Two Trains and a Fly

Question

Two trains 100 km apart drive toward each other, each at 50 km/h. A fly starts on the nose of one train, flies at 75 km/h straight to the other's nose, turns around instantly, and shuttles back and forth until the trains meet. How far does the fly fly in total?

Accepted Answer

The trains close at 50 + 50 = 100 km/h, so they meet after exactly 1 hour. The fly flies at 75 km/h for that hour, regardless of its zigzag path: 75 text km/h times 1 text h = mathbf75 text km. The point: distance = speed × time needs total time, not the trajectory. The infinite bouncing is decoration designed to bait you into a series. Approach 2, sum the series anyway First leg: fly and oncoming train close at 125 km/h over 100 km → 0.8 h → 60 km flown; the trains are then 20 km apart, and the geometry has shrunk by a factor of 5. Legs form a geometric series: 60 cdot left(1 + tfrac15 + tfrac125 + cdotsright) = frac601 - 1/5 = 75. checkmark Knowing both routes is the flex: the shortcut shows you hunt for the invariant (elapsed time), the series shows the shortcut isn't a fluke. The meta-lesson for trading interviews, when a process looks recursive, first ask what aggregate quantity is already determined.

Two Trains and a Fly

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