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Passing Trains
Mr. William Johnson, a retired railroad man, was puzzled by the fact that most of the trains he saw at the crossing at night traveled eastward. This almost drove him mad, for he knew very well there should be an equal number of trains going west. At last he got the solution to this puzzle from an amateur mathematician. What was the solution and how did he get it?
In a small midwestern town there lived a retired railroad engineer named William Johnson. The main line on which he had
worked for so many years passed through the town. Mr. Johnson suffered from insomnia and would often wake up at any odd hour
of the night and be unable to fall asleep again. He found it helpful, in such cases, to take a walk along the deserted
streets of the town, and his way always led him to the railroad crossing. He would stand there thoughtfully watching the
track until a train thundered by through the dead of the night. The sight always cheered the old railroad man, and he would
walk back home with a good chance of falling asleep.
After a while he made a curious observation; it seemed to him that most of the trains he saw at the crossing were traveling eastward, and only a few were going west. Knowing very well that this line was carrying equal numbers of eastbound and westbound trains, and that they alternated regularly, he decided at first that he must have been mistaken in this reckoning. To make sure, he got a little notebook, and began putting down 'E' or 'W', depending on which way the first train to pass was traveling. At the end of a week, there were five 'E' s' and only two 'W' s' and the observations of the next week gave essentially the same proportion. Could it be that he always woke up at the same hour of night , mostly before the passage of eastbound trains?
Being puzzled by this situation, he decided to undertake a rigorous statistical study of the problem, extending it also to
the daytime. He asked a friend to make a long list of arbitrary times such as 9:35 a.m. , 12:00 noon, 3:07 p.m. , and so
on, and he went to the railroad crossing punctually at these times to see which train would come first. However, the result
was the same as before. Out of one hundred trains he met, about seventy-five were going east and only twenty-five west.
In despair, he called the depot in the nearest big city to find whether some of the westbound trains had been re-routed
through another line, but this was not the case. He was, in fact, assured that the trains were running exactly on schedule,
and that equal numbers of trains daily were going each way.
This mystery brought him to such despair that he became completely unable to sleep and was a very sick man.
The local doctor whom Mr. Johnson consulted concerning his health was also an amateur mathematician and a collector of puzzles.
'This is a new one to me, ' said he, when Mr. Johnson described the cause of his troubles. 'But wait a minute, there must be a rational answer. ' And, after a few minutes of reflection , the doctor had the answer ready.
'You see, ' said he, the whole thing depends on the fact that the trains are running on a schedule, even though you were
arriving at the crossing at any odd time. Let us suppose that eastbound trains pass our town every hour on the hour,
whereas the westbound ones pass by at a quarter past each hour. There are, of course, an equal number of trains in each
direction. But let us see which train wilI be first to pass when you arrive at the crossing. If you arrive between an even
hour and a quarter past the hour, say between l :00 and I: 15 p. m. , the first train to pass will be that going west,
that is, the l :15 p. m. train.
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