6. A regular dodecahedron has 20 vertices, 30 edges, and each vertex is the intersection of three edges, the other ends of which are three other vertices of the regular dodecahedron, and these three vertices are called adjacent to the first vertex. Place a real number at each vertex, such that the real number placed at each vertex is exactly the arithmetic mean of the real numbers placed at the three adjacent vertices. Let $M$ and $m$ be the largest and smallest of these 20 real numbers, respectively, then $M-m=(\quad$.
(A)0
(B) 1
(C) 12
(D) cannot be determined
6. (A).

Let the total at $P_{1}$ be $m$, if $P_{2}, P_{3}, P_{4}$ and $P_{2}$ H $_{1}$ jaw, $P_{3}, P_{4}$ have a number greater than $m$, then $P_{1}>m$. Otherwise, $P_{2}, P_{3}$, $P_{4}$ can only have $m$. Considering the points adjacent to $P_{2}, P_{3}, P_{4}$, they also cannot have $m$. Thus, all points can only have $m$, hence $M=m$, which means $-m=0$.