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?

Clockwise trains never encounter other clockwise trains, and counter clockwise trains never encounter other counterclockwise train, so all encounters are between a clockwise and a counterclockwise train. Since there is one of each involved in each encounter, clockwise trains (in total) have the same number of encounters as do counterclockwise trains in any given time period. And since they have the same starting frequency, the have the same number of trips per day (in total). Therefore, clockwise and counterclockwise trains have the same number of encounters per trip. Charlie has shown that this number is 20.

The traveler on a clockwise train therefore encounters 20 counterclockwise trains (including one at the start). And he sees his own train, for a total of 21. We do not count, per puzzle instructions, the counterclockwise train that he sees at the end of the trip. This is good, I think, because it means that the answer would be the same even if the clockwise trains started 5 minutes after the counterclockwise trains.

It has been pointed out that these are not distinct trains, but we don't have any information on that point. I prefer not to make assumptions about how frequently trains get pulled out of service and replace by other trains. Each traveler sees 21 trains, including his own. That's my answer, and I'm sticking to it.