Max number of clues

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

Postby coloin » Fri Nov 03, 2006 11:03 pm

JPF wrote:Glad you finally got 35s in this grid.

Aye, I had given up ! The post on the largest easy minimal puzzles promted me to do a suexrat......
JPF wrote:Why did you pick the highest rated 34 ?

At the time there was only one puzzle with a suexrat of 106, the next one had a rating of 86......

It seems somewhat obvious now - It might be that a puzzle with less room for an extra clue is easier to solve - eg the chameleon puzzles. These all have a suexrat of 47.

Looking at the data from the chameleon work you did, it does seem that the 106 rated puzzle has a low mutable clue count and also a high number of grid solutions ........

Code: Select all
rating 106        ..........6...1352.5..436.1.1..579.....4.9..7..9.1..6...1..57.6.721..5...85.742.3
19 mutable clues  0,0,1,1,1,0,2,0,1,0,0,0,0,1,1,1,0,0,3,0,0,0,1,2,2,0,1,1,0,1,1,1,1,1
gridsols          5,7,2,2,9,*,5,8,2,7,5,*,*,2,9,2,4,*,4,5,5,7,2,3,6,*,2,2,5,2,4,2,2,2


there were some clues not represented........so not perfect...but it paved the way for more 35s to emerge

22 35s in the dukuso15 grid
Code: Select all
..........6...135..5..436.1.1..5793.5.64.9.....9.12.....1.257...7213.5...8..742.3
..........6...135..57...6.1.1..5793....4.9..7..931..6...1.257.6.7213.5...85.742..
..........6...135..57..36.1.1..5793.5.64.9.....9.1..6...1.257...7213.5...8..742.3
..........6...1352.57...6.1.18.5793....4.9..7..931..6...1..57...7.13.5...85.742.3
..........64..135..57.4.6.1.18.5793......9..7..931..6...1..5.4..7.13.5...85.742.3
..........64..135..57.4.6.1.18.5793......9..7..931..6...1..57...7.13.5...85.742.3
..........64..135..57.4.6.1.18.5793......9..7..931..65..1..5.4..7.13.....85.742.3
..........64..135..57.4.6.1.18.5793......9..7..931..65..1..57...7.13.....85.742.3
..........64..135..57.4.6.1.18.5793.5....9..7..931..6...1..5.4..7.13.5...8..742.3
..........64..135..57.4.6.1.18.5793.5....9..7..931..6...1..57...7.13.5...8..742.3
..........64..135..57.4.6.1.18.5793.5....9..7..931..65..1..5.4..7..3.....8..74213
..........64..135..57.4.6.1.18.5793.5....9..7..931..65..1..5.4..7.13.....8..742.3
..........64..135..57.4.6.1.18.5793.5....9..7..931..65..1..57...7..3.....8..74213
..........64..135..57.4.6.1.18.5793.5....9..7..931..65..1..57...7.13.....8..742.3
..........64..135..57.4.6.1.18.5793.5....91.7..93...65..1..5.4..7..3.....8..74213
..........64..135..57.4.6.1.18.5793.5....91.7..93...65..1..5.4..7.13.....8..742.3
..........64..135..57.4.6.1.18.5793.5....91.7..93...65..1..57...7..3.....8..74213
..........64..135..57.4.6.1.18.5793.5....91.7..93...65..1..57...7.13.....8..742.3
..........64..135..57.436.1.1..5793......9..7...31..6...1..57.6.7213.5...85.742..
..........64..135..57.436.1.1..5793......9..7...31..65..1..57.6.7213.....85.742..
..........64..135..57.436.1.1..5793.5....9..7...31..6...1..57.6.7213.5...8..742..
..........64..135..57.436.1.1..5793.5....9..7...31..65..1..57.6.7213.....8..742..


C
Last edited by coloin on Tue Dec 19, 2006 11:08 pm, edited 4 times in total.
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Postby coloin » Mon Nov 20, 2006 1:35 am

Update again......

The 35s from the dukuso15 grid gave me an idea.....trying to crush the clues together......

The MC grid with 54 clues
no 9s, no row 9, no column 9 and no box 9.

Code: Select all
+---+---+---+
|123|456|78.|
|78.|123|45.|
|456|78.|12.|
+---+---+---+
|231|564|8..|
|564|8.7|23.|
|8.7|231|56.|
+---+---+---+
|312|645|...|
|645|.78|...|
|...|...|...|
+---+---+---+


Four more 35s
Code: Select all
..3..678.78.12.45...678..2..31..48...648..23.8..2..5..312645...645...............
..3..678.78.12.45...67..12..3..648...648.72..8..2..5..312645...645...............
..3..678.78.1..45...678.12..31.648...6.8..2..8..2.15..312645...645...............
..3..678.78..2.45...678.12..31.648....48..2..8..23.5..312645...645...............


Amazingly this puzzle came out very easily....at last

Code: Select all
+---+---+---+
|..3|..6|78.|
|78.|12.|45.|
|..6|78.|12.|
+---+---+---+
|.31|.64|8..|
|.64|8..|2..|
|8..|2..|5..|
+---+---+---+
|312|645|...|
|645|...|...|
|...|...|...|
+---+---+---+

Code: Select all
..3..678.78.12.45...678.12..31.648...648..2..8..2..5..312645...645...............


Its a 36

Every clue has 2 grid solutions.....therefore a chameleon puzzle

C
Last edited by coloin on Sun Nov 19, 2006 9:59 pm, edited 1 time in total.
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Postby udosuk » Mon Nov 20, 2006 1:58 am

Congrats coloin...:) Persistance pays off, as always...
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Postby JPF » Mon Nov 20, 2006 2:12 am

Congratulations !

And you had the élégance to find a Chameleon:D
Thanks !

JPF
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Postby coloin » Mon Nov 20, 2006 2:38 am

Thank-you ......

I was just miffed with the near symmetry until......the column swap made it ....

Code: Select all
+---+---+---+
|.3.|..6|78.|
|7.8|12.|45.|
|.6.|78.|12.|
+---+---+---+
|.13|.64|8..|
|.46|8..|2..|
|8..|2..|5..|
+---+---+---+
|321|645|...|
|654|...|...|
|...|...|...|
+---+---+---+


it looks like a "water-boatman"

C
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Postby JPF » Mon Nov 20, 2006 8:39 am

coloin wrote:Its a 36
Every clue has 2 grid solutions.....therefore a chameleon puzzle

And removing any clue gives a 35-puzzle with 2 solutions.

In addition, it is a 3-steppers:)

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Postby Ocean » Mon Nov 20, 2006 9:04 am

coloin wrote:
Code: Select all
..3..678.78.12.45...678.12..31.648...648..2..8..2..5..312645...645...............


Its a 36

Congratulations!
Amazing puzzle! A slightly different symmetric variaton is by swapping row 2 and 3 - thus keeping the canonical form:
Code: Select all
 *-----------*
 |..3|..6|78.|
 |..6|78.|12.|
 |78.|12.|45.|
 |---+---+---|
 |.31|.64|8..|
 |.64|8..|2..|
 |8..|2..|5..|
 |---+---+---|
 |312|645|...|
 |645|...|...|
 |...|...|...|
 *-----------*
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Postby tarek » Mon Nov 20, 2006 9:11 am

Congratulations, 36 clues symmetrical minimal puzzle........Amazing:D

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Postby coloin » Mon Nov 20, 2006 11:05 pm

Thankyou again...to all. Heres how I did it !

Normally generating minimal sudoku puzzles with 32 and 33 and above clues is very slow and each clue is harder to add.

I had given up on the MC grid where the first 35 was found by Ocean - despite a wide search around it. But in retrospect the hints to find the 36 were in the 35 puzzle - I cant help feeling that dukuso would have taken the extra step very quickly.

One of the two known 35s in the MC grid
Code: Select all
+---+---+---+
|1..|.56|78.|
|78.|12.|45.|
|4..|7..|1.3|
+---+---+---+
|2.1|.64|8..|
|.6.|.9.|..1|
|8..|2..|5..|
+---+---+---+
|3.2|645|...|
|64.|.7.|...|
|...|..2|...|
+---+---+---+

It is a long way from our 36 but not that far !

It has the empty box - we all know now the effect of an empty box in increasing difficulty of puzzles !

The sole 9 clue in the r5c5 position has 21 grid solutions - this appeared to be an essential clue ! [It was a clue in all the 33s and 34s we had!]

Removing all of one clue undoubtably is a correct manouvere - but on its own it did not point to the region where our 36 was.

The 3 at r3c9 has 36 grid solutions.....that might have been a hint to removing a whole row and column.....and this pattern was present in many of the new 35s found in the dukuso15 grid [above].

In a "normal grid "there are 9^3 =729 ways us to remove a box, a row and column and all of one clue number. [In many of these if there is a 4 clue unavoidable exposed - there will be 2 solutions]

The reason that we found the right clues to remove so quickly in the MC grid is that many of the ways to do this in the MC grid are equivalent.

The removal of a box is equivalent. [*9]
The removal of all occurances of one clue is equivalent [*9]
The removal of 8 out the 9 ways to remove a row and column [coinciding over the box removal] allows our puzzle. [9*9*8 = 648]

This agrees with the 648 isomorphic equivalents to this puzzle.

On running the suex9s program - which removes superfluous clues from the sub grid text file,
Code: Select all
12345678.78.12345.45678.12.2315648..5648.723.8.723156.312645...645.78............
33s were generated easily - this was unusual given so many initial clues.
Code: Select all
C:\suxx>suex9s mctest.txt seed 30000 32
..3..678.78..2.45...6.8.12..3..648....4..72..8..23.5..31.645...645..8............
..3..678.78.1..45...678.12..31.648...6.8..2..8..2.15..312645...645...............
.23..678.7....345....7..12..3..6.8....48.72..8.7.315..312645...645...............
..3..678.78..2..5...678.12...1.648....48..2..8..23.56.312645...645...............
..3..678.78..2.45...67..12..3..648....4..72..8..23.5..31.645...645..8............
..3..678..8.12.45...67..12..3..648...6.8.72....72..5..312645...645...............
..3..678.7....345..5.78.12..3..648.....8..2..8.7.315..312645...645...............
Removing a few clues which never appeared in these 33s
Code: Select all
.23..678.78.12345.45678.12.2315648..5648.723.8.723156.312645...645.78............
gave me these 34s
Code: Select all
C:\suxx>suex9s mctest.txt seed 30000 33
..3..678.78..2345....78.12..31.648...6.8..2..8....15..312645...645...............
..3..678.78.1..45...678.12..31.648...6.8..2..8..2.15..312645...645...............
..3..678.78..2..5...678.12...1.648....48..2..8..23.56.312645...645...............
Removing a few more clues from the string
Code: Select all
..3..678.78.12345.45678.12.2315648..5648.723.8.723156.312645...645...............
gave me the 35s and along with these was our 36!
Code: Select all
C:\suxx>suex9s mctest.txt seed 30000 34
..3..678.78.1..45...678.12..31.648...6.8..2..8..2.15..312645...645...............   35
..3..678.78..2.45...678.12..31.648....48..2..8..23.5..312645...645...............   35
..3..678.78.12.45...678..2..31..48...648..23.8..2..5..312645...645...............   35
..3..678.78.12.45...678.12..31.648...648..2..8..2..5..312645...645...............   36


All in all, it took less than 1/2 an hour !

In a puzzle with 36 clues the chances of every clue having 2 grid solutions[all mutable and therefore a chameleon puzzle] was high on the cards - and indicates that there is no more room for extra clues!

The low stepper rate was also a predictable feature and a bonus - unlike the previous 35 - which is a difficult puzzle !.

The symmetry - perhaps it is not surprizing given the repeating nature of this grid - it probably reflects theunavoidable sets which each clue uniquely cover. I think that each clue covers a 6-clue unavoidable.

I dont even consider the possibility of a 3* in this grid !

Well...........you wont believe it !

5 more 36s.......
Code: Select all
..34.678.78.12.45...678.12..31.648...648.....8..2..5..3126.5...645....1..........
..3.5678.78.12.45...678.12..31.648...648.....8..2..5..3126.5...645....1..........
..3..678.78.12.45...678.12.231.648...648.....8..2..5..3126.5...645....1..........
..3..678.78.12.45...678.12..315648...648.....8..2..5..3126.5...645....1..........
..3..678.78.12.45...678.12..31.648..5648.....8..2..5..3126.5...645....1..........


Present known maximums in a few specified grids.
Code: Select all
                        Ran 1   Dukuso15   SFB    Coloinmax   Canon [MC]  SF     RW     Ran11
Min. number of clues    19      20         17       19        20          17     20      20
Number of 36s found      0       0          0        0         6           0      0       0 yet
Number of 35s found      4      22         57        2        38           0      2     681
Number of 34s found      -    4332          -        -         -          42      -       -

Code: Select all
ran11
592148367734269158168357249321975486645821793879634512213486975456793821987512634

C
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Posts: 1629
Joined: 05 May 2005

Postby coloin » Mon Jan 01, 2007 4:12 pm

A recently foundgrid with over 200 [certainly more] 35s is
Code: Select all
123456789457189236968372514291738465374265198685941327546813972732694851819527643
It has 36 U4s and no U6s

The U4s are
Code: Select all
                 disjoint solitary clue
{11,14,21,24}            11
{12,13,82,83}            13
{12,16,32,36}            32
{14,16,84,86}            84
{15,16,55,56}            55
{15,17,35,37}            17
{16,19,26,29}            29         
{18,19,58,59}            18
{21,22,71,72}            22
{23,25,33,35}            23
{24,25,74,75}            75
{24,26,64,66}            66
{24,28,34,38}            38
{27,28,67,68}            27
{31,33,91,93}            31
{34,35,44,45}            44
{35,36,95,96}            96
{37,39,47,49}            39
{41,48,61,68}            41
{42,43,92,93}            42
{43,47,53,57}            53
{45,47,65,67}            65
{46,49,56,59}            46
{47,48,97,98}            98
{51,52,81,82}            51
{52,59,62,69}            62
{54,58,64,68}            54
{57,59,87,89}            87
{61,63,71,73}            63
{68,69,78,79}            79
{71,74,91,94}            94
{71,78,81,88}            88           
{72,76,82,86}            76
{73,77,93,97}            97
{82,89,92,99}            99
{83,85,93,95}            85

This pseudopuzzle [with 2 solutions] has the above 36 solitary clues, each of which occurs only once in the above list of unavoidables
Code: Select all
+---+---+---+
|1.3|...|78.|
|.57|...|2.6|
|96.|...|.14|
+---+---+---+
|29.|7.8|...|
|3.4|26.|...|
|.85|.41|...|
+---+---+---+
|...|.13|9.2|
|...|69.|85.|
|...|5.7|.43|
+---+---+---+

there are 36 minimal 35 clue solutions to this pseudopuzzle [1 clue added 2 clues removed]
Code: Select all
12....78..57...2.69......1429.7.8...3.426.....85.41.......139.2...69.85....5.7.43
..34..78..57...2.696.....1429.7.8...3.426.....85.41.......139.2....9.85....5.7.43
1.3.5..8..57...2.696.....1429.7.8...3.42......85.41.......139.2...69.85....5.7.43
1.3...7.9.57...2..96.....1429.7.8...3.426.....85.41.......139.2...69.85....5.7.43
..3...78.4.7...2.696.....1429.7.8...3.426.....85.41.......139.2...69.85....5.7.43
1.3...78..5..8.2.696.....1429.7.8...3.426.....85.41........39.2...69.85....5.7.43
1.3...78..57..92..96.....1429.7.8...3.426.....85.4........139.2...69.85....5.7.43
1.3...78..57....3696......429.7.8...3.426.....85.41.......139.2...69.85....5.7.43
1.3...78..5....2.6.68....1429.7.8...3.426.....85.41.......139.2...69.85....5.7.43
1.3...78..57...2.696.3....429...8...3.426.....85.41.......139.2...69.85....5.7.43
1.3...78..57...2.69....2.1429.7.8...3.426.....85.41.......139.2...69.85....5...43
1.3....8..57...2.696....51.29.7.8...3.426.....85.41.......139.2...69.85....5.7.43
1.3...78..57...2.696.....142.17.8...3..26.....85.41.......139.2...69.85....5.7.43
1.3...78..57...2.696.....1429..38...3.426.....85..1.......139.2...69.85....5.7.43
1.3...78..57...2.696.....14.9.7.8.6.3.426.....85.41.......139.2...69.85....5.7..3
1.3...78..57...2.696.....1.29.7....53.426.....85.41.......139.2...69.85....5.7.43
1.3...78..57...2.696.....1429.7.8....7426......5.41.......139.2...69.85....5.7.43
1.3...78..57...2.696.....1429.7.....3.42.5....85.41.......139.2...69.85....5.7.43
1.3...78..57...2.696.....1429.7.8...3..26.1...85.41.......139.2...69..5....5.7.43
1.3...7...57...2.696.....1429.7.8...3.4.6..9..85.41.......139.2...69.85....5.7.43
1.3...78..57...2.696.....14.9.7.8...3.426....68..41.......139.2...69.85....5.7.43
1.3...78..57...2.696.....1429.7.8...3.4.6.....8594........139.2...69.85....5.7.43
1.3...78..57.....696.....1429.7.8...3.426.....85..13......139.2...69.85....5.7.43
1.3...78..57...2.696.....1429.7.8...3.426......5.41..7....139.....69.85....5.7.43
1.3...78...7...2.696.....1429.7.8...3.426.....85.41....4..1.9.2...69.85....5.7.43
1.3...78..57...2.696.....1429.7.8...3.426.....8..41.....6.13..2...69.85....5.7.43
1.3...78..57...2.696.....1429.7.8...3.426.....85.41......8.39.2...69.85......7.43
1.3...78..57...2.696.....1429.7.8...3.426.....85.41.......1397....69.8.....5.7.43
1.3...78..57...2.696.....1429.7.8.....426.....85.41.......139.27..69.8.....5.7.43
1.....78..57...2.696.....1429.7.8...3.426.....85.41.......139.2..26..85....5.7.43
1.3...78..57...2.696.....1429.7.8...3.426.....85.41.......1.9.2....9485....5.7.43
1.3...78..57...2.696.....1429.7.8...3.426.....85.41.......139.2...69..51...5.7.4.
1.3...78..57...2.6.6.....1429.7.8...3.426.....85.41.......139.2...69.85.8....7.43
1.3...78..57...2.696.....142..7.8...3.426.....85.41.......139.2...69.85..1.5.7.4.
1.3...78..57...2.696.....1429.7.8...3.426.....85.41.......139.2...6..85....52..43
1.3...78..57...2.696.....1429.7.8...3.426.....85.41.......13..2...69.85....5.76.3

The original 36 clue pseudopuzzle has 2 solutions
Code: Select all
123456789457189236968372514291738465374265198685941327546813972732694851819527643 - our "No U6" grid
143926785857134296962875314296758431314269578785341629578413962431692857629587143 - the other solution

This other solution grid is the MC most canonical grid !!!!![150 U6s]

Amazing ?

If we do the same to the dukuso15 grid
We get a pseudopuzzle with 2 solutions
Code: Select all
+---+---+---+
|...|...|...|
|.64|.91|.52|
|.57|.43|.81|

+---+---+---+
|.18|.57|.34|
|.36|.89|.27|
|...|...|...|
+---+---+---+
|.91|.25|.46|
|.72|.36|.98|
|...|...|...|
+---+---+---+ 2 sols

.......the MC grid is also the other solution grid !

Leading to a least 70 more different 35s in dukuso15

Present known maximums in a few specified grids.
Code: Select all
                        Ran 1   Dukuso15   SFB    Coloinmax   Canon [MC]  SF     RW     Ran11   No6grid
Min. number of clues    19      20         17       19        20          17     20       20       20
Number of 36s found      0       0          0        0         6           0      0       12        0
Number of 35s found      4      94         63        2        38           0      2     1550      288
Number of 34s found      -    4332          -        -         -          42      -       -       -


Eventually after finding more and more 35s in Ran11 I found a 36.....
Here are some minimal 36s in Ran11
Code: Select all
....48...7..2.9.........2.93....5486..5.....38...345.22.34869.5.56.938..9..5.2.34
....48...7..2.9.........2.93....5486..5.....38...345..2.34869.5.56.9382.9..5.2.34
....48...7..2.9.........2.93....54.6..58....38...345.22.34869.5.56.938..9..5.2.34
....48...7..2.9.........2.93....54.6..58....38...345..2.34869.5.56.9382.9..5.2.34
....48...7....9.........2493....5486..5.....38...345.22.34869.5.56.938..9..5.2.34
....48...7....9.........2493....5486..5.....38...345..2.34869.5.56.9382.9..5.2.34      ER 6.6


I think the limit is reached.....

EDIT Well not quite..........

Here are 5 more minimal 36s - all with an ER of 8.4 !
Code: Select all
..2.4.3.......9....6....249..197.48...58...938...345..2...8.9.5.5679382..8.5.2.3.    8.4 
..2.4.3.......9....6...7249..197.48...58...93....345..2..48.9.5.56.9382..8.5.2.3.    8.4 
..2.4.3.......9....6...7249..197.48...58...938...345..2..48.9.5.56.9382....5.2.3.    8.4 
..2.4.3.......9....68..724...197.48...58...93....345..2..48.9.5.56.9382..8.5.2.3.    8.4 
..2.4.3.......9....68..724...197.48...58...938...345..2..48.9.5.56.9382....5.2.3.    8.4 
:D
C

EDIT And theres one more !

Code: Select all
..2...3.......9....6.3..24....9.548...58..7938...345..2..48.9.5.5679382..8.5.2.3.    7.7

coloin
 
Posts: 1629
Joined: 05 May 2005

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