CAT 2023 Slot 3 DILR Set 3: Routes & Networks (5 Questions) with Solutions

The streets are A-B, A-C, A-E, A-F, E-F, C-D and D-E. Team 4 avoids B, D and F, so it can only use A, C and E. Because C and E are not joined to each other, it must alternate between A and one of C or E.

Team 2 is at E at 10:00. The A-E street belongs only to Teams 1 and 4 (fact 5), so Team 2 arrives from F: A, F, E. Team 3 is at D at 10:00, which forces A, C, D. Fact 3 puts Team 3 at E at 10:30.

Team 3 moves A to C in the first leg, so Team 4 cannot use the same street in the same direction (fact 1). Team 4 therefore goes A, E, A. Fact 3 says only Teams 1 and 3 are in E at 10:30, so Team 4 is at C then. Fact 4 puts it in E at 11:30. Team 4 is A, E, A, C, A, E, A.

Fact 4 puts Team 1 in B at 11:30, so it is at E at 10:30, A at 11:00 and B at 11:30. At 9:30 it can only be at B. C and F are taken by Teams 3 and 2 on the same streets, and E is taken by Team 4. Team 1 is A, B, A, E, A, B, A.

Team 3 is at E at 10:30. It cannot use A-E, and F leads nowhere useful. It is at D at 11:00 and C at 11:30: A, C, D, E, D, C, A.

Team 2 is at F at 11:30, because B, E and C are excluded for it. Its 10:30 position is F or D and its 11:00 position is E or A, so its E visits are one or two.

Visits to E: Team 4 twice, Teams 1 and 3 once each, and Team 2 at least once. That is at least 5. Station C is visited 3 times and D at most 3 times. F is visited at most 4 times (Team 2 at 9:30 and 11:30, plus at most two more if it uses the F route at 10:30).

B connects only to A, so any visit to B is a round trip from A.

Fact 4 says Team 1 is the only team in B at 11:30. So Team 1 goes E at 10:30, A at 11:00, B at 11:30, A at 12:00.

Team 1 is at E at 10:30 and at D, E or A at 10:00 is not available to it. It is at A at 10:00, so its 9:30 stop is a neighbour of A. C and F are used by Teams 3 and 2 on the same streets at that time, and E is used by Team 4. That leaves B, so Team 1 visits B at 9:30.

Team 4 never visits B (fact 6). Team 2 is A, F, E at 9:00 to 10:00 and is at F at 11:30. Team 3 is A, C, D, E, D, C, A. Neither visits B.

B is therefore visited only by Team 1, at 9:30 and 11:30.

Fact 2 says Team 3 is the only team in D at 10:00. D is two legs from A, so Team 3 is A, C, D at 9:00, 9:30 and 10:00.

Fact 3 says Team 3 is in E at 10:30. The only street from D to E is the one in question, so Team 3 travels it between 10:00 and 10:30.

At 10:15 Team 3 is therefore midway along the D-E street.

The other teams are elsewhere at that time. Team 4 is on the A-E and A-C streets (A, E, A, C, A, E, A). Team 1 is on the A-B and A-E streets. Team 2 is at E at 10:00 and is on the E-F street or the E-D street in the opposite direction, but not towards E.

Fact 1 allows only one team per direction on a street, and Team 3 is the one heading D to E.

Fact 6 bars B, D and F. C and E are not joined to each other, so Team 4 is at A at 10:00 and 11:00, and at C or E at 9:30, 10:30 and 11:30.

Fact 4 puts Team 4 in E at 11:30.

Fact 3 says only Teams 1 and 3 are in E at 10:30, so Team 4 is at C at 10:30.

At 9:30 Team 4 is at C or E. Team 3 must be at D at 10:00 (fact 2), so it goes A to C in the first leg. Fact 1 forbids a second team on that street in the same direction, so Team 4 goes to E at 9:30.

Team 4's route is A, E, A, C, A, E, A. It is in E at 9:30 and 11:30.

Fact 2 puts Team 3 at D at 10:00. D is two legs from A, so Team 3 goes A, C, D, and it visits C at 9:30.

Fact 3 puts Team 3 in E at 10:30. It cannot use A-E, which is reserved for Teams 1 and 4. At 11:00 it is at D, and at 11:30 it is at C, because B and E are excluded and F cannot return to A. So Team 3 visits C twice, but it is still one team.

Team 4 uses A, C and E. Fact 3 keeps it out of E at 10:30, so it is at C then. Team 4 visits C.

Team 1 is A, B, A, E, A, B, A, so it never visits C.

Team 2 is A, F, E at 9:00 to 10:00 and at F at 11:30. Reaching C at 11:00 would need D at 10:30 and then a move to A or D at 11:30, not F, so Team 2 never visits C.

The teams that pass through C are Teams 3 and 4.