Given an $n\times n\times n$ cube, each of the six faces is divided into $n^2$ squares. So there are $6n^2$ squares in total. A rook placed on a square can attack any square, including squares on the other faces, that can be reached vertically or horizontally from its square. In particular, any rook can choose one of the two directions "vertical" or "horizontal", and walk in that direction across other faces until it comes back, spanning a total of $4n$ squares in each direction. What is the maximum number of non-attacking rooks that can be placed?
We can place $n$ rooks on the diagonal of one face -- this already leaves no square unattacked. Unlike the $2$-dimensional case, however, it is unclear if this is the best we can do.