2. Betya and Vasya are playing a game on an $n \times n$ chessboard. At the start, all squares are white except for one corner square which is black and contains a "rook". Betya starts, and the two players take turns moving the rook several squares horizontally or vertically. The squares the rook passes over and lands on immediately turn black, and the rook cannot pass over or land on black squares. The player who cannot make a move according to the rules loses, and the game ends. Question: Who has a winning strategy?
2. Betya has a winning strategy.

Betya's strategy: At each move, move the rook vertically to the farthest possible square.
Below is an explanation of how this strategy can lead to victory.
If at some point the rook is in the same connected region of white squares, then the white squares at that moment are called "reachable."

It is easy to see that whenever Betya finishes a move, the reachable squares form no more than two rectangles, each of which has more rows than columns; the rook is in the first or last row and is horizontally adjacent to a corner of each rectangle of reachable squares. Next, no matter how Vasya moves (only horizontally), Betya can continue to make moves.