Mounting A Defense Against Sheldon

Discrete Mathematics Level 4

We say that a set of rooks dominate the chessboard if each square is either occupied by a rook, or can be directly attacked by a rook. Obviously the smallest number of rooks which can dominate a \( 12 \times 12 \) chessboard is 12, which we can achieve by placing them along the main diagonal.

In 3-dimensional chess, 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 \( 12 \times 12 \times 12 \) chessboard?

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