Question 111
Some bishops and knights are placed on an infinite chessboard, where each square has side length $1$ unit. Suppose that the following conditions hold:
- For each bishop, there exists a knight on the same diagonal as that bishop (there may be another piece between the bishop and the knight).
- For each knight, there exists a bishop that is exactly $\sqrt{5}$ units away from it.
- If any piece is removed from the board, then at least one of the two conditions above is no longer satisfied.
If $n$ is the total number of pieces on the board, find all possible values of $n$.
- For each bishop, there exists a knight on the same diagonal as that bishop (there may be another piece between the bishop and the knight).
- For each knight, there exists a bishop that is exactly $\sqrt{5}$ units away from it.
- If any piece is removed from the board, then at least one of the two conditions above is no longer satisfied.
If $n$ is the total number of pieces on the board, find all possible values of $n$.