A rectangular cuboid has the sum of the three edges starting from one vertex as $p$, and the length of the space diagonal is $q$. What is the surface area of the body? - Can its volume be calculated?
I. For a cuboid, the lengths of the edges meeting at a vertex are denoted by $a, b, c$ respectively, and according to the given data,

$$
\begin{gathered}
a+b+c=p \\
a^{2}+b^{2}+c^{2}=q^{2}
\end{gathered}
$$

These two pieces of data are not sufficient to calculate the lengths of the edges. However, the surface area $F$ of the body can still be calculated, as follows:

$$
F=2(a b+a c+b c)=(a+b+c)^{2}-\left(a^{2}+b^{2}+c^{2}\right)=p^{2}-q^{2}
$$

II. The volume $V=a b c$, and this cannot be calculated from the given data. It is sufficient to provide a counterexample. For $p=15, q=9$, there are even two solutions for the edge triplet using integers:

$$
a_{1}=4, b_{1}=4, c_{1}=7 ; \quad a_{2}=3, b_{2}=6, c_{2}=6
$$

and from these, $V_{1}=112$, and $V_{2}=108$.