On specific grids and difficulty of their puzzles

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On specific grids and difficulty of their puzzles

Postby Mauricio » Wed Dec 06, 2006 2:45 am

What is the maximum difficulty for a sudoku with a given solution (ie, grid). Are the ultra hard sudokus special for their final grid?

Searching in specific grids, I found these in the Strangely Familiar grid, MC grid and one nice grid that I'll call M grid, unless it already has a name.
Their SE rating is 9.3.

SF grid
Code: Select all
. . . 2 . . . 8 .
. . 4 7 . . 1 . .
5 . . . . 3 6 . 4
1 . . . 5 . 9 . 6
. . . . . . 8 5 .
. . . . . 9 . . 7
3 . 2 . . . . . .
8 . . 5 9 . . . .
. 7 . . . 6 . . .

SE 9.3, gsfr 99365, suexr9 482


MC grid
Code: Select all
. . . . . 6 . 8 .
. . . 7 8 . . 2 3
7 . . . . . 4 . .
. 1 . . 4 . 9 7 .
. . 5 9 . . . . .
. 7 . 3 . 2 . . .
2 . 1 . . . . . .
5 6 . . . 7 2 . .
. 9 . . 3 . . 6 .
SE 9.3 gsfr 99345 suexr9 414


M grid
Code: Select all
. . . . . 6 7 . .
. 5 . 7 8 . . . 3
. . . 1 2 . . . .
. . 2 3 . . 6 7 8
. 4 . . . . . 1 .
. . 8 . . . 3 . .
8 . . . . . . . 7
. . 4 . . . . 9 .
. 6 7 . . 1 2 . .
SE 9.3, gsfr 99357, suexr9 596

solution, M grid
Code: Select all
1 2 3 4 5 6 7 8 9
4 5 6 7 8 9 1 2 3
7 8 9 1 2 3 4 5 6
9 1 2 3 4 5 6 7 8
3 4 5 6 7 8 9 1 2
6 7 8 9 1 2 3 4 5
8 9 1 2 3 4 5 6 7
2 3 4 5 6 7 8 9 1
5 6 7 8 9 1 2 3 4


Among those sudokus I found several 9.2 grids, but i don't think it is worth posting them.

Can we find more difficult sudokus in these grids? I would think we can. But if we can't, what are the characteristics of the grid that make sudokus hard?
Mauricio
 
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Postby RW » Wed Dec 06, 2006 10:25 am

This issue has been discussed earlier in the "hardest" thread and the "structures" thread. The common belief is that there is no specific characteristics that make it more likely for difficult puzzles to appear. The solutions to ravel's list have been checked for many measurable grid characteristics, but none has showed unusual behavior. I've checked the the list for small unavoidables and two digit unavoidables, and they are as average as average can be.

In a recent test, 1/1 million randomly generated puzzles was hard enough to qualify for the list. However, the amount of minimal puzzles in a random grid is estimated to 10^15, which would give 10^9 superhard puzzles per grid...

RW
RW
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