The best symmetry

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

Postby tarek » Thu May 22, 2008 8:41 am

This is great...

So all of our batch postings are going to change forever:(

I will be using it for postings in that regard.

I hope that it is not too slow, in any case that would be an incentive towards limiting the number of posted puzzles.

Thanx,

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Postby gsf » Thu May 22, 2008 9:26 am

tarek wrote:I will be using it for postings in that regard.

I hope that it is not too slow, in any case that would be an incentive towards limiting the number of posted puzzles.

too slow : ~2 puz/min/Ghz
but SE is still the slow part in hardest analysis
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Postby ronk » Thu May 22, 2008 11:42 am

Code: Select all
# Golden Nugget
 . . . | . . 9 | . . 5                . . . | . . 1 | . . 5
 . 3 . | . . . | . 7 .                . 3 . | . . . | . 7 .
 . . 2 | . 8 . | 6 . .                . . 2 | . 8 . | 6 . .
-------+-------+-------              -------+-------+-------
 . . 8 | . . 2 | 4 . .                . . . | 6 . 8 | 9 . .
 . . . | 6 4 . | . . .                . . 8 | 4 . . | . . .
 1 . . | 8 . . | . . .                9 . . | . 2 . | . . .
-------+-------+-------              -------+-------+-------
 . . 6 | 9 . . | 2 . .                . . 6 | . 4 . | 2 . .
 . 7 . | . . . | . 1 .                . 7 . | . . . | . 1 .
 5 . . | . . . | . . 3                5 . . | . . . | . . 3
# d-3 (including b5)                  # d-2 (excluding b5)


gsf, since I'm the bloke that suggested finding best symmetry excluding the center box, I find that result somewhat unsatisfactory.

Since both permutations are d-2 when ignoring b5, shouldn't including the effect of b5 be the tie-breaker? Or is row minlex a higher priority tie-breaker?
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Postby gsf » Thu May 22, 2008 3:03 pm

ronk wrote:
Code: Select all
# Golden Nugget
 . . . | . . 9 | . . 5                . . . | . . 1 | . . 5
 . 3 . | . . . | . 7 .                . 3 . | . . . | . 7 .
 . . 2 | . 8 . | 6 . .                . . 2 | . 8 . | 6 . .
-------+-------+-------              -------+-------+-------
 . . 8 | . . 2 | 4 . .                . . . | 6 . 8 | 9 . .
 . . . | 6 4 . | . . .                . . 8 | 4 . . | . . .
 1 . . | 8 . . | . . .                9 . . | . 2 . | . . .
-------+-------+-------              -------+-------+-------
 . . 6 | 9 . . | 2 . .                . . 6 | . 4 . | 2 . .
 . 7 . | . . . | . 1 .                . 7 . | . . . | . 1 .
 5 . . | . . . | . . 3                5 . . | . . . | . . 3
# d-3 (including b5)                  # d-2 (excluding b5)


gsf, since I'm the bloke that suggested finding best symmetry excluding the center box, I find that result somewhat unsatisfactory.

Since both permutations are d-2 when ignoring b5, shouldn't including the effect of b5 be the tie-breaker? Or is row minlex a higher priority tie-breaker?

the center box is ignored period
I'll keep tiebreakers in mind
thanks
gsf
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Postby ronk » Thu May 29, 2008 2:33 pm

gsf wrote:
ronk wrote:Since both permutations are d-2 when ignoring b5, shouldn't including the effect of b5 be the tie-breaker? Or is row minlex a higher priority tie-breaker?

the center box is ignored period

gsf, a tangentially related observation:

Using -qFN -f"%#cc # %#C#sc" on coloin's new Platinum Blonde ...

Code: Select all
+---+---+---+
|...|...|.12|
|...|...|..3|
|..2|3..|4..|
+---+---+---+
|..1|8..|..5|
|.6.|.7.|8..|
|...|..9|...|
+---+---+---+
|..8|5..|...|
|9..|.4.|5..|
|47.|..6|...|
+---+---+---+ Platinum Blonde

... produces ...

Code: Select all
 . . 3 | . . . | . . .
 . 4 . | . . 2 | . . 3
 1 . 2 | . . . | . . .
-------+-------+-------
 . . . | . . . | . 9 .
 . . 5 | . . 1 | . . 8
 . 8 . | 6 . . | 7 . .
-------+-------+-------
 . . . | . . 8 | . . 5
 . . . | 7 4 . | . 6 .
 . 5 . | . 9 . | 4 . .  # d-2

Vertical reflection would move the single clue in row 1 from box 1 to box 3. Isn't row minlex of the pattern supposed to be a factor here:?:
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Postby gsf » Thu May 29, 2008 7:30 pm

ronk wrote:Isn't row minlex of the pattern supposed to be a factor here:?:

only in a tie break with another symmetry with same distance
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Postby ronk » Thu May 29, 2008 7:55 pm

gsf wrote:
ronk wrote:Isn't row minlex of the pattern supposed to be a factor here:?:

only in a tie break with another symmetry with same distance

I agree, but this looks like "d-2" versus "a-2" to me ... where the row minlex should vote for "a-2."
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Postby gsf » Thu May 29, 2008 10:41 pm

ronk wrote:
gsf wrote:
ronk wrote:Isn't row minlex of the pattern supposed to be a factor here:?:

only in a tie break with another symmetry with same distance

I agree, but this looks like "d-2" versus "a-2" to me ... where the row minlex should vote for "a-2."

right you are
the code did not match the documentation
the code used dihedral index which is the index in the table of dihedral elements
it should use dihedral order, and will in the next release
with this fix it produces a-2
thanks
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Re:

Postby m_b_metcalf » Fri May 06, 2022 1:02 pm

JPF wrote:Could we find minimal puzzles with symmetry modulo n ; n>9 ?

Have I understood your question correctly? Does this puzzle have symmetry modulo 11 according to your definition?
Code: Select all
  .  .  .  .  *  .  *  .  1
  .  .  .  .  *  2  .  3  .
  .  4  5  *  .  *  2  6  7
  *  .  .  .  4  8  *  .  .
  *  8  .  3  .  5  .  9  4
  .  .  4  9  2  .  .  .  3
  *  9  8  5  .  4  3  *  .
  .  3  .  *  8  .  .  .  .
  5  .  2  .  9  .  .  .  .   rotational symmetry 'off-by'11'
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Re: The best symmetry

Postby JPF » Fri May 06, 2022 4:32 pm

I think your puzzle has a diagonal symmetry modulo 7:
Code: Select all
+---+---+---+
|...|...|..5|
|..4|..5|.82|
|.8.|..4|39.|
+---+---+---+
|...|..2|.3.|
|.9.|.36|...|
|.43|1.7|...|
+---+---+---+
|.39|...|.5.|
|4.2|...|8.9|
|85.|.2.|.4.|
+---+---+---+

result given by gsf's program:
Code: Select all
........5..4..5.82.8...439......2.3..9..36....431.7....39....5.4.2...8.985..2..4. # d-7

In my definition, we have first to find the symmetry having the best symmetry .

JPF
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Re: The best symmetry

Postby m_b_metcalf » Sat May 07, 2022 11:48 am

Jean-Pierre,
After a longer search, this is the best I can come up with, a few puzzles that I think have n > 9 in rotational, diagonal, anti-diagonal and up-down symmetries, but fall at the last, left-right hurdle with n = 6:

Code: Select all
.87.........2.19....1.87......5642..84....5.92..8..3.6..81....3.6........7.4....5     
986.....2....3.........76.41..3.4...4....98...6.7.....61..7.2....742.1.88....1.36     
9...5...37...3..8132....6.4.7...8.6..3.29......5......84...1..9......1....2..98.6     
.9.5........63..813..9.76....9...........4.....51..3.847.3.9.25.1.......96......7     


Regards,

Mike
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Re: The best symmetry

Postby JPF » Sat May 07, 2022 7:33 pm

For each of those puzzles could you show the morphs with n=6 ?

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Re: The best symmetry

Postby m_b_metcalf » Sun May 08, 2022 10:19 am

JPF wrote:For each of those puzzles could you show the morphs with n=6 ?

Actually, no, because I appear to have made an error in the left-right check, which now gives values of 11, 13, 11, and 12, respectively. If there is no further error (it's all new code), then these four puzzles would have n > 9 for all five symmetries. In fact, I found more, all together eight:
Code: Select all
...456.8..5.7...32.8.....46........5....12.....156....36....9..5.8.9..649......58     
1.....78......91.278........7.5....4.6.......8..3....1..26.3..8.489.5........2465     
...4.67.........3.6892.......863.1..7..8.1.24.16..2.7..6......7..4..89....1...4.3     
.234.6.....718..3...93...5.2..6.89.......1....489..1..5.........3.....92.7..6.8..     
.234.6.8..5.1....6.98............9.....91236..4.....7...47.5....6....8..7.2.9....     
..34..7..4...8.2.6..8...1.523.............6...648.7..3.1.7....8...9......8.3..92.     
..3..67.9...18..36.9......5.697......1........7452.9.35..8.3..1......4..9........   
.23.5.......1892..86.3.....27....195.......2...4...3.7.......4....29..7.7.6....5.   


Regards,

Mike

P.S. The puzzles come from Denis Berthier's collection.
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Re: The best symmetry

Postby JPF » Sun May 08, 2022 8:43 pm

Excellent!

Here are other minimal puzzles with best symmetry = 10:
Code: Select all
.1....2344.2..516.........7....4.781..7.1.9...8.29............5.5.7.96.21.64.....
123.4..5.....6.1.4......2.7....825.1............3...6...6...4..3.5.748...42.5..76
........12.1..3456..6.74..8.....138.7....5....29......4...6.5..915...........9..4
.1.......2.3.45.67..68..1536........1..354..6...7....8..2..6371....7.9.44........
1...23........4...25.67.....4523.8.99.......27.1.5.3...7..4..2.....9.....9.36..5.
123.4........5.....5.6...7271....8.....4.6...296.1...59..8.2.14......6........958
...1....23.1.4.5..6..3..17.8..43...97..9.1...4....2..6.84...9....5...64.1.....2..
.....1....23.4....1..5632...6..75.2.2.......8.5..8.67.3.21.8....4...65.....4...8.
.1..234....5...........627.8....1.5.......8...5.2.4.1..835..62....3..7.1..146...8
......1234....56....3..2.7.3...74.5.........76..58..94..9...5....642.9.88.1......
.1...2.3...4.3.....5....678.79.......3..1....1......9...2....6.....2475..978.312.
.....1.......23.4.564......6.3..2....51.672..7.....1.33....86.9.9.17.5...1.......
....123.4...56..781..47.....4.....296...49....2............5..321..9.4..3.5.....7

Now, the new target is 11.

JPF
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Re: The best symmetry

Postby m_b_metcalf » Mon May 09, 2022 9:44 am

JPF wrote:Now, the new target is 11.

I think I'll pass on that. But here are a few more puzzles with what I regard as the most chaotic patterns:
Code: Select all
...456.8..5.7...32.8.....46........5....12.....156....36....9..5.8.9..649......58     
...4.67.........3.6892.......863.1..7..8.1.24.16..2.7..6......7..4..89....1...4.3     
.234.6.8..5.1....6.98............9.....91236..4.....7...47.5....6....8..7.2.9....     
.23.5.......1892..86.3.....27....195.......2...4...3.7.......4....29..7.7.6....5.     
12.4.....4.7.8....69....4.1.3........61.....5..5.7..12..45..1.....8...2...276453.     


Mike
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