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


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Explaining the Simplified Elevator Puzzle with Simple Logic

If you are near the bottom of the building, the elevator is more likely to be anywhere above you than below. Therefore you are more likely to see it going down rather than up.

The same holds if you start near the top: the more floors below, the more likely the elevator is below you and needs to go up before it can return down.

Since your random destination is more likely to be where there are more floors, it makes perfect sense that you want to go where the elevator already is! So the elevator has to come back past you.

The General Multiple-Elevator Puzzle in the Real World

Multiple Simple Elevators

As Weisstein notes in the Wolfram “Elevator Paradox” article, the situation is more complex if there are more elevators.


Suppose each elevator has its own “brute force” approach of opening on every floor in each direction on every round trip. Each elevator has the same probability distribution. However, queueing theory now asks whether the elevators start synchronized from different starting points, and whether they are delayed by passengers entering or leaving.

Multiple Smart Elevators

In addition, smart programming can alleviate some of the problems, because the goal would be to minimize wait times. Empty elevators could be dispatched to different floors, in order to minimize waiting times.

Elevators might serve a few adjacent floors until someone signals for a long ride; this skews the probability of where this elevator is likely to be.

Tall buildings may dedicate a bank of elevators to low floors and another bank to high floors plus the main lobby. Others allocate banks to serve odd or even floors.

Any of these techniques change the range of possible states the elevator could be in, or are likely to be in, when you first signal for an elevator.

Programming elevators with dedicated tasks can be seen as training the passengers to behave in ways that benefit the collective: think of it as “enforced car pooling”.

Passengers’ Destinations are Not Random

Someone who works only on the top floor would never notice the elevator paradox, since he or she only uses the first and nth floors. Perhaps this guarantee of a “happy” state is why the penthouse suite in a condominium tower commands a premium price.

Trained Passengers Reduce Elevator Delays

This author remembers working in one office tower where the elevator puzzle was visible because there were two main floors: one at street level and one on the lower shopping concourse. I, and many others, who wanted to go “up” from street level found it easier to take the first elevator down to the food concourse and stay on board. The alternative was to jam into an elevator overcrowded with workers returning from dining and shopping underground.

Perform Your Own Experiments on the Elevator Puzzle

Since scientific theories must be tested by experiment, try it for yourself. Take notes for a day, week or month. Besides the “happy” or “annoyed” result, note your starting floor and the range of possible floors the elevator could be on.

Many elevator lobbies display the state of each elevator, so you don’t really have to wait for the first to open. In a lobby with many elevators, this is impractical. The security people may ask probing questions if you record the elevator status using a video device.

Decoded Science would love to hear from our readers: did you experience the “Elevator Puzzle” yourself?

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