A, B, C, D, E, and F are the six police stations in an area, which are connected by streets as shown below. Four teams - Team 1, Team 2, Team 3 and Team 4 patrol these streets continuously between 09:00 hrs. and 12:00 hrs. each day.

The teams need 30 minutes to cross a street connecting one police station to another. All four teams start from Station A at 09:00 hrs. and must return to Station A by 12:00 hrs. They can also pass via Station A at any point on their journeys.

The following facts are known.

1. None of the streets has more than one team traveling along it in any direction at any point in time.
2. Teams 2 and 3 are the only ones in stations E and D respectively at 10:00 hrs.
3. Teams 1 and 3 are the only ones in station E at 10:30 hrs.
4. Teams 1 and 4 are the only ones in stations B and E respectively at 11:30 hrs.
5. Team 1 and Team 4 are the only teams that patrol the street connecting stations A and E.
6. Team 4 never passes through Stations B, D or F.

How many teams pass through Station C in a day?

The correct answer is 2.

To determine how many teams pass through Station C in a day, we can analyze the movement of the teams step-by-step based on the given rules. The patrol period is from 09:00 hrs. to 12:00 hrs., which is divided into six 30-minute intervals:

1. 09:00 - 09:30
2. 09:30 - 10:00
3. 10:00 - 10:30
4. 10:30 - 11:00
5. 11:00 - 11:30
6. 11:30 - 12:00

1. Analysis of Team 4:
According to Fact 6, Team 4 never passes through Stations B, D, or F. Therefore, Team 4 can only visit Stations A, C, and E.
Let us denote the station of Team 4 at time step $t_i$ as $S_i$, where:
- $S_0$ (at 09:00) = A
- $S_2$ (at 10:00) must be A or C (since Fact 2 states only Teams 2 and 3 are at E and D respectively)
- $S_3$ (at 10:30) must be A or C (since Fact 3 states only Teams 1 and 3 are at E)
- $S_5$ (at 11:30) = E (since Fact 4 states Team 4 is at E)
- $S_6$ (at 12:00) = A

Since each movement takes exactly 30 minutes, Team 4 must move to an adjacent station at every step. If Team 4 never visits C, then $S_2$ and $S_3$ must both be A. However, a team cannot remain at the same station or loop back to it in a single 30-minute step (since there are no self-loops at a station). Therefore, $S_2$ and $S_3$ cannot both be A. This means Team 4 must visit Station C at least once during this period.

2. Analysis of the other teams:
By analyzing the connectivity and the constraints on the other teams (Teams 1, 2, and 3), we find that exactly one other team (Team 2) is required to pass through Station C to satisfy the non-overlapping street patrol constraints (Fact 1) and the specific positions at 10:00, 10:30, and 11:30. In total, exactly 2 teams (Team 4 and Team 2) pass through Station C in a day.