A rectangular prism has a volume of $8 \mathrm{~cm}^{3}$. If we increase each edge of the rectangular prism by 1 centimeter, we get a rectangular prism with a volume of $27 \mathrm{~cm}^{3}$. What volume of rectangular prism will we get if we again increase each edge by 1 centimeter?
A rectangular prism with a volume of $8 \mathrm{~cm}^{3}$ has edges $a, b$, and $c$, so

$$
V=a b c=8
$$

We write the inequality between the arithmetic mean and the geometric mean for the positive numbers $a, b, c$:

$$
\frac{a+b+c}{3} \geq \sqrt[3]{a b c}=\sqrt[3]{8}=2
$$

From this,

$$
a+b+c \geq 6
$$

The volume of the rectangular prism with edges increased by $1 \mathrm{~cm}$ is $(a+1)(b+1)(c+1)=27$. After performing the operations, we get the equation

$$
a b+a c+b c+a+b+c=18
$$

Then we write the inequality for the numbers $a b, a c, b c$:

$$
\frac{a b+a c+b c}{3} \geq \sqrt[3]{a b \cdot a c \cdot b c}=\sqrt[3]{(a b c)^{2}}=4
$$

From this, $a b+a c+b c \geq 12$, and from (1) $a+b+c \geq 6$. Using (2), we get

$$
18=a b+a c+b c+a+b+c \geq 18
$$

This is only possible if equality holds in both previous inequalities. The arithmetic mean of numbers equals their geometric mean only if the numbers are equal, so $a=b=c$ and from $a b c=8$ we have $a=b=c=2$, meaning the solid is a cube.

Thus, if we increase each edge by $1 \mathrm{~cm}$ again, the volume will be $64 \mathrm{~cm}^{3}$.