I’ve been thinking some more about this problem, mainly trying to prove that the maximum number of stones grows linearly, but no success so far.
A friend of mine wrote a program using simulated annealing that can find good solutions quite efficiently.
Here is a solution with 41 stones on an 18x18 board which doesn’t use any of the corners:
(most optimal solutions so far use at least one corner so I found this one a little surprising)
And here is a solution with 240 stones on a 100x100 board:
241:
Another update, 246:
