Is the Elevator Puzzle a Math Paradox or a Paranoid Delusion?

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Why is the first elevator you see more likely to be going in the wrong direction? Two physicists reported this phenomenon in 1958.

"Pacific Center Building Elevators" by ricardodiaz11

Pacific Center Building Elevators: Photo by ricardodiaz11

It is sometimes called the “elevator paradox,” but “elevator puzzle” better expresses the surprise or annoyance that everyone feels when faced with an uncooperative elevator that always seems to be going the wrong direction. No, this is not a paranoid delusion. Just follow the mathematics into probability theory, or skip to the logic at the end.

Defining the Elevator Puzzle

It seems likely that a person on a lower floor of a multistory building wants to go up, since there are more floors above than below. Similarly, a person on a higher floor has more potential destinations below than above.

If the likelihood of the desired direction of travel depends on the relative starting position, then it seems reasonable that the elevators are more likely to be going in that direction.

However, if you request both an “up” and a “down” elevator, you would observe that the first elevator to open its doors is going in the direction where there are fewer floors!

The Simplified Elevator Puzzle

The simplest version of the elevator puzzle is an abstraction from the real world.

Suppose you work in a ten-story building. Your company rents offices on the second and ninth floors. Your job requires frequent visits to each of these floors. Each visit takes a random amount of time: much longer than one elevator ride, but usually much less than one thousand rides. So you ride the elevator, and make observations, many times a day.

One elevator services the entire building from floors one through ten. When it opens its door, it also signals whether it is heading up or down.

The building’s management implemented a cost-cutting measure; it happens to make this puzzle easier to explain. Rather than having sophisticated computer controls, the elevator simply goes from one floor to the next. It stops to open its door on each floor, but people can hold the elevator open as needed when passengers need more time to get in or out.

Therefore the elevator goes all the way up, and all the way down, visiting each floor in turn.

Behold the Simplified Elevator Puzzle!

"Elevator Puzzle 2 or 9 of 10" by Mike DeHaan

Elevator Puzzle: 2 or 9 of 10. Image by Mike DeHaan

To your annoyance, you notice that usually the elevator is going in the “wrong” direction. If you are on the second floor wanting to go up, the first elevator is heading down. When you are finished with the ninth floor, the first elevator you see is heading up to the tenth.

What’s going on? Are you very unlucky? Are “they” trying to drive you insane? Or is there a mathematical explanation?

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