOCET pattern could have been one way to do it, but as it is one of the "nice pieces of logic", it would not have created the "final hardest".

In fact, the characteristics should be extracted of the short list of hardest as in the 18 puzzles published just above.

To close that post, in my database, more than 50% of the puzzles have the EXOCET pattern as in Golden Nugget

Code: Select all
`1= .......392= .....1..53= ..3.5.8..4= ..8.9...65= .7...2...6= 1..4.....7= ..9.8..5.8= .2....6..9= 4..7.....  1247 `

In that puzzle, the EXOCET pattern is in stack 3 and the base is made of candidates 1247 in r12c7.

In the stack, the candidates 1247 are not there
All other digit are there.

This is I think a property shared by all EXOCT patterns an it is far from your 2 rules.

Just a side remark at the end.

We publish here only minimal puzzles. Many puzzles in that list have a single as start

sorry for that long post where you should find, I hope, some answers to your questions

champagne
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### Re: The hardest sudokus (new thread)

LordRot wrote:1) There are a maximum of 3 clues in a 3x3 box none of which share a row or column with any other clue in that box.
2) There are no repeated clues of the same clone in any vertical or horizontal band of three 3x3 boxes.

This is an old idea. In gsf's man page you can read:
gsf wrote: Generate using the hardest sudoku formula: at most one clue
per minirow/minicol and no repeated values in bands/chutes,
at most three clues per value, and minimal.

In fact puzzles with these properties have a very high average rating. But it turned out, that the hardest - according to different ratings - dont comply with these rules.
eleven

Posts: 1864
Joined: 10 February 2008

### Re: The hardest sudokus (new thread)

Glad to be in sudoku cyber space.

I do not think 27 clues in a puzzle is a severe limit to making hard puzzles. Maybe 21 might be. 17's must break down efficiently (they're easy puzzles) to the solution. It's as if the set of solution grids is a 17 dimensional space, with each clue being a coordinate in that multidimensional space. These are incredibly complex things here.

Take the six 11.9's on the list. One has 21 clues, four have 22, one has 23. The form of these puzzle is radically different than mine but I have noticed very hard puzzles fall exactly in that range, from 21 to 23 clues which means I don't have to resort to more than 3 of a kind. This suggests a maximum of 27 clues that my rules imply does not seem unreasonable, indeed there is some room for juggling things around.

All my puzzle will be minimized as well, I don't like the artificial symmetry added to number place by the Japanese. It makes for non-minimal, and I think, less hard puzzles. Try this one at 10.2:

8.........2..9...7..4....6...62..4......1..8..9...3..5...8..3..5...4..9..1...7..2

Glad to be here and thank you.
LordRot

Posts: 2
Joined: 05 September 2011

### Re: The hardest sudokus (new thread)

LordRot wrote:Glad to be in sudoku cyber space.

I do not think 27 clues in a puzzle is a severe limit to making hard puzzles. Maybe 21 might be. 17's must break down efficiently (they're easy puzzles) to the solution. It's as if the set of solution grids is a 17 dimensional space, with each clue being a coordinate in that multidimensional space. These are incredibly complex things here.

Take the six 11.9's on the list. One has 21 clues, four have 22, one has 23. The form of these puzzle is radically different than mine but I have noticed very hard puzzles fall exactly in that range, from 21 to 23 clues which means I don't have to resort to more than 3 of a kind. This suggests a maximum of 27 clues that my rules imply does not seem unreasonable, indeed there is some room for juggling things around.

All my puzzle will be minimized as well, I don't like the artificial symmetry added to number place by the Japanese. It makes for non-minimal, and I think, less hard puzzles. Try this one at 10.2:

8.........2..9...7..4....6...62..4......1..8..9...3..5...8..3..5...4..9..1...7..2

Glad to be here and thank you.

Hi,

Just a difficult puzzle, as the 10.2 rating lets expect.
No special property, "easily" solved by my solver.

Regarding "hardest" puzzles, I think we could find them most in the 21/24 area.

take that list of puzzles found by eleven

Code: Select all
`.2.4..7...5...9.3.6...7......5..8.9.7...2.....4.6..3.......1..85......1...1...9.3   11.8   1.2   341....6.8....18.2...9..7.......7....37...6..1...9..54...4......25..6...7...1...3..   11.6   1.2   120.......894....92.....7...4..35.7......651....8....2....7..6.5..5.......8..13.5...   11.5   1.2   233.......894....92.....7...4..35.7......6.1....8....2.5..7..6.5..5.......8..13.5...   11.5   1.2   231.......894....92.....7...4..3..7......651....8....2.5..7..6.5..5....7..8..13.....   11.5   1.2   230.......894....92.....7...4..3..7......651....8....2.5..7..6.5..5.......8..13.5...   11.5   1.2   229..3..6.8.45...9....8.1..5.....6...7.....7.8.2..7.....3.1.5.......8.6..2.9....4...   11.5   1.2   2281......8...71....6.9.....5...56....7..17.4..5......34.57.2.............2..2.61...   11.5   1.2   226..3.....94....91......3.54....6.....3...9...1.7...2.6...2...8....8..7.9.9...4...5   11.5   1.2   215.2..5......67.9.2.7.....5..2...64.7.6...9..5....3....6..4......5...7..9......18..   11.5   1.2   203.23..6....567...2.7.........1.3..4..6....5.3...9.1.......8..9..5....3.7.....4...8   11.5   1.2   202.2.4..7....6.8...3.......6.2..9....7..4.2.9...9.....3.5.8.....1.7.2..3......15...   11.5   1.2   195.2.4....9..7.8....6....21..2....13....47......8..9....5......13..2.1.5.......5.6.   11.5   1.2   194`

they all have 23 clues but the start is a single (may be more) which mean that the "non minimal" form including the singles would

. have more than 23 clues
. be rated higher than the minimal form

and would have the same final difficulty.

champagne

EDIT I did a little more on the list of puzzles just above

most of them are equivalent to 24 clues
2 of them are equivalent to a 25 clues
one is equivalent to a 26 clues
champagne
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### Re: The hardest sudokus (new thread)

Hi champagne,

and thank you for your work with the collection of "potential hardest puzzles".
Can you recommend a program that can solve all the puzzles in your list?

/SudoQ
SudoQ

Posts: 39
Joined: 09 September 2011

### Re: The hardest sudokus (new thread)

SudoQ wrote:Hi champagne,

and thank you for your work with the collection of "potential hardest puzzles".
Can you recommend a program that can solve all the puzzles in your list?

/SudoQ

Hi SudoO,

Welcome in the forum.

It seems that several newcomers are watching waiting for the good timing to step in.

It's not easy to answer your question and I am not expert in all programs available on the web. I am exclusively working with my own code.

AFAIK, all the old programs commonly used for rating solve all the puzzles. Some of them have a kind of escape lane, which means that they switch to T&E when they fail to find other logic rules.

Sudoku Explainer solves all known puzzles, but using some very complex logic involving sometimes more than 100 candidates. I doubt that such logic can still be seen as valid for a player.

My own solver, solves all puzzles as well, but again, sometimes using logic constructions involving to many candidates to be qualified as "reasonable" solution.

The "nice pieces of logic" described in that thread {SK loop, EXOCET, Symmetry of given, Multi fish) appeared later.

I don't know who included them in it's solver. These ones are in mine.

Other "Nice pieces of logic" have been described in that forum.

Some as chaining XWings left established are included in the old Sudoku Explainer.

My solver is relatively poor in that field.

Another point is the availability of programs.

I am for example ready to share my code, it's not a problem, but it is basically a personal tool and the design is

1) in French for most of the comments, names ...
2) somewhere in between a debugging print and a path explanation
3) It's a multi purpose tool including all the code I use in generation of code.

I hope that other will react to your post to come to a wider view..

If you are interested in knowing more about my own solver, I can develop;

champagne
champagne
2017 Supporter

Posts: 6542
Joined: 02 August 2007
Location: France Brittany

### Re: The hardest sudokus (new thread)

Hi champagne,

champagne wrote:If you are interested in knowing more about my own solver, I can develop

I downloaded version labeled "Version: V1.0.0 dated June 24th 2011", and after some reverse engineering found the basic command line syntax. It runs about 30-200 times faster than SE, processing a batch of 300K non-average puzzles in 1h 23'.

Some basic documentation is welcome. It is up to you in choosing where or whether to open a discussion.

Cheers,
MD
dobrichev
2016 Supporter

Posts: 1615
Joined: 24 May 2010

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