Two Trains Puzzle

Two trains are on the same track a distance 100 km apart heading towards one another, each at a speed of 50 km/h. A fly starting out at the front of one train, flies towards the other at a speed of 75 km/h.

Upon reaching the other train, the fly turns around and continues towards the first train. How many kilometers does the fly travel before getting squashed in the collision of the two trains?

Now, the trains take one hour to collide (their relative speed is 100 km/h and they are 100 km apart initially). Since the fly is traveling at 75 km/h and flies continuously until it is squashed (which it is to be supposed occurs a split second before the two oncoming trains squash one another), it must therefore travel 75 km in the hour's time.
The position of the fly at time is plotted above.

However, a brute force method instead solves for the position of the fly along each traversal between the trains. For example, the fly reaches the second train when

 (1)

or h, at which point it has traveled a distance km. It then turns around and reaches the first train again when

 (2)

or . Continuing, the total distance traveled by the fly is given by summing the series

 (3)

When posed with a variant of this question involving a fly and two bicycles, John von Neumann is reputed to have immediately answered with the correct result. When subsequently asked if he had heard the short-cut solution, he answered no, that his immediate answer had been a result of explicitly summing the series (MacRae 1992, p. 10; Borwein and Bailey 2003, p. 42).

In Ron Howard's 2001 film A Beautiful Mind, John Nash (played by Russell Crowe) can be overheard discussing this problem with a group of students in the library.

REFERENCES:

Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, p. 42, 2003.

Flannery, S. and Flannery, D. In Code: A Mathematical Journey. London: Profile Books, pp. 13-14, 2000.

MacRae, N. John von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More. Providence, RI: Amer. Math. Soc., 1992.

Πηγή