There is a family of chess problems where you try to dominate a board with as few copies of a given piece as possible. The chessboard is dominated if every square either contains a piece, or is attacked by one.
For example, to dominate a usual chessboard with $8$ rooks, simply place a rook on every square of a diagonal. You can easily verify this is optimal, that is, no $7$ rooks can dominate a chessboard. This result easily extends to an $n\times n$ board, for any $n$.
You can also ask about rook domination in three dimensions, where rooks can slide left/right, back/front, or up/down. This problem is harder, but there is a nice solution: $\lceil n^2/2\rceil$ rooks are necessary and sufficient to dominate an $n\times n\times n$ board. See here for a proof.
My question is about the four dimensional case. Here, there are four axes, and rooks can slide along any axis. Specifically,
How many rooks does it take to dominate a $4\times4\times4\times4$ board?
I was wondering if anyone had studied this problem before, or had any good ideas of an attack.
I know that $32$ rooks are sufficient, while $22$ rooks are insufficient. I've filled pages trying to succeed with $31$ rooks, to no avail. If your curious, the optimal number for a 2 x 2 x 2 x 2 board is 4 rooks, and for a 3 x 3 x 3 x 3 board is 9 rooks.