In the Valley of Five Lakes, there are five identical lakes, some of which are connected by streams (on the diagram, dashed lines indicate possible "routes" of the streams). Small carp are born only in lake $S$. While growing up, a carp crosses from one lake to another exactly four times via some stream (the carp chooses a stream at random), and then stays to live in the lake where it ends up. Out of every thousand carp, on average 375 stay to live in lake $S$, and the rest stay to live in lake $B$, with no one staying to live in the other lakes. Determine how many streams there are in the Valley of Five Lakes.

![](https://cdn.mathpix.com/cropped/2024_05_08_22585010b2534c7ab9ebg-31.jpg?height=572&width=349&top_left_y=826&top_left_x=863)
The transition from one lake to another will be called a route of length $n$ if it passes through $n$ streams. Let's prove several statements.

1. There is no route of length 2 from $S$ to any of the lakes $A, C$, and $D$.

Proof. Suppose it is possible to sail from lake $S$ to lake $A$ through one intermediate lake (say $C$). Then the route $S-C-A-C-A$ would be possible. In this case, there would be a non-zero probability of reaching lake $A$ in four transitions. Therefore, there is no route of length 2 from $S$ to $A$.

![](https://cdn.mathpix.com/cropped/2024_05_08_22585010b2534c7ab9ebg-31.jpg?height=569&width=349&top_left_y=1802&top_left_x=863)

2. Lake $S$ is connected to lake $B$ by a route of length 2.

Proof. Clearly, $S$ and $B$ are connected. Suppose they are connected by a common stream. Then there are no other exits from $B$, otherwise condition (1) would not be satisfied. This means one can only arrive at $B$ from $S$ on the first or third move. But then, after the fourth transition, the carp cannot end up in $B$. Contradiction.

Suppose $S$ and $B$ are connected by a route of length 3 or 4. But then there is a route of length 2 from $S$ to one of the lakes $A$, $C$, or $D$, which cannot exist.

Therefore, there is a route from $S$ to $B$ passing through one intermediate lake $A, C$, or $D$, and one is no better than the other. We can assume, for definiteness, that there is a route $S-A-B$.

From this and statement 1, it follows that there are no routes $A-C$ and $A-D$.

3. There is at least one of the routes $B-C$ or $B-D$.

Proof. Suppose there is neither a route $B-C$ nor $B-D$. We get one of the following diagrams:

![](https://cdn.mathpix.com/cropped/2024_05_08_22585010b2534c7ab9ebg-32.jpg?height=597&width=1466&top_left_y=-1&top_left_x=296)

In the first case, the carp makes the first transition to lake $A$, and it has three transitions left. Lakes $S$ and $B$ are equally positioned relative to $A$, so their chances should be equal. It is easy to see that in the other three cases, the probability of remaining in lake $S$ is greater than in $B$.

Therefore, one of the routes $B-C$ or $B-D$ must exist.

From this, it follows that there is no route $C-D$.

4. The constructed configuration of three streams satisfies the condition.

Proof. For the constructed configuration, the following routes of length 4 are possible:

![](https://cdn.mathpix.com/cropped/2024_05_08_22585010b2534c7ab9ebg-32.jpg?height=526&width=1129&top_left_y=935&top_left_x=470)

Above each transition, the probability of this transition is indicated, and on the right, the probability of the entire route, obtained by multiplying the probabilities of transitions between lakes.

The probability that the carp, after four transitions, will end up in lake $S$ is $0.25 + 0.125 = 3/8$ - exactly what is needed.

Therefore, the probability that $B$ is the final point of the route is $1 - 3/8 = 5/8$.

5. There cannot be more streams.

Proof. By adding any streams $S-C, S-D, B-D$ individually or together, we will change the probabilities. This can be verified directly by making seven more diagrams corresponding to possible configurations and calculating the probabilities for each of the resulting schemes.

Thus, with the accuracy of the vertex labeling, the scheme shown in the last diagram is the only possible one, and the total number of streams is 3.

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