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 times does Team 4 pass through Station E in a day?

The correct answer is 2.

To determine the number of times Team 4 passes through Station E, we analyze the constraints step-by-step:
1. Vertex/Station Constraints for Team 4:
According to Fact 6, Team 4 never passes through Stations B, D, or F. Since there are only six stations (A, B, C, D, E, and F) in the network, Team 4 is strictly restricted to visiting and traveling between stations A, C, and E.

2. Time and Duration of Patrol:
The patrol runs from 09:00 hrs to 12:00 hrs (total of 180 minutes). Since crossing any street takes exactly 30 minutes, Team 4 will make exactly 6 movements (edges) between stations during this 3-hour period.

3. Tracking Team 4's Route:
- All teams start at Station A at 09:00 hrs and must return to Station A at 12:00 hrs.
- According to Fact 4, Team 4 is at Station E at 11:30 hrs.
- Since Team 4 must return to Station A by 12:00 hrs and the trip takes 30 minutes, Team 4 must travel from E to A during the interval 11:30 to 12:00.
- Since Team 4 is restricted to stations A, C, and E, we can trace its route backward from 11:30 hrs:
- At 11:30 hrs, Team 4 is at E.
- Since it arrived at E at 11:30 hrs, it must have started its journey from A at 11:00 hrs (as only Team 1 and Team 4 patrol the A-E street, per Fact 5).
- Before being at A at 11:00 hrs, Team 4 must have traveled from C to A in the 10:30 to 11:00 hrs slot (since it cannot go from E to A at this time without violating other schedules/constraints).
- This means Team 4 was at C at 10:30 hrs.
- Prior to being at C, Team 4 traveled from A to C during the 10:00 to 10:30 hrs slot.
- Thus, Team 4 was at A at 10:00 hrs.
- Prior to 10:00 hrs, Team 4 traveled from E to A during the 09:30 to 10:00 hrs slot.
- Thus, Team 4 was at E at 09:30 hrs.
- Prior to 09:30 hrs, Team 4 traveled from A to E during the 09:00 to 09:30 hrs slot.

The complete sequential path of Team 4 from 09:00 hrs to 12:00 hrs is:
$\text{A (09:00)}\to \text{E (09:30)}\to \text{A (10:00)}\to \text{C (10:30)}\to \text{A (11:00)}\to \text{E (11:30)}\to \text{A (12:00)}$

From this route, we count the number of times Team 4 visits/passes through Station E:
1. First visit is at 09:30 hrs.
2. Second visit is at 11:30 hrs.

Therefore, Team 4 passes through Station E exactly 2 times in a day.