Final answer:
The second player is the winner in the game of placing bishops on a chessboard. The winning strategy for the second player is to strategically place their bishops to restrict the first player's options and limit their number of possible moves.
Step-by-step explanation:
In the game of placing bishops on a chessboard, the player who cannot make a move loses. We can determine the winner by considering the number of bishops that can be placed on the board without threatening each other. Each bishop can attack the squares that are diagonally aligned with it. Since the bishops are placed one at a time, the first player can place their bishop in any of the 8x8 squares, giving them a total of 64 possible moves. After the first move, the second player can place their bishop in any of the remaining 49 squares (since they cannot place it in the same row, column, or diagonal as the first bishop), giving them a total of 49 possible moves. The game continues in this manner, with the number of possible moves decreasing by 15 each turn. Ultimately, the player who makes the last move will be unable to make any further moves, and therefore will lose the game. Therefore, the second player is the winner.
A winning strategy for the second player is as follows:
- On their first move, the second player should place their bishop in one of the four corners of the chessboard. This allows them to control two diagonals, reducing the number of remaining squares for the first player.
- On subsequent moves, the second player should continue to strategically place their bishops in positions that restrict the first player's options and limit the number of possible moves.
- The second player should always prioritize blocking the first player's access to diagonal lines, as this prevents the first player from placing bishops in certain positions.