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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

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    $\begingroup$ 3x3 is obviously impossible, and 4x4 has two simple solutions. For 3 dimensional and higher, you will have to define what a queen is, i.e. which cells it attacks. $\endgroup$ Apr 13, 2017 at 12:53
  • $\begingroup$ 4x4 is the smallest such board. There are 2 such placements of queens and they look the same. See here : lia.disi.unibo.it/Staff/MicheleLombardi/or-tools-doc/… $\endgroup$ Apr 13, 2017 at 13:12

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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.

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