CAT 2023 Slot 3 · DILR · Set 2: Police Teams Patrolling Six Stations

Solving the set

Streets (from the map): A–B, A–C, A–E, A–F, E–F, E–D, C–D. Every 30 minutes each team moves to a neighbouring station, so each team visits a station at 9:30, 10:00, …, 11:30 and is back at A at 12:00 (6 moves).

Step 1 – Team 3 is at D at 10:00, two moves after leaving A, without using A–E (fact 5): its route starts A–C–D. Team 2 is at E at 10:00, two moves after leaving A, without A–E: the only such route is A–F–E.

Step 2 – Team 4 moves only among A, C and E (fact 6) and must use A–E (fact 5). It cannot be at E at 10:00 or 10:30 (facts 2, 3), and it is at E at 11:30. Starting A–C would put it on street A–C together with Team 3 between 9:00 and 9:30 (fact 1), so it starts A–E. Its route is A–E–A–C–A–E–A.

Step 3 – Team 1 at 9:30: streets A–C, A–F and A–E are taken in the first half hour (Teams 3, 2, 4), so Team 1 goes to B and returns to A at 10:00. It reaches E at 10:30 by A–E, and must be back at A at 11:00 to be at B at 11:30. Route: A–B–A–E–A–B–A.

Step 4 – Team 3 after E at 10:30: going to F would leave it unable to reach A at exactly 12:00 without passing E at 11:30 (reserved for Team 4), so it returns E–D–C–A. Route: A–C–D–E–D–C–A.

Step 5 – Team 2 after E at 10:00: E–D would clash with Team 3 on street D–E (10:00–10:30) and E is reserved at 10:30, so it goes to F. From F it goes to A or E at 11:00. At 11:30 it cannot be at E (Team 4 only) or B (Team 1 only); going A–C would make it share street C–A with Team 3 between 11:30 and 12:00. So it is at F at 11:30 and back at A at 12:00.

This question

From the visits table, Station E is visited 5 or 6 times, more than C (3), F (3) and D (2).

Hence, the answer is option 2.

Solving the set

Streets (from the map): A–B, A–C, A–E, A–F, E–F, E–D, C–D. Every 30 minutes each team moves to a neighbouring station, so each team visits a station at 9:30, 10:00, …, 11:30 and is back at A at 12:00 (6 moves).

Step 1 – Team 3 is at D at 10:00, two moves after leaving A, without using A–E (fact 5): its route starts A–C–D. Team 2 is at E at 10:00, two moves after leaving A, without A–E: the only such route is A–F–E.

Step 2 – Team 4 moves only among A, C and E (fact 6) and must use A–E (fact 5). It cannot be at E at 10:00 or 10:30 (facts 2, 3), and it is at E at 11:30. Starting A–C would put it on street A–C together with Team 3 between 9:00 and 9:30 (fact 1), so it starts A–E. Its route is A–E–A–C–A–E–A.

Step 3 – Team 1 at 9:30: streets A–C, A–F and A–E are taken in the first half hour (Teams 3, 2, 4), so Team 1 goes to B and returns to A at 10:00. It reaches E at 10:30 by A–E, and must be back at A at 11:00 to be at B at 11:30. Route: A–B–A–E–A–B–A.

Step 4 – Team 3 after E at 10:30: going to F would leave it unable to reach A at exactly 12:00 without passing E at 11:30 (reserved for Team 4), so it returns E–D–C–A. Route: A–C–D–E–D–C–A.

Step 5 – Team 2 after E at 10:00: E–D would clash with Team 3 on street D–E (10:00–10:30) and E is reserved at 10:30, so it goes to F. From F it goes to A or E at 11:00. At 11:30 it cannot be at E (Team 4 only) or B (Team 1 only); going A–C would make it share street C–A with Team 3 between 11:30 and 12:00. So it is at F at 11:30 and back at A at 12:00.

This question

Only Team 1 passes through B (Team 4 never does, and Teams 2 and 3 have no time to). Team 1 is at B at 9:30 and 11:30: 2 times.

Hence, the answer is 2.

Solving the set

Streets (from the map): A–B, A–C, A–E, A–F, E–F, E–D, C–D. Every 30 minutes each team moves to a neighbouring station, so each team visits a station at 9:30, 10:00, …, 11:30 and is back at A at 12:00 (6 moves).

Step 1 – Team 3 is at D at 10:00, two moves after leaving A, without using A–E (fact 5): its route starts A–C–D. Team 2 is at E at 10:00, two moves after leaving A, without A–E: the only such route is A–F–E.

Step 2 – Team 4 moves only among A, C and E (fact 6) and must use A–E (fact 5). It cannot be at E at 10:00 or 10:30 (facts 2, 3), and it is at E at 11:30. Starting A–C would put it on street A–C together with Team 3 between 9:00 and 9:30 (fact 1), so it starts A–E. Its route is A–E–A–C–A–E–A.

Step 3 – Team 1 at 9:30: streets A–C, A–F and A–E are taken in the first half hour (Teams 3, 2, 4), so Team 1 goes to B and returns to A at 10:00. It reaches E at 10:30 by A–E, and must be back at A at 11:00 to be at B at 11:30. Route: A–B–A–E–A–B–A.

Step 4 – Team 3 after E at 10:30: going to F would leave it unable to reach A at exactly 12:00 without passing E at 11:30 (reserved for Team 4), so it returns E–D–C–A. Route: A–C–D–E–D–C–A.

Step 5 – Team 2 after E at 10:00: E–D would clash with Team 3 on street D–E (10:00–10:30) and E is reserved at 10:30, so it goes to F. From F it goes to A or E at 11:00. At 11:30 it cannot be at E (Team 4 only) or B (Team 1 only); going A–C would make it share street C–A with Team 3 between 11:30 and 12:00. So it is at F at 11:30 and back at A at 12:00.

This question

At 10:00 Team 3 is at D and at 10:30 it is at E, so at 10:15 it is on the D–E street.

Hence, the answer is option 4.

Solving the set

Streets (from the map): A–B, A–C, A–E, A–F, E–F, E–D, C–D. Every 30 minutes each team moves to a neighbouring station, so each team visits a station at 9:30, 10:00, …, 11:30 and is back at A at 12:00 (6 moves).

Step 1 – Team 3 is at D at 10:00, two moves after leaving A, without using A–E (fact 5): its route starts A–C–D. Team 2 is at E at 10:00, two moves after leaving A, without A–E: the only such route is A–F–E.

Step 2 – Team 4 moves only among A, C and E (fact 6) and must use A–E (fact 5). It cannot be at E at 10:00 or 10:30 (facts 2, 3), and it is at E at 11:30. Starting A–C would put it on street A–C together with Team 3 between 9:00 and 9:30 (fact 1), so it starts A–E. Its route is A–E–A–C–A–E–A.

Step 3 – Team 1 at 9:30: streets A–C, A–F and A–E are taken in the first half hour (Teams 3, 2, 4), so Team 1 goes to B and returns to A at 10:00. It reaches E at 10:30 by A–E, and must be back at A at 11:00 to be at B at 11:30. Route: A–B–A–E–A–B–A.

Step 4 – Team 3 after E at 10:30: going to F would leave it unable to reach A at exactly 12:00 without passing E at 11:30 (reserved for Team 4), so it returns E–D–C–A. Route: A–C–D–E–D–C–A.

Step 5 – Team 2 after E at 10:00: E–D would clash with Team 3 on street D–E (10:00–10:30) and E is reserved at 10:30, so it goes to F. From F it goes to A or E at 11:00. At 11:30 it cannot be at E (Team 4 only) or B (Team 1 only); going A–C would make it share street C–A with Team 3 between 11:30 and 12:00. So it is at F at 11:30 and back at A at 12:00.

This question

Team 4's route A–E–A–C–A–E–A passes through E at 9:30 and 11:30: 2 times.

Hence, the answer is 2.

Solving the set

Streets (from the map): A–B, A–C, A–E, A–F, E–F, E–D, C–D. Every 30 minutes each team moves to a neighbouring station, so each team visits a station at 9:30, 10:00, …, 11:30 and is back at A at 12:00 (6 moves).

Step 1 – Team 3 is at D at 10:00, two moves after leaving A, without using A–E (fact 5): its route starts A–C–D. Team 2 is at E at 10:00, two moves after leaving A, without A–E: the only such route is A–F–E.

Step 2 – Team 4 moves only among A, C and E (fact 6) and must use A–E (fact 5). It cannot be at E at 10:00 or 10:30 (facts 2, 3), and it is at E at 11:30. Starting A–C would put it on street A–C together with Team 3 between 9:00 and 9:30 (fact 1), so it starts A–E. Its route is A–E–A–C–A–E–A.

Step 3 – Team 1 at 9:30: streets A–C, A–F and A–E are taken in the first half hour (Teams 3, 2, 4), so Team 1 goes to B and returns to A at 10:00. It reaches E at 10:30 by A–E, and must be back at A at 11:00 to be at B at 11:30. Route: A–B–A–E–A–B–A.

Step 4 – Team 3 after E at 10:30: going to F would leave it unable to reach A at exactly 12:00 without passing E at 11:30 (reserved for Team 4), so it returns E–D–C–A. Route: A–C–D–E–D–C–A.

Step 5 – Team 2 after E at 10:00: E–D would clash with Team 3 on street D–E (10:00–10:30) and E is reserved at 10:30, so it goes to F. From F it goes to A or E at 11:00. At 11:30 it cannot be at E (Team 4 only) or B (Team 1 only); going A–C would make it share street C–A with Team 3 between 11:30 and 12:00. So it is at F at 11:30 and back at A at 12:00.

This question

From the routes, Team 3 (9:30 and 11:30) and Team 4 (10:30) pass through C. Teams 1 and 2 do not. So 2 teams.

Hence, the answer is option 4.