3-4. In the city, there are 57 bus routes. It is known that:

1) from any stop to any other stop, one can travel without transferring;
2) for any pair of routes, there is one, and only one, stop where one can transfer from one of these routes to the other;
3) each route has no fewer than three stops.

How many stops does each of the 57 routes have?
Solution 4. Answer: 8. We will prove that if the given conditions are satisfied, then the number of stops $n$ and the number of routes $N$ are related by the formula $N=n(n-1)+1$. First, we will show that if a route has $n$ stops, then any other route also has $n$ stops, and furthermore, each stop is served by $n$ routes. Let's take a stop $B$ that is not on the considered route $a$. From $B$, there is a route to each of the $n$ stops on route $a$, and such a route is unique, since two different routes cannot have two common stops. Each route passing through $B$ intersects route $a$. Therefore, exactly $n$ routes pass through $B$. Now it is clear that a route $b$ that does not pass through stop $B$ also has exactly $n$ stops. Indeed, exactly $n$ routes pass through $B$, each intersecting $b$ at one point, two routes cannot intersect $b$ at the same point, and each stop on route $b$ is served by one of these routes. Now it is clear that each stop is served by $n$ routes. In particular, each stop on route $a$ is served by $n-1$ routes other than $a$ itself. In total, we get $n(n-1)$ different routes and the route $a$ itself.