Two travelers set out at the same time to travel opposite ways round a circular railway. Trains start each way every fifteen minutes: on the hour, fifteen minutes past, half past, and forty-five minutes past. Clockwise trains take two hours for the journey, counterclockwise trains take three hours.

Including trains seen at the starting point and the ones they are traveling on, how many trains did each traveler see on his journey?

Counterclockwise trains take three hours and leave at 15-minute intervals, so there are 12 counterclockwise trains on the track at any given time. Similarly there are 8 clockwise trains on the railway (presumably a different track) at any one time.

Between the time the clockwise train sets out and the time it returns to its starting station, someone on it will see all the trains currently on the other track (that's 12) plus all those that set out during that 2-hour interval, which is 8, making 20. Of course he also sees his own train, bringing the total up to 21, but we could also say that he sees the train of the other traveler in the question twice, making 22 in all.

If we count each individual train only once, no matter how many times one sees it, then of course the clockwise traveler sees only the 12 other-direction trains plus his own.

For the counterclockwise traveler, he sees all 8 clockwise trains plus all that start out in his 3-hour trip, which is 12, giving the same 20 we found for the clockwise traveler. The same added-train considerations apply. But in the case of counting any given physical train only once, he sees only the 8 trains going in the other direction plus his own train.

Edited on July 12, 2013, 1:09 pm

Posted by Charlie on 2013-07-12 13:07:46