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?
I think 72 rooks is a minimum.
On South East of the bottom 6 boards, arrange the Rooks as Follows:
Board 1
R-----
-R----
--R---
---R--
----R-
-----R
Board 2
-R----
--R---
---R--
----R-
-----R
R-----
Board 3
--R---
---R--
----R-
-----R
R-----
-R----
Board 4
---R--
----R-
-----R
R-----
-R----
--R---
Board 5
----R-
-----R
R-----
-R----
--R---
---R--
Board 6
-----R
R-----
-R----
--R---
---R--
----R-
That covers all but the Northwest corner of boards 1-6, plus it covers the southeast corner of boards 7-12.
Now do the same in the Northwest corner of boards 7 - 12. Everything is covered.
I expect that this is the minimum. 72 is my final answer.
Domination (spoiler)
