12.131 Given 16 identical marbles, we want to select $n$ marbles to form a spatial structure, such that each marble is in contact with exactly 3 other marbles. Find all possible values of $n$, and explain how the $n$ marbles should be arranged.
[Solution] If there are $n$ balls that can be placed to meet the requirements, then each ball has 3 contact points, making a total of $3n$ points. However, in this counting process, each point is counted exactly twice, so the number of distinct contact points is $\frac{3}{2} n$. Therefore, $n$ must be even and $n>2$.

When $n=4$, the 4 ball centers can be placed at the 4 vertices of a regular tetrahedron with edge length equal to the diameter of the ball. When $n=2k, k \geqslant 3$, first place $k$ balls on a plane such that their centers are exactly the $k$ vertices of a regular $k$-sided polygon with side length equal to the diameter of the ball, and then place 1 ball directly above each ball so that the two balls are tangent. In summary, all $n$ that can meet the requirements are $\{4,6,8,10,12,14,16\}$.