I've been making symmetrical (reflection) sudokus and working my way down to fewer and fewer clues. (I found this with 18 clues):
"18 clue reflection / shape#21"
- Code: Select all
. . . . . . . . .
. . 1 . 3 . 5 . .
. . . 2 . 4 . . .
. 2 . . . . . 8 .
. . 3 . 9 . 1 . .
. . . . 5 . . . .
. 4 . 6 . 8 . 2 .
. . 5 . 7 . 9 . .
. . . . . . . . .
I have also been trying to count how many total distinct shapes exist for 18 clues/reflection symmetry (this one plus others on these forums and elsewhere on-line).
I counted 20 distinct patterns (if I made no mistakes) plus the one above for 21 shapes total. Note I am counting shapes not total sudokus, so if two have isomorphic shapes I just count it once.
(these 2 links lead to places where I counted them):
this forum:
http://forum.enjoysudoku.com/viewtopic.php?t=4147&postdays=0&postorder=asc&start=510
elsewhere:
http://www.flickr.com/photos/npcomplete/
So now here is where my question gets more detailed:
What would be the ratio of symmetrical 18 sudokus compared to all total (for reflection symmetry).
My math skill are OK but not not superb. Here is my attempt to answer it mathematically:
Take a sudoku with 18 clues, and assume there are 2 clues per row. The number of ways to place 2 clues in a row is:
C=9!/(7!2!)=36
Of these combination, 4 are symmetrical. That would be when the clues are in positions 1,9; 2,8; 3;7, and 4,6.
So for a single row only, the chance of symmetry is 4/36 = 1/9.
Now considering a full sudoku (9 rows), whats the chance of finding one row which already has symmetry as is (no moving columns around).
I believe that would be:
P = 1-(1-1/9)^9 = 65.4%
So there is a fairly good chance that 18 randomly placed clues will already have one row with symmetry (but very unlikely that there will be more than one).
So to preserve this symmetry, we now have to assume the column positions are locked (no switching column positions to find symmetry).
So with one row having symmetry, here is the chance that ALL 8 OTHER rows will also have symmetry:
P = (1/9)^8 = 2.323 x 10-8
=0.0000023%
or
one in 43,046,721 chances.
I guess this is stating the obvious, that reflection symmetry relative to all sudokus is rare.
I know this logic is using some assumptions (like assuming every row has 2 clues), but I wonder if this is in the ballpark for the right answer.
Does anyone see errors or bad assumptions in my math, or a more accurate way to calculate it?
Also, if anyone has any similar analysis for rotational symmetry, or with a different number of clues that will be interesting too.
Regards
