8.2. Two people are playing a game, with a checker piece placed in the middle square of a $1 \times 2005$ strip of squares. They take turns moving the piece. First, Player A moves the piece 1 square in any direction, then Player B moves the piece 2 squares in any direction, then Player A moves 4 squares, then Player B moves 8 squares, and so on (on the $k$-th turn, the piece is moved $2^{k-1}$ squares in any direction). If a player cannot move the piece according to the rules on their turn, they lose. Who can win regardless of the other player's actions?
8.2. Answer: Player A can win without relying on Player B's actions.

Notice that, Player A's movement distances are $1, 4, 16, \cdots, 1024$.
We will prove that: Player A can successfully move a distance of 1024 (this way, Player B loses, as he would need to move the piece 2048 spaces, which is impossible).

Since $1+2+4+\cdots+128=1024)$.