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.
- None of the streets has more than one team traveling along it in any direction at any point in time.
- Teams 2 and 3 are the only ones in stations E and D respectively at 10:00 hrs.
- Teams 1 and 3 are the only ones in station E at 10:30 hrs.
- Teams 1 and 4 are the only ones in stations B and E respectively at 11:30 hrs.
- Team 1 and Team 4 are the only teams that patrol the street connecting stations A and E.
- Team 4 never passes through Stations B, D or F.
Solution
The most important part of any LR question is to get the initial table right. Reading the very first question gives us clarity that to answer the questions, we would have to know at time , which team was at what station. From the question we know that every path takes minutes which means that a team reaches a station every minutes.
Hence, our table would look like (we know that they start and end at A):
| 9.00 | 9.30 | 10.00 | 10.30 | 11.00 | 11.30 | 12.00 | |
|---|---|---|---|---|---|---|---|
| Team 1 | A | A | |||||
| Team 2 | A | A | |||||
| Team 3 | A | A | |||||
| Team 4 | A | A |
| 9.00 | 9.30 | 10.00 | 10.30 | 11.00 | 11.30 | 12.00 | |
|---|---|---|---|---|---|---|---|
| Team 1 | A | E | B | A | |||
| Team 2 | A | E | A | ||||
| Team 3 | A | D | E | A | |||
| Team 4 | A | E | A |
Once you have gotten the table right, fill it in with the information shared in the question.
From 2, at 10:00, T2 is at E and T3 is at D
From 3, at 10:30, T1 and T3 are at E
From 4, at 11:30, T1 is at B and T4 is at E
At this point, you should also make a note on your sheet that:
- A -> E only T1 and T4 can go thru
- T4 never goes to B, D, and F
- At a time, there is only one team at one street
We will refer to this information while filling out the table. Points 2, 3, and 4 indicate that only the teams mentioned are present at that station at the specified time. However, instead of verifying this during each step, we will complete the table first and then perform a final check based on this information.
| 9.00 | 9.30 | 10.00 | 10.30 | 11.00 | 11.30 | 12.00 | |
|---|---|---|---|---|---|---|---|
| Team 1 | A | E | B | A | |||
| Team 2 | A | F | E | A | |||
| Team 3 | A | D | E | A | |||
| Team 4 | A | E | A |
Let's figure out the paths where there is one station missing in the middle.
Team 2: A -> -> E
There is only one way to get to E: A -> F -> E
Others are too long. Hence, T2 is at F at 9:30.
| 9.00 | 9.30 | 10.00 | 10.30 | 11.00 | 11.30 | 12.00 | |
|---|---|---|---|---|---|---|---|
| Team 1 | A | E | B | A | |||
| Team 2 | A | F | E | A | |||
| Team 3 | A | C | D | E | A | ||
| Team 4 | A | E | A |
Team 3: There are two ways from A -> D:
A -> C -> D
A -> E -> D
Team 3 is not allowed to use the A-> E path. Hence, at 9:30, T3 is at C.
| 9.00 | 9.30 | 10.00 | 10.30 | 11.00 | 11.30 | 12.00 | |
|---|---|---|---|---|---|---|---|
| Team 1 | A | E | A | B | A | ||
| Team 2 | A | F | E | A | |||
| Team 3 | A | C | D | E | A | ||
| Team 4 | A | E | A |
Team 1 goes from E -> -> B
There is only one way such that there is one station from E -> B:
E -> A -> B
Team 1 is allowed to use the A-> E path. Hence, at 11:00, T1 is at A.
| 9.00 | 9.30 | 10.00 | 10.30 | 11.00 | 11.30 | 12.00 | |
|---|---|---|---|---|---|---|---|
| Team 1 | A | B | E | A | B | A | |
| Team 2 | A | F | E | A | |||
| Team 3 | A | C | D | E | A | ||
| Team 4 | A | E | E | A |
Let's try to fill the 9:30 column. Let's use the constraints we defined:
- At a time, there is only one team at one street:
We can also see that from A, there are 4 streets.
From the constraint, we can't use F or C as they are already being used.
Remaining are B & E.
- T4 never goes to B, D, and F
Since, T4 can't go to B, it has to go to E.
Since, F, C and E are in use, T1 has to go to B
| 9.00 | 9.30 | 10.00 | 10.30 | 11.00 | 11.30 | 12.00 | |
|---|---|---|---|---|---|---|---|
| Team 1 | A | B | A | E | A | B | A |
| Team 2 | A | F | E | A | |||
| Team 3 | A | C | D | E | A | ||
| Team 4 | A | E | E | A |
Team 1: B -> -> E
Only one way, B -> A -> E
Others would be too long.
| 9.00 | 9.30 | 10.00 | 10.30 | 11.00 | 11.30 | 12.00 | |
|---|---|---|---|---|---|---|---|
| Team 1 | A | B | A | E | A | B | A |
| Team 2 | A | F | E | A | |||
| Team 3 | A | C | D | E | A | ||
| Team 4 | A | E | A | E | A |
At 10:00, let's find where T4 is.
There are 3 paths from E:
E -> F
E -> A
E -> D
T2 is already using the E -> F path.
From constraints, we know that T4 never goes to station D.
Hence, at 10:00, T4 goes to A.
| 9.00 | 9.30 | 10.00 | 10.30 | 11.00 | 11.30 | 12.00 | |
|---|---|---|---|---|---|---|---|
| Team 1 | A | B | A | E | A | B | A |
| Team 2 | A | F | E | A | |||
| Team 3 | A | C | D | E | A | ||
| Team 4 | A | E | A | C | A | E | A |
Since, T4 has a lot of contraints, we can easily fill that row.
A -> -> -> E
Possiblities:
- A -> E-> A -> E
- A -> B -> A -> E
- A -> C -> A -> E
We reject 1, as at the same time T1 is using the A -> E path.
We reject 2, as T4 never passes B.
Hence, T4 follows A -> C -> A -> E
| 9.00 | 9.30 | 10.00 | 10.30 | 11.00 | 11.30 | 12.00 | |
|---|---|---|---|---|---|---|---|
| Team 1 | A | B | A | E | A | B | A |
| Team 2 | A | F | E | A | |||
| Team 3 | A | C | D | E | D | C | A |
| Team 4 | A | E | A | C | A | E | A |
Filling for T3 now.
E -> -> -> A
Possiblities:
- E -> D -> C -> A
- E -> A -> E -> A
We reject 2, as T3 is not allowed on E -> A.
Hence, T3 follows E -> D -> C -> A
| 9.00 | 9.30 | 10.00 | 10.30 | 11.00 | 11.30 | 12.00 | |
|---|---|---|---|---|---|---|---|
| Team 1 | A | B | A | E | A | B | A |
| Team 2 | A | F | E | F | A | ||
| Team 3 | A | C | D | E | D | C | A |
| Team 4 | A | E | A | C | A | E | A |
T2 has to be at A at 12:00.
So, it must be at F, B, C or E at 11:30.
However, we know from point 4 of the question that only T1 and T4 are at B and E respectively.
Moreover, we know that T3 is using the C -> A path.
Hence, the only option that remains is F.
| 9.00 | 9.30 | 10.00 | 10.30 | 11.00 | 11.30 | 12.00 | |
|---|---|---|---|---|---|---|---|
| Team 1 | A | B | A | E | A | B | A |
| Team 2 | A | F | E | F | F | A | |
| Team 3 | A | C | D | E | D | C | A |
| Team 4 | A | E | A | C | A | E | A |
Where all can Team 2 at 10:30?
- E -> F
- E -> A
- E -> D
Reject 2 as T2 not allowed on that path
Reject 3 as T3 is using the path bw 10:00 & 10:30.
Hence, at 10:30, Team 2 is at F
| 9.00 | 9.30 | 10.00 | 10.30 | 11.00 | 11.30 | 12.00 | |
|---|---|---|---|---|---|---|---|
| Team 1 | A | B | A | E | A | B | A |
| Team 2 | A | F | E | F | A/E | F | A |
| Team 3 | A | C | D | E | D | C | A |
| Team 4 | A | E | A | C | A | E | A |
Team 2: F -> -> F
- F -> A-> F
- F -> E-> F
Both are possible
| 9.00 | 9.30 | 10.00 | 10.30 | 11.00 | 11.30 | 12.00 | |
|---|---|---|---|---|---|---|---|
| Team 1 | A | B | A | E | A | B | A |
| Team 2 | A | F | E | F | A/E | F | A |
| Team 3 | A | C | D | E | D | C | A |
| Team 4 | A | E | A | C | A | E | A |
From the table, we can see that 2 teams (teams 3 and 4) have passed through station C on
the given day.
The correct option is C