Prove that it is possible to place
parallelepipedic (rectangular) pieces of soap of dimensions
in a cubic box with edge
if and only if
is even or
.
Remark. It is assumed that the edges of the pieces of soap are parallel to the edges of the box.
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Prove that it is possible to place $2n(2n + 1)$ parallelepipedic (rectangular) pieces of soap of dimensions $1 \times 2 \times (n + 1)$ in a cubic box with edge $2n + 1$ if and only if $n$ is even or $n = 1$.
Remark. It is assumed that the edges of the pieces of soap are parallel to the edges of the box.
Školjka