Interesting thread....so there are no fully symmetric minimal puzzles in the SF grid.
This has implications for the total number of fully symmetric puzzles . Perhaps also explains why
mauricio never found the elusive 36 clue fully symmetric minimal puzzle !
Each grid has around 10^14 - 10^15 minimal puzzles according to estimates - which is in line with
JPFs estimates of total minimal and non-minimal puzzles.
In an effort to explore the extent of the distribution of "hard puzzles".
Many of our hardest puzzles have 21 clues.
I estimate, that on average, there are about 50 million 21-puzzles per grid. [estimated by the frequency of duplicate generation]
The SF grid probably has significantly more than this.
From the archives this is a distibution of the clue counts of puzzles generated by
dukuso- Code: Select all
dukoso wrote:
here is some statistics, starting from a full grid and generating 1e6
random locally minimal sudokus from it.
1) one grid from each G-class at random
2) Gordon's grid with 29 17s, the SF grid
3) our canonical grid,(1,1,1-1,1,1)
Code:
clues , 1) 2) 3)
----------------------------
17, 0 0 0
18, 0 0 0
19, 0 4.3 0
20, 59 182 0
21, 2428 6051 85
22, 33966 61826 1775
23,170727 227480 21648
24,342620 352289 116766
25,298349 248568 286836
26,122691 86061 329853
27, 25237 15908 185028
28, 2733 1547 50469
29, 205 74 7040
30, 7.6 8.6 486
31, 0 0 12
32, 0 0 2.4
-------------------------------
aver.24.38 24.10 25.72
The method of puzzle production skews the distribution away from large puzzles.
From this The SF appears to have approx x2 more 21-puzzles than average.
At the other end of the scale there are numerous 34-puzzles per grid to be found - if you try hard enough. In the random grid I studied - I gave up when it became obvious there was an increasingly expanding amount of puzzles - and this was only in one region in the grid. Interestingly many of the 34-puzzles have a highish ER.
- Code: Select all
+---+---+---+
|6.9|.4.|...|
|.8.|.65|..3|
|5.7|983|6..|
+---+---+---+
|...|...|..6|
|79.|.3.|85.|
|...|..9|23.|
+---+---+---+
|.4.|..8|...|
|8..|.94|37.|
|..5|326|4..|
+---+---+---+ 34 clue minimal in SF grid
My [hardly earth-shattering] conclusion [based on the estimate of 5*10^7 * 5*10^9 ~ 2*10^17 for the total number of 21-puzzles] was that it was unlikely that we have fully explored all the possible 21-puzzles !
C
