## Symmetric 18s

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

### Re: Symmetric 18s

Hi, colleagues!
I found an arithmetic error in my calculations (manual calculations are error-prone). eleven, you are right - total numbers of 18-clue symmetric patterns is still 2,402,330,656.

About filtering out patterns not complying to composition rules. I have proper C code, but embedded in another program. I'll try to code autonomous utility (reading eleven's file) ASAP.

Serg
Serg
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### Re: Symmetric 18s

eleven wrote:With 2 automorphisms each puzzle has 3 equivalents with the same pattern.

Understanding automorphism eluded me for a long time, so I could still be wrong, but ... shouldn't that be "1 equivalent?"
ronk
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### Re: Symmetric 18s

I think its achieved by swapping c46 = 2
swapping stacks 1 and 3 and swapping c123 and c789 = 2

so at least 4 equivalents for each pattern
C
coloin

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### Re: Symmetric 18s

coloin wrote:I think its achieved by swapping c46 = 2
swapping stacks 1 and 3 and swapping c123 and c789 = 2

so at least 4 equivalents for each pattern

OK, I should have been thinking about vertical symmetry, then automorphisms >= 4, not 2.

here gsf wrote:my solver has code to canonicalize fish exemplars
a fallout of that process is the number of automorphisms

Use gsf's program on a file containing all known symmetric 18s to verify automorphisms >= 4.
Last edited by ronk on Tue Apr 17, 2012 5:13 pm, edited 1 time in total.
ronk
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I currently have analyzed about 3000 patterns and I can confirm that the first 2147 patterns have no uniquely solvable 18 clue Sudoku.

eleven, I look into your question.

Edit: This one has the most essentially different puzzles of all patterns with a total of 4,693,836,890 essentially different puzzles and 4 automorphisms.
Code: Select all
`....X......X...X...X.....X....X.X.....X...X..X.......X...XXX....X.....X.X.......X`

It should take about a day to solve all those puzzles. An honourable mention goes to
Code: Select all
`....X......X...X....X...X.....X.X....X.....X..X.....X....XXX...X.......XX.......X`

with 1,146,653,411 essentially different puzzles and 32 automorphisms.
Afmob

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Joined: 28 June 2011

ronk wrote:
coloin wrote:
eleven wrote:With 2 automorphisms,
each puzzle has 3 equivalents with the same pattern

• swapping c46
• swapping stacks 1 and 3 and swapping c123 and c789
so at least 4 equivalents

OK, I should have been thinking about vertical symmetry, then automorphisms >= 4, not 2.

eleven + coloin are right

at least 2 automorphisms
==> each puzzle is part of a class of at least 4 automorphs
i.e. has at least 3 automorphs

Pat

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Joined: 18 July 2005

### Re:

Afmob wrote:This one has the most essentially different puzzles of all patterns with a total of 4,693,836,890 essentially different puzzles and 4 automorphisms.
Code: Select all
`....X......X...X...X.....X....X.X.....X...X..X.......X...XXX....X.....X.X.......X`

It should take about a day to solve all those puzzles. An honourable mention goes to
Code: Select all
`....X......X...X....X...X.....X.X....X.....X..X.....X....XXX...X.......XX.......X`

with 1,146,653,411 essentially different puzzles and 32 automorphisms.

Thanks Afmob,
the first pattern is not very encouraging for this project. To solve all puzzles would probably take me 1 1/2 days on my PC on a single core.

I came across this one with 48 automorphs (dont know yet, if this is the maximum in the list).
Code: Select all
`.............1.......1.1.....1...1....1...1....1...1....11111...1.....1.1.......1`
eleven

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Joined: 10 February 2008

### Re: Symmetric 18s

Do you mean maximum number of automorphisms? While searching for the one with the most essentially different puzzles I found one with 576 automorphisms if I remember correctly but there might be patterns in your file with more automorphisms.

By the way, I mentioned the other pattern with 32 automorphisms because if you disregard the automorphisms it would have the highest number of puzzles (over 32 billion).
Afmob

Posts: 130
Joined: 28 June 2011

### Re: Symmetric 18s

Hi, colleagues!
I filtered eleven's list of patterns by applying composition rules. Here is my program report.
Code: Select all
`Composition rules filter (8 rules), version 1.0Total number of processed patterns: 294742Filtering statisticsEmpty rows (columns): filtered out      0 patternsComposition rule 1:   filtered out  22670 patternsComposition rule 2:   filtered out  40853 patternsComposition rule 3:   filtered out    379 patternsComposition rule 4:   filtered out      0 patternsComposition rule 5:   filtered out      0 patternsComposition rule 6:   filtered out    420 patternsComposition rule 7:   filtered out      0 patternsComposition rule 8:   filtered out   5567 patternsTotally filtered out patterns:  69889               Valid patterns: 224853Total program execution time: 6 seconds`

Number of "rectified" patterns - 224853. So, eleven was right when he predicted not less than 200000 patterns after filtering out invalid patterns. I hoped for the better ... . On the other hand we can imagine that we have already done exhaustive search for 70000 patterns. Not so bad .

Zipped file with filtered patterns has size almost 1 MB. You know this forum limits alone attached file size to 256 Kb. So, I had to split entire file onto 4 parts - the first part is attached to this post, other 3 parts are attached to my 3 next (dummy) posts.

Serg
Attachments
sym18pat_filtered.zip
Last edited by Serg on Tue Apr 17, 2012 9:47 pm, edited 1 time in total.
Serg
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### Re: Symmetric 18s

Part 2 of 4
You must rename this file after downloading to "sym18pat_filtered.z01" and move to the directory, containing 1st part of archive.
Attachments
sym18pat_filtered_z01.zip
Serg
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Location: Russia

### Re: Symmetric 18s

Part 3 of 4
You must rename this file after downloading to "sym18pat_filtered.z02" and move to the directory, containing 1st part of archive.
Attachments
sym18pat_filtered_z02.zip
Serg
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Location: Russia

### Re: Symmetric 18s

Part 4 of 4
You must rename this file after downloading to "sym18pat_filtered.z03" and move to the directory, containing 1st part of archive.
Now you can unzip entire archive.

Serg
Attachments
sym18pat_filtered_z03.zip
Serg
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Location: Russia

### Re: Symmetric 18s

Fine, Serg,

I removed the remaining 304 equivalents and uploaded the file there.
eleven

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Joined: 10 February 2008

### Re: Symmetric 18s

Hi, eleven!
I loaded your file (without isomorphic duplicates). I have some ideas concerning additional filtering out patterns (but I don't expect significant search space reduction). I need several days to implement these ideas.

Serg
Serg
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Location: Russia

I've analyzed the first 658 patterns of the new list and none of them have a uniquely solvable 18 clues Sudoku. As you can see the filtered list takes much longer than the original list. So every pattern before

Code: Select all
`.............1.......111................1......11111............1.1.1.1..11...11.`

can be omitted.
Afmob

Posts: 130
Joined: 28 June 2011

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