Rooks in 3D chess board

Question

How many rooks are needed for a 3D chess board of size NxNxN so that every empty cube on the board can be reached by a rook in a single move?

Answer 1

I have also seen this as the 'broken combination lock' problem, and found that a simple 'cube' visualization helped to show how the minimum number of rooks actually looks. See http://iantotman.blogspot.ca/2014/09/the-broken-combination-lock-problem.html

Answer 2

I think your question is equivalent to finding the maximum number of rooks to place in a 3D chessboard such that none of them are non-attacking. This is because having the maximum number of rooks such that they are non attacking means that any other rook placed will result in at least a pair of rooks attacking each other. Thus, this also mean that the maximum number of rooks such that none are non-attacking is the minimum number of rooks that is needed such that all cubes can be reached in a single move.

And the answer hence is N^2. (Reference to a paper by Arthur low, Shalin Mehta. Rook Polynomial in 3D)