4.65 A city has a total of 57 bus routes, and the following conditions are met:
(1) From any station, it is possible to reach any other station without transferring;
(2) Any two routes have exactly one common station, from which one can transfer from one route to another;
(3) Each route has no fewer than 3 stations.

How many stations are there on each of the 57 routes?
[Solution] First, we prove that the number of stations on each line is the same. If not, suppose there are two lines $l_{1}$ and $l_{2}$ with different numbers of stations:
$l_{1}: M, A_{1}, A_{2}, \cdots, A_{m}, l_{2}: M, B_{1}, B_{2}, \cdots, B_{n}$ and $m>n$. Since the number of stations on each line is no less than 3, there is a third line $l_{3}$ passing through station $M$. Choose a station $C \neq M$ on $l_{3}$, then $C$ is connected to each $A_{i}$ by a line. Since every two lines have a common station, each of the $m$ lines mentioned above has one of the stations $\left\{B_{1}, B_{2}, \cdots, B_{n}\right\}$. Because $m>n$, by the pigeonhole principle, there must be two lines among these $m$ lines that pass through the same $B_j$, which is a contradiction.

Suppose the station $A$ with the most lines passing through it has $m$ lines, then by (2), each of the other lines has exactly $m$ stations. Therefore, from the previous proof, each line has $m$ stations, and each station has $m$ lines passing through it. Hence, the number of lines is equal to the number of stations, which is $n=m(m-1)+1$. Given that the total number of lines is 57, we have $m(m-1)=56$, solving this gives $m=8$, meaning each line has exactly 8 stations.