Quote:
Originally Posted by petrw1
There's a game that is played on a 9x9 grid with a free space in the middle. Each player has 40 pieces to be played in any of the 80 remaining tiles, one at a time, in alternation. I'm interested in the proportion (or equivalently the number) of (A) arrangements in which the 80 tiles can be placed on the board such that no player gets 5 in a row, orthogonally or diagonally. And, as a more complicated version, (B) how many ways such that neither player gets more than one row of 5 on the board.

if the rules are >=5 and that hole in the middle, and players have no interest in outcome and play randomly, then you can sample quite accurately by:
 enum all lines of 5
 repeat 10^8 times {
. . . throw 40 random black darts (don't count if you hit a past dart, go on until 40), the rest assumed white,
. . . sum up all preenum'd lines of 5 (is it or is it not) and
. . . record both of the wanted outcomes (A) and (B)
. . . clean up
}
and you will have a fairly accurate estimate. You can estimate a CI of that random process.