Enough fuel for exactly one lap

Question

Several fuel stations are placed at arbitrary points around a circular road. Between them they hold exactly the amount of fuel needed for a car to drive once around the whole circle. The car has an unlimited-capacity tank but starts empty, and it may only take fuel from a station when it is at that station.

Prove that there is at least one station from which the car can start, driving in the clockwise direction, and complete the entire lap without ever running out of fuel.

Accepted Answer

Yes. Start at the station where an imaginary fuel gauge reaches its lowest point. The imaginary lap. Pick any station and imagine starting there with a huge amount of borrowed fuel, so that running out is impossible. Drive the lap clockwise, at each station picking up all its fuel, and record the fuel level just before arriving at each station. Because the stations hold exactly one lap's worth of fuel, the gauge reads the same at the end of the lap as at the beginning: the fuel picked up equals the fuel burned. Choose the minimum. Among the readings "just before arriving at station S", pick t…

Enough fuel for exactly one lap

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