3: There is an $8 \times 8$ chessboard, and at the start, each of the 64 small squares contains a "castle" chess piece. If a castle chess piece can attack an odd number of other castle chess pieces still on the board, it is removed. Question: What is the maximum number of castle chess pieces that can be removed (a castle chess piece can attack other pieces in the same row or column without any obstruction)?


First, prove that the pieces on the four corners of the chessboard will not be taken away. Since a castle on a corner can attack at most two other castles, if this castle is taken away, it means that the castle can only attack one. Without loss of generality, we can assume that the castle in the upper left corner is the first to be taken away among these four, which means that either the first column or the first row has been completely cleared. In other words, the castle in the upper right corner or the lower left corner has been taken away, which is impossible.

When there are five castles left, the castles outside these four corners can only attack two or zero castles, so at least five castles will remain. Taking the castles in the order of the numbers in Figure 5 is one way to end up with five castles left. Therefore, the maximum number of castles that can be taken away is $64-5=59$.