A $2020\times 2020\times 2020$ grid is made of unit cubes. A beam is a $1\times 1\times 2020$ rectangular prism. Beams are placed in the grid parallel to the grid axes such that their interiors are disjoint, and every beam is stable (meaning its four $1\times 2020$ lateral faces each touch either a face of the outer grid box or the lateral face of another beam). Find the minimum number of beams required to satisfy these stability conditions.

Step 1 (General Formula): In an $n\times n\times n$ grid, the minimum number of beams required is $\frac{3n}{2}$ for even $n$, and $\frac{3n-1}{2}$ for odd $n$.
For $n=2020$ (which is even), the minimum number of beams is:

Step 2 (Lower Bound): To prove that at least $\frac{3n}{2}$ beams are required, we define a boundary constraint. Each beam must have all 4 of its lateral faces supported. If we count the number of faces of beams parallel to each coordinate plane, every beam parallel to the x-axis requires support from beams parallel to the y-axis or z-axis. By setting up a system of inequalities representing the mutual support constraints, we show that the number of beams in the three orthogonal directions ($B_x,B_y,B_z$) must satisfy $B_x+B_y\ge n$, $B_y+B_z\ge n$, and $B_z+B_x\ge n$. Summing these gives $2(B_x+B_y+B_z)\ge 3n$, hence $B_x+B_y+B_z\ge \frac{3n}{2}$.

Step 3 (Construction): We can achieve exactly $3030$ beams by constructing an interlocking "staircase" or cycle. We place $1010$ beams parallel to the x-axis, $1010$ parallel to the y-axis, and $1010$ parallel to the z-axis. These are arranged along the diagonal of the cube in a cyclic, overlapping fashion where each beam supports its neighbors, locking the entire structure in place against the walls of the outer grid box. $■$