3 3.7 A regular dodecahedron has 20 vertices and 30 edges. Each vertex is the intersection of 3 edges, and the other endpoints of these 3 edges are 3 other vertices of the regular dodecahedron. We call these 3 vertices adjacent to the first vertex. Place a real number at each vertex, such that the number placed at each vertex is exactly the arithmetic mean of the numbers placed at the 3 adjacent vertices. Let $M, m$ be the largest and smallest of these 20 numbers, respectively. What is the range of $M-m$?
［Solution］Let $m$ be placed at point $P_{1}$.
If $P_{2}, P_{3}, P_{4}$ are adjacent to $P_{1}$, then the numbers placed at $P_{2}, P_{3}, P_{4}$ are all not less than $m$. If one of the numbers placed at $P_{2}, P_{3}, P_{4}$ is greater than $m$, then $m$ should not be placed at $P_{1}$. Therefore, $P_{2}, P_{3}, P_{4}$ can only place $m$.

Now consider the points adjacent to $P_{2}, P_{3}, P_{4}$, and similarly analyze, they can only place $m$ as well. Thus, all vertices of the regular dodecahedron can only place $m$.
Therefore, $M=m, M-m=0$.