The picture above shows a map of 9 train stations. These stations are connected by a single track running from station 1 to 9 as represented by the solid line (the trains runs both ways from 1->9 and 9->1). Let's say each station is 1 km distance apart from the adjacent station, along the railway track (solid line). Distance between 4 and 6, 4 and 7, 3 and 7, 3 and 8, 2 and 8, 2 and 9, 1 and 9 is also 1 km.
Now people from station 1 are only 1km far from station 9, but they have to go all the way round traveling 8km to reach station 9. Similarly, it is inefficient for people at other stations as well.
So the engineers decide to connect the stations on both sides. But they can only use a 1km railway track, due to budget constraints.
Which two stations do they connect so that it benefits all the people the most? They are supposed to consider the collective benefit. More specifically, they are to optimize the average travel time of the passengers, assuming that people embark and disembark at all stations with equal probability.
Note: I am new on this forum, and new at creating puzzles so please bear with me if I didn't give sufficient information for this puzzle to work. And I don't know the answer to the puzzle yet, I myself am trying to solve it.