Ugh -- turns out I had a bug in the program, as revealed by a Twitter correspondent who fortunately did independent counting for 9x9. For the record, the correct values are: All grids: 1, 15, 397, 35324, 18446134, 41752489852, 409764131469787 No black edges: 1, 12, 312, 31187, 17438702, 40575832476, 404139015237875 Apologies, Neil; I've proposed edits for A323838 and A323839. Dan: Hmm, I don't immediately see how I would count the rot90-symmetric grids, now that you mention it... --Michael On Mon, Feb 4, 2019 at 1:59 PM Dan Asimov <dasimov@earthlink.net> wrote:
Nice!
Now let's find the counts for the grids that *also* have symmetry group G, for each subgroup G of Isom([0,1]x[0,1]) = D_4, the dihedral group of order 8.
(There are 10 subgroups altogether: the trivial group, the whole group, and 5 isomorphic to Z_2, 1 to Z_4, 2 to Z_2 + Z_2.)
—Dan
----- The 538 Riddler from two and a half weeks ago asked for the number of 15x15 black-and-white grids that are legal for crossword puzzles.
https://fivethirtyeight.com/features/how-many-crossword-puzzles-can-you-make...
Rules: Every white square must be "checked" (in both a horizontal and vertical word), words must be at least 3 letters long, the whole grid must be symmetric under 180-degree rotation, and the set of white squares must be connected.
Evidently nobody solved it, so I decided to :-)
Answer: there are 347804238364806 legal 15x15 grids according to those rules (347 trillion). And if you don't like the fact that those rules allow some all-black rows around the edges of the puzzle, and want them to be "really" 15x15, then that cuts down the count to 342935406100702.
To convey the information to the outside world (including the 538 Riddler guy), I've posted some more about this on Twitter. Please forgive me.
https://twitter.com/Log3overLog2/status/1092472516571000839
NJAS, the sequence of counts of nxn grids for n=3,5,7,9,11,13,15 is: All grids: 1, 15, 397, 35184, 17431781, 37147554097, 347804238364806 No black edges: 1, 12, 312, 31047, 16459558, 36076951460, 342935406100702 -----
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