How many such square columns are there, in which the edge length measured in cm is an integer, and the surface area measured in $\mathrm{cm}^{2}$ is the same as the volume measured in $\mathrm{cm}^{3}$?
Let the base side of the square column be $a$, and its height be $m$. Then

$$
V=a^{2} m \quad \text { and } \quad F=2 a^{2}+4 a m
$$

From the equality of the surface area and the volume:

$$
a^{2} m=2 a^{2}+4 a m
$$

Rearranging the equation and dividing both sides by $a \neq 0$:

$$
a m-2 a-4 m=0
$$

Adding 8 to both sides, the left side can be factored:

$$
a m-2 a-4 m+8=a(m-2)-4(m-2)=(a-4)(m-2)=8
$$

The factorizations of 8 on the right side are: $8 \cdot 1, 4 \cdot 2, 2 \cdot 4, 1 \cdot 8$. (Negative factors cannot be used, because $a-4>-4$ and $m-2>-2$.) From this, we get four values for $a$ and $m$: $(12,3); (8,4); (6,6)$ and $(5,10)$.

Thus, four square columns satisfy the conditions.