First, we put a rook on the 8*8 board. ● Then we put a second one on the board, so any of the two rooks can take the other. ● This continues with a third one, so any of the three rooks can take any other one, and so on…
How many rooks can be put on the chessboard and How many configurations can be found with the maximal number of towers?
If the positions of the two first rooks are imposed, can we reach the maximal number of towers in any case?
Here's my approach to the question, Since a rook can attack another rook only if it is placed in the same horizontal row or vertical column as the first rook. (For eg. If the rook is placed at e4, to begin with, the other rook must be placed somewhere between h4 to a4 or e1 to e8). So for the third rook, it can be in either of the vertical and horizontal columns of the first two. Since this is true for all other rooks that follow, the entire chessboard could be filled with rooks attacking any other rook? Once a row is completely filled, rooks can be placed on vertically either side and thus the entire chessboard will be filled. This seems extremely odd to me as the second part of the question would essentially become pointless and would appreciate it if someone could point out the flaw in this answer.
Normally rooks cannot move through other rooks, so you cannot place the third one so that each can take all the others. If you have three in a line the end two cannot take each other. Maybe you want each successive rook to be able to take at least one other? Then, as you say, you can fill the board.