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Beams inside a cube

Source: USOMO 2020 Problem 2, USOJMO 2020 Problem 3

June 21, 2020
AMCUSA(J)MOgeometry3D geometryHi

Problem Statement

An empty 2020×2020×20202020 \times 2020 \times 2020 cube is given, and a 2020×20202020 \times 2020 grid of square unit cells is drawn on each of its six faces. A beam is a 1×1×20201 \times 1 \times 2020 rectangular prism. Several beams are placed inside the cube subject to the following conditions: [list=] [*]The two 1×11 \times 1 faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are 3⋅202023 \cdot {2020}^2 possible positions for a beam.) [*]No two beams have intersecting interiors. [*]The interiors of each of the four 1×20201 \times 2020 faces of each beam touch either a face of the cube or the interior of the face of another beam.
What is the smallest positive number of beams that can be placed to satisfy these conditions?
Proposed by Alex Zhai
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