The elevator paradox is an apparent paradox first noted by George Gamow and Moritz Stern, physicists who had offices on two different floors of a multi-story building. Gamow, who had an office near the bottom of the building, noted that the first elevator to stop at his floor was most often going down, while Stern, who had an office near the top, noticed that the first elevator to stop at his floor was most often going up.

At first sight, this created the impression that perhaps elevators were being manufactured in the middle of the building and sent upwards to the roof and downwards to the basement to be dismantled. Clearly this was not the case. But how could the observation be explained?

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## Modelling the elevator problem

Several attempts (beginning with Gamow and Stern) were made to analyze the reason for this phenomenon, which is more difficult to analyze than it at first seems.

Essentially, the explanation seems to be this: a single elevator spends most of its time in the larger section of the building, and thus is more likely to approach from that direction when the prospective elevator user arrives. An observer who remains by the elevator doors for hours or days, observing every elevator arrival, rather than only observing the first elevator to arrive, would note an equal number of elevators traveling in each direction.

A similar effect can be observed in railway stations near the end of a railway line running a shuttle service, or watching a single car going round an oval racetrack from a point near one of the ends of the racetrack.

## More than one elevator

Interestingly, if there is more than one elevator in a building, the bias decreases - since there is a greater chance that the intending passenger will arrive at the elevator lobby during the time that at least one elevator is below him; with an infinite number of elevators, the probabilities would be equal. Watching cars pass on an oval racetrack, one perceives little bias if the time between cars is small compared to the time required for a car to return past the observer.

## The real-world case

In a real building, there are complicating factors such as the tendency of elevators to be frequently required on the ground or first floor, and to return there when idle. These factors tend to shift the frequency of observed arrivals, but do not eliminate the paradox entirely. In particular, a user very near the top floor will perceive the paradox even more strongly, as elevators are infrequently required, or already present, above his floor.

• Martin Gardner, Knotted Doughnuts and Other Mathematical Entertainments, chapter 10. W H Freeman & Co.; (October 1986). ISBN 0716717999.

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