Problem #6

Recall that in chess, a queen can move an arbitrary number of (unobstructed) squares in a horizontal, vertical, or diagonal direction. It is not too difficult to show that for n > 3, the maximum number of queens which can be placed on an n x n board so that no queen attacks another is n.

Consider an n x n x n cube. Define a hyperqueen to be a piece which can move an arbitrary number of (unit) cubes forward or backward, left or right, up or down, along any of the six diagonal directions parallel to a face of the cube, or along any of the directions parallel to the four space diagonals adjoining opposite vertices of the cube.

1. What is the maximum number of hyperqueens that can be placed on a 3 x 3 x 3 cube so that no hyperqueen attacks another?
2. What about a 4 x 4 x 4 cube?
3. An n x n x n cube?

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