the 8 queens on the chessboard: variations

Question

i've heard of the 8 Queens problem: put 8 queens onto a standard chessboard without any of them able to capture any other queen. my question is: what is the smallest 2-D chessboard that can have queens? (as in the chessboard is $x^2$, with x queens on it. and what about cubic and hyper-cubic chessboards? A queen can move in any direction (up, down, left, right, diagonal, forward, backward, and the movement in any hyper-cubic space

Answer 1

Trivially, a $1\times1$ chessboard has $1$ nonattacking queen. And the number of $n\times n$ chessboards with $n$ nonattacking queens is given by OEIS sequence A000170

1, 0, 0, 2, 10, 4, 40, 92, 352, 724, 2680, 14200, 73712, 365596, 2279184, ...

With regard to a $n\times n\times n$ chessboard, one could simply place $n$ queens at the "bottom board" of the cube, and use the $n \times n$ configuration. This argument shows that A000170 would serve as a lower bound on the number of configurations on the $n\times n\times \cdots \times n$ analog of the nonattacking queens problem.