\(n=\)

An empty \(n\times n\times n\) cube is given, and a \(n\times n\) grid of square unit cells is drawn on each of its six faces. A beam is a \(1 \times 1 \times n\) rectangular prism.

Several beams are placed inside the cube subject to the following conditions:

Several beams are placed inside the cube subject to the following conditions:

- The two \(1 \times 1\) faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are \(3n\) possible positions for a beam.)
- No two beams have intersecting interiors.
- The interiors of each of the four \(1 \times n\) faces of each beam touch either a face of the cube or the interior of the face of another beam.

© 2023 __Peter E. Francis__