Limited v. Local Buses

Doug is interested in traveling to Brilliantia . From his house, he walked to the starting bus terminal, where there are two different buses both about to depart simultaneously. Before his travel, he went inside both buses to ask about the bus stops between his current location and the final stop. He learned that:

  • Each of the stations is named with the consecutive positive integer, so the ordered stations are { 0 , 1 , 2 , 3 , , 35 } \{0,1,2,3,\dots,35\} from closest to farthest. Doug is currently at 0 0 -th station, so there are 35 stations ahead.
  • A station is said to be available if there is no bus currently waiting, or if a bus instantly departs at the end of its waiting time.
  • Local buses travel to any closest available station and then, wait for 1 minute.
  • Limited buses travel to any available even numbered station and then, wait for 2 minutes. If both local and limited buses arrive to one of these stations at the same time, only limited buses stop there.
  • Both buses travel exactly 2 minutes between i i -th and ( i + 1 ) (i+1) -th stations, where i { 0 , 1 , 2 , , 34 } i\in \{0,1,2,\dots , 34\} .
  • Both buses respectively repeat their steps until they reach their final stop at 35 35 -th station.

Which bus should he take in order to minimize the total time elapsed between the starting and ending terminals - local or limited buses?

Note: Neglect heavy traffic, crowds, detours and accidents throughout the trip.

Local bus Limited bus Both buses arrive at the same time Not enough information

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1 solution

Richard Xu
Apr 3, 2018

If you hand-solve for the first several minutes, you'll find a pattern that, for every k 3 k\ge 3 , at T = 6 k 3 T=6k-3 , the local bus arrives at station 2 k 2k and stay for 1 minute, and at T = 6 k 2 T=6k-2 , the local bus leaves while the limited bus arrives and stay for 2 minutes.

The following table serves as an induction.

Time Local Bus Limited Bus
6k-3 2k way
6k-2 2k 2k
6k-1 way 2k
6k 2k+1 2k
6k+1 2k+1 way
6k+2 way 2k+1
6k+3 2k+2 way
6k+4 2k+2 2k+2
... ... ...

Therefore, the local bus would arrive at station 35 two minutes earlier than the limited bus, as is shown in the table above.

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