Patterns with a unique sudoku

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Patterns with a unique sudoku

Postby Mauricio » Thu Jun 07, 2007 9:43 pm

Are there sudokus such that its pattern is unique, ie, that there is only one sudoku for such pattern?

If there is, what is the maximum number of clues of a pattern that has only 1 valid sudoku?

For instance this sudoku is interesting:
Code: Select all
1 . .|. . .|. . 2
. 3 .|4 . 5|. . .
. . .|. . .|6 . .
-----+-----+-----
. . 5|. . .|7 . .
. . .|3 . 8|. . .
. . 6|. . .|2 . .
-----+-----+-----
. . 2|. . .|. . .
. . .|9 . 1|. 3 .
7 . .|. . .|. . 4


If you change the values of up to five clues and get a valid sudoku, then the sudoku obtained is isomorphic to the starting sudoku, though I have not proved yet that this pattern has just a valid sudoku.

I'll do a research on gordon's list to see if I can find anything interesting.


Addendum: In gordon's list of 17's there are a lot of patterns with a unique sudoku.
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Postby JPF » Fri Jun 08, 2007 1:47 pm

Mauricio wrote:Addendum: In gordon's list of 17's there are a lot of patterns with a unique sudoku.

Yes, It was known when the list had 32930 items :

gfroyle wrote:I now have 32930 distinct 17-clue sudokus...
...
There are 20717 different clue-patterns with the most popular pattern occurring 36 times


With these informations, I'm sure that as long as the list has less than 41433 puzzles, it will exist one pattern with only one sudoku:)

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Postby Mauricio » Fri Jun 08, 2007 2:56 pm

JPF wrote:With these informations, I'm sure that as long as the list has less than 41433 puzzles, it will exist one pattern with only one sudoku:)

In the list of 40535 17's, there are 26308 different patterns and 18797 have only 1 sudoku, so we need at least 18797 more 17's. So it is a sure thing (probabilisticly speaking) that patterns of 17 clues with a unique sudoku exist (maybe someone has already done a complete search for a specific pattern and found just 1). My question now is what is the maximum number of clues of a pattern with a unique sudoku.

Edit: Corrected wrong statements I said.
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Postby Havard » Fri Jun 08, 2007 9:12 pm

JPF wrote:With these informations, I'm sure that as long as the list has less than 41433 puzzles, it will exist one pattern with only one sudoku:)
JPF


I am confident that we will pass 41433 17 puzzles. (even 50000!) There might be new patterns among those though...:)

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Postby ronk » Fri Jun 08, 2007 9:52 pm

JPF wrote:I'm sure that as long as the list has less than 41433 puzzles, it will exist one pattern with only one sudoku:)

What's the origin, the magic, the significance of that number:?:
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Postby Mauricio » Fri Jun 08, 2007 10:49 pm

ronk wrote:
JPF wrote:I'm sure that as long as the list has less than 41433 puzzles, it will exist one pattern with only one sudoku:)

What's the origin, the magic, the significance of that number:?:


41433=2* 20717-1.

As long as there are less than twice the numbers of patterns (back then), then there is at least one puzzle such that its pattern is unique.


Havard wrote:I am confident that we will pass 41433 17 puzzles. (even 50000!)


(50000!) is a lot, more than the number of possible latin squares, I'd think:D . (Don't take this comment seriously, I am just making fun of the factorial simbol)
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Postby Havard » Fri Jun 08, 2007 11:08 pm

Mauricio wrote:(50000!) is a lot, more than the number of possible latin squares, I'd think:D . (Don't take this comment seriously, I am just making fun of the factorial simbol)


hehe, I'm optimistic, but maybe not that optimistic...:D
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