Similar to 2-dimensional chess board imagine a 3-dimensional chess box-board, rooks can attack in a straight line in the direction of the coordinate axis.
What is the minimum number of rooks which can dominate a 12 x 12 x 12 chessboard?
Upper Bound:
I can do it with 108 rooks. Place 12 rooks on the first row of bottom board, 12 on the 2nd row of the 2nd board, 12 on the 3rd row of the 3rd board, etc for the first 6 boards. This leaves 6 rows uncovered on each of the boards 7 - 12, and these can be covered by 6 rooks on each board, for a total of 108. I do not expect that this is optimal, but it might be. I look forward to lower answers.
Lower bound:
A rook can cover 34 squares, including the one it is on. Since there are 12*12*12 = 1728 squares, at least 1728/34 rooks = 50.82 rooks are required. So 51 rooks is an absolute minimum. This is not really achievable.
A start
