Fully symmetrical puzzles

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

Postby coloin » Thu Sep 06, 2007 10:20 pm

mauricio wrote:
Code: Select all
+-------+-------+-------+
| . . . | . 1 . | . . . |
| . . 1 | 2 3 4 | 5 . . |
| . 2 6 | 5 . 7 | 4 1 . |
+-------+-------+-------+
| . 3 4 | . . . | 6 2 . |
| 2 1 . | . . . | . 5 4 |
| . 6 8 | . . . | 1 3 . |
+-------+-------+-------+
| . 8 2 | 7 . . | 3 4 . |
| . . 3 | 1 2 8 | 7 . . |
| . . . | . 4 . | . . . |
+-------+-------+-------+ 35 clue minimal


Well done, Ive played around and cant get near this....its not far at all to get a 36 now...... [:) I jest]

However you might want the solution grid of your eventual puzzle to have the 9 clues arranged similarly to this - so as they dont feature in your puzzle !
Code: Select all
+---+---+---+
|...|9..|...|
|.9.|...|...|
|...|...|..9|
+---+---+---+
|9..|...|...|
|...|...|9..|
|...|..9|...|
+---+---+---+
|...|.9.|...|
|...|...|.9.|
|..9|...|...|
+---+---+---+

Code: Select all
....8......53.76...782.4.3..86...57.53.....26.47...81..524.136...4835............
....8......57..6...784.312..86...5..53.....46.47...31..523.7.6...48152......4....

two 32 clue puzzles with the 9s sited advantageously.
Code: Select all
....4......56.83...863.154..64...83.2......64.37...12..781.425...27.64......5....
....4......56.83...862..54..34...82.2......64.67...13..781.425...28674......5....

Im improving:!:
two 33 clue minimal puzzles

C
coloin
 
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Postby Mauricio » Sat Sep 08, 2007 1:34 am

I tried a different pattern:
Code: Select all
+-------+-------+-------+
| . . . | . x . | . . . |
| . x x | x . x | x x . |
| . x x | x . x | x x . |
+-------+-------+-------+
| . x x | . . . | x x . |
| x . . | . . . | . . x |
| . x x | . . . | x x . |
+-------+-------+-------+
| . x x | x . x | x x . |
| . x x | x . x | x x . |
| . . . | . x . | . . . |
+-------+-------+-------+

A 36 with 35 matching clues:
Code: Select all
+-------+-------+-------+
| . . . | . 1 . | . . . |
| . 1 2 | 3 . 4 | 5 6 . |
| . 5 3 | 6 . 2 | 1 7 . |
+-------+-------+-------+
| . 8 5 | . . . | 4 3 . |
| 3 . . | . . . | . . 7 |
| . 2 7 | . . . | 6 . . |
+-------+-------+-------+
| . 7 4 | 8 5 3 | 9 1 . |
| . 3 8 | 1 . 9 | 7 4 . |
| . . . | . 4 . | . . . |
+-------+-------+-------+

A 35 with diagonal symmetry
Code: Select all
+-------+-------+-------+
| . . . | . 1 . | . . . |
| . 1 2 | 3 . 4 | 5 6 . |
| . 7 3 | 2 . 5 | 4 1 . |
+-------+-------+-------+
| . 2 6 | . . . | 1 4 . |
| 3 . . | . . . | . . 6 |
| . 8 1 | . . . | 3 9 . |
+-------+-------+-------+
| . 3 8 | 7 . 9 | . 5 . |
| . 6 5 | 1 . 8 | 9 7 . |
| . . . | . 5 . | . . . |
+-------+-------+-------+

A 34 with double diagonal symmetry
Code: Select all
+-------+-------+-------+
| . . . | . 1 . | . . . |
| . . 1 | 2 . 3 | 4 5 . |
| . 2 3 | 6 . 4 | 1 7 . |
+-------+-------+-------+
| . 1 8 | . . . | 9 4 . |
| 7 . . | . . . | . . 5 |
| . 3 5 | . . . | 7 2 . |
+-------+-------+-------+
| . 5 7 | 4 . 8 | 2 9 . |
| . 8 2 | 9 . 1 | 5 . . |
| . . . | . 2 . | . . . |
+-------+-------+-------+

But unfortunately no 36.
Mauricio
 
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Joined: 22 March 2006

Postby coloin » Sat Sep 08, 2007 8:51 pm

Very respectable effort, I think you will get it one day......just may take a little longer.

The pattern changes look interesting, although with my method I dont think I will get furthur than this 34 without expanding into other grid solutions, which is indeed what we will have to do. Anyhow it is again without the "9" clues [in the template] which I believe is where you have hit solid rock.

Code: Select all
+---+---+---+
|...|.1.|...|
|..1|284|7..|
|.27|5.6|41.|
+---+---+---+
|.74|...|32.|
|21.|...|..4|
|.63|...|1..|
+---+---+---+
|.82|3.5|64.|
|..5|168|2..|
|...|.4.|...|
+---+---+---+ 34 clues


Perhaps you can expand on this one ?

C
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Postby ravel » Sun Sep 09, 2007 2:36 pm

Nice work, Mauricio.

I start to believe in a fully symmetric 36:)
ravel
 
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Postby Mauricio » Tue Sep 11, 2007 12:40 am

The nature of the patterns of those high clues sudokus is strange, sometimes I can't find a 32, for some patterns I can randomly find a 31, but not a 34 (even with specialized algorithms).
Recently I have expanded my search space not just to valid minimal puzzles, but to invalid (multisolutional) "minimal" puzzles too, that has helped greatly.

Many patterns don't look like snow flakes:
Code: Select all
+-------+-------+-------+
| . 1 2 | . . . | 3 4 . |
| 3 4 . | . . . | . 5 6 |
| 7 . 6 | . 3 . | 2 . 1 |
+-------+-------+-------+
| . . . | . 8 6 | . . . |
| . . 4 | 3 . 5 | 6 . . |
| . . . | 1 4 2 | . . . |
+-------+-------+-------+
| 4 . 1 | . 5 . | 8 . 2 |
| 2 3 . | . . . | . 6 5 |
| . 8 5 | . . . | 1 7 . |
+-------+-------+-------+
Mauricio
 
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Postby Mauricio » Mon Sep 17, 2007 5:22 am

My morale goes down with every pattern "failed". But I have to admit, generating a minimal puzzle with high number of clues and fixed pattern is not the easiest of things, and I am very proud of my improvement in that matter.
Code: Select all
+-------+-------+-------+
| . . . | . . . | . . . |
| . . 1 | 2 3 4 | 5 . . |
| . 2 4 | 6 . 7 | 3 1 . |
+-------+-------+-------+
| . 4 5 | 3 . 8 | 7 6 . |
| . 8 . | . . . | . 5 . |
| . 7 6 | 4 . 5 | . 8 . |
+-------+-------+-------+
| . 1 2 | 7 . 3 | 8 4 . |
| . . 8 | 5 4 2 | 1 . . |
| . . . | . . . | . . . |
+-------+-------+-------+
Code: Select all
+-------+-------+-------+
| . . . | . 1 . | . . . |
| . 2 . | 3 . 4 | . . . |
| . . 1 | 2 5 6 | 3 . . |
+-------+-------+-------+
| . 5 2 | . 7 . | 6 3 . |
| 4 . 7 | 6 . 5 | 2 . 8 |
| . 8 3 | . 2 . | 7 4 . |
+-------+-------+-------+
| . . 4 | 7 6 3 | 8 . . |
| . 3 . | 5 . 1 | . 7 . |
| . . . | . 4 . | . . . |
+-------+-------+-------+
Mauricio
 
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Postby TTHsieh » Thu Sep 27, 2007 5:06 am

    28 clues, Hard Pattern ?
    Is there any minimal puzzle ?
Code: Select all
 x . . | . . . | . . x
 . x . | x x x | . x .
 . . . | . . . | . . .
 - - - + - - - + - - -
 . x . | x x x | . x .
 . x . | x . x | . x .
 . x . | x x x | . x .
 - - - + - - - + - - -
 . . . | . . . | . . .
 . x . | x x x | . x .
 x . . | . . . | . . x

    100000001010111010000000000010111010010101010010111010000000000010111010100000001
TTHsieh
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Postby gsf » Thu Sep 27, 2007 7:20 am

TTHsieh wrote:
    28 clues, Hard Pattern ?
    Is there any minimal puzzle ?
Code: Select all
 x . . | . . . | . . x
 . x . | x x x | . x .
 . . . | . . . | . . .
 - - - + - - - + - - -
 . x . | x x x | . x .
 . x . | x . x | . x .
 . x . | x x x | . x .
 - - - + - - - + - - -
 . . . | . . . | . . .
 . x . | x x x | . x .
 x . . | . . . | . . x


here's one, not minimal
Code: Select all
6 . . | . . . | . . 3
. 7 . | 6 4 9 | . 1 .
. . . | . . . | . . .
------+-------+------
. 5 . | 9 3 7 | . 6 .
. 2 . | 1 . 6 | . 9 .
. 1 . | 2 8 4 | . 7 .
------+-------+------
. . . | . . . | . . .
. 9 . | 3 2 1 | . 5 .
4 . . | . . . | . . 9
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Postby TTHsieh » Thu Sep 27, 2007 8:04 am

Up to now, I can't find any minimal puzzle of this pattern.
The minimal rate of this pattern maybe is less than 1/1000000, 1/2000000, ..., or no minimal puzzle.

who knows...

TTHsieh
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Joined: 13 July 2007

Postby JPF » Thu Sep 27, 2007 3:13 pm

TTHsieh wrote:
    28 clues, Hard Pattern ?
    Is there any minimal puzzle ?
Code: Select all
 x . . | . . . | . . x
 . x . | x x x | . x .
 . . . | . . . | . . .
 - - - + - - - + - - -
 . x . | x x x | . x .
 . x . | x . x | . x .
 . x . | x x x | . x .
 - - - + - - - + - - -
 . . . | . . . | . . .
 . x . | x x x | . x .
 x . . | . . . | . . x



Here are 2 fully symmetric sub-patterns :

20 clues, minimal
Code: Select all
 1 . . | . . . | . . 2
 . 3 . | 4 . 5 | . 6 .
 . . . | . . . | . . .
-------+-------+-------
 . 5 . | . 2 . | . 7 .
 . . . | 1 . 8 | . . .
 . 6 . | . 3 . | . 9 .
-------+-------+-------
 . . . | . . . | . . .
 . 4 . | 7 . 6 | . 5 .
 3 . . | . . . | . . 8



24 clues, not minimal
Code: Select all
 
 1 . . | . . . | . . 2
 . 3 . | 4 5 6 | . 1 .
 . . . | . . . | . . .
-------+-------+-------
 . 1 . | 7 . 2 | . 6 .
 . 2 . | . . . | . 4 .
 . 5 . | 6 . 8 | . 9 .
-------+-------+-------
 . . . | . . . | . . .
 . 4 . | 3 2 7 | . 5 .
 6 . . | . . . | . . 1
 

Any minimal for this last one ?

JPF
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Postby TTHsieh » Thu Sep 27, 2007 3:59 pm

JPF wrote:Any minimal for this last one ?

Here are some minimal puzzles.
Code: Select all
 1 . . | . . . | . . 2
 . 3 . | 4 5 6 | . 1 .
 . . . | . . . | . . .
 - - - + - - - + - - -
 . 6 . | 7 . 2 | . 8 .
 . 2 . | . . . | . 4 .
 . 5 . | 8 . 9 | . 3 .
 - - - + - - - + - - -
 . . . | . . . | . . .
 . 4 . | 2 1 8 | . 5 .
 3 . . | . . . | . . 6   ER=2.0


minimal puzzles wrote:
100000002030456010000000000060702080020000040050809030000000000040218050300000006 # 2.0
100000002030456010000000000070803060020000040050901080000000000010382050600000008 # 2.3
100000002020345060000000000010602050070000030040508090000000000030291040500000009 # 2.5
100000002030245060000000000010702050060000080040506030000000000020193040500000003 # 2.6
100000002030145060000000000070403050060000010040708020000000000010327040500000006 # 2.8
100000002020345010000000000010602050070000030040501080000000000030827040500000008 # 6.6
...


TTHsieh
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Postby m_b_metcalf » Thu Sep 27, 2007 6:15 pm

JPF wrote:Any minimal for this last one ?

JPF


I came up with these:
Code: Select all
800000001060491030000000000010207080050000040070608090000000000040836050600000009 1 SE 2.0
800000005060491030000000000010205080050000040070608090000000000040836050600000009 2 SE 2.8
800000005060194030000000000010205080050000040070608090000000000040836050600000009 3 SE 2.8
800000001030561080000000000010902060020000040050704030000000000040387050600000003 4 SE 7.1
400000001070259060000000000020705030010000040050802010000000000040371020500000007 5 SE 2.8
400000001070258060000000000020705030010000040050806010000000000040371020500000007 6 SE 2.8
400000001070258060000000000020805030010000040050906010000000000040371020500000007 7 SE 2.8

I haven't checked for isomorphs.

Regards,

Mike Metcalf
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Postby Mauricio » Thu Sep 27, 2007 8:21 pm

TTHsieh wrote:
    28 clues, Hard Pattern ?
    Is there any minimal puzzle ?
Code: Select all
 x . . | . . . | . . x
 . x . | x x x | . x .
 . . . | . . . | . . .
 - - - + - - - + - - -
 . x . | x x x | . x .
 . x . | x . x | . x .
 . x . | x x x | . x .
 - - - + - - - + - - -
 . . . | . . . | . . .
 . x . | x x x | . x .
 x . . | . . . | . . x


The best I could do is this minimal 27:
Code: Select all
+-------+-------+-------+
| 1 . . | . . . | . . 2 |
| . 2 . | 1 3 4 | . 5 . |
| . . . | . . . | . . . |
+-------+-------+-------+
| . 1 . | . 2 6 | . 7 . |
| . 4 . | 5 . 1 | . 3 . |
| . 5 . | 4 8 7 | . 6 . |
+-------+-------+-------+
| . . . | . . . | . . . |
| . 3 . | 7 6 2 | . 1 . |
| 7 . . | . . . | . . 4 |
+-------+-------+-------+

All minimal puzzles that I found with 27 clues, have the same pattern, and I was not looking for that.
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Postby JPF » Thu Sep 27, 2007 9:48 pm

Thanks TTHsieh, m_b_metcalf

Here is a fully symmetric and minimal puzzle
Code: Select all
 1 . . | . . . | . . 2
 . 3 . | 1 . 4 | . 5 .
 . . . | . . . | . . .
-------+-------+-------
 . 5 . | 6 . 3 | . 4 .
 . . . | . . . | . . .
 . 6 . | 4 . 7 | . 3 .
-------+-------+-------
 . . . | . . . | . . .
 . 4 . | 3 . 5 | . 6 .
 8 . . | . . . | . . 9


... with 3 solutions:(

JPF
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Postby m_b_metcalf » Thu Sep 27, 2007 10:11 pm

JPF wrote:Here is a fully symmetric and minimal puzzle
Code: Select all
 1 . . | . . . | . . 2
 . 3 . | 1 . 4 | . 5 .
 . . . | . . . | . . .
-------+-------+-------
 . 5 . | 6 . 3 | . 4 .
 . . . | . . . | . . .
 . 6 . | 4 . 7 | . 3 .
-------+-------+-------
 . . . | . . . | . . .
 . 4 . | 3 . 5 | . 6 .
 8 . . | . . . | . . 9


... with 3 solutions:(

JPF

Can a 'puzzle' really be described as both minimal and as having several solutions? One with multiple solutions is surely sub-minimal.

Regards,

Mike Metcalf
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