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?

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 remained 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.

Further reading:

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

External links

• A detailed treatment, part 1 (http://www.kwansei.ac.jp/hs/z90010/english/sugakuc/toukei/elevator/elevator.htm) by Tokihiko Niwa
• Part 2: the multi-elevator case (http://www.kwansei.ac.jp/hs/z90010/english/sugakuc/toukei/elevator2/elevator2.htm)
• MathWorld article (http://mathworld.wolfram.com/ElevatorParadox.html) on the elevator paradox

There is also another (unrelated) elevator paradox relating to the position of a hydrometer floating in water, which, in a classic demonstration, remains at an equilibrium position as the entire system is placed in an elevator which moves up and down, changing the air pressure.

This is due to the fact that the change in air pressure is applied to the entire hydrometer flask; the underwater portion of the flask receives a transmitted force through the water, thus the flask receives no net force due to the change in air pressure. A cartesian diver, on the other hand, has an air space which, unlike a hydrometer, is not sealed, and thus can change its displacement as increasing external air pressure compresses the air in the diver. It should be noted that not all closed containers are immune from this change; a non-rigid container like a toy balloon will be changed, as will the rib cage of a human SCUBA diver, and such systems will vary in buoyancy. A glass hydrometer is rigid under normal pressure, for all practical purposes.

If a cartesian diver (instead of being in the classic plastic bottle) were placed in a beaker in an elevator, and the elevator could be lowered or raised enough to significantly change the volume of the air, the diver would respond.

It is interesting to note that the upward or downward acceleration of the elevator, as long as the net force is directed downward, will not change the equilibrium point either - the force due to acceleration acts on the hydrometer exactly as it would on an equal mass of water.