Alice and Bob are playing an unusual game of chess.
We begin with a piece on a square of an infinite chessboard.
On each of her turns, Alice moves the piece.
On each of his turns, Bob destroys a square of the chessboard.
Bob wins if he can trap Alice (i.e., create a 'moat' of destroyed squares too big for Alice to cross).
Alice wins if she can prove that Bob has no winning strategy.
Assume perfect play.
Now for the pieces: Let's make up pieces called n-kings, and say that an n-king leaps to any square within n squares of it (so can fly over destroyed squares).
Clearly Bob wins from the get-go if Alice starts with a 0-king.
What is the highest order of king for which Bob has a winning strategy?
What is the lowest order of king for which Alice has a winning strategy (if any)?