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Author: Josh
This game is highly symmetrical, which motivates the following strategy known as 'strategy stealing'.
We can instruct Josh to go first. Whenever Josh marks a cell, we can simply mark the same cell but rotated 
~180^{\circ}~ about the center of the board. Because of rotational symmetry, and the fact that a cell is never adjacent to the cell which is rotated ~180^{\circ}~ about the center of the board from it, it is easy to see that whenever Josh makes a valid move, then our move will also be valid. Thus, we can never run out of moves first, so we will always win.
The only situation where this does not work is if 
~N~ is odd and Josh marks the center cell, since the same cell can't be marked twice. For odd ~N~, we can choose to go first and mark the center cell, and then perform the strategy stealing as described before.
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