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

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: 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. 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?

Resolving the Simplified Elevator Puzzle

Consider this a simple probability exercise. At any moment, the most recent “status” of each elevator is a floor and a direction. For example, “On the first floor and about to go up” is the state “1 : Up”. There are 18 such states, as shown in the first image.

These states are {(1 : up), (2 : up),…,(9 : up), (10 : down), (9 : down), (8 : down),…,(3 : down), (2 : down)}.

When the elevator first opens on your floor, if it is going in the direction you want, you will be “Happy!” and board the elevator. If it is going the other way, you will be “Annoyed!” by the extra waiting time.

At the moment when you step into the second-floor elevator lobby, the elevator is equally likely to be in any one of these 18 states.

But only under those 2 conditions, out of 18, will you observe the elevator first open its door on the second floor while heading up. In the other 16 cases, you will first see the elevator open its door before going down to the first floor. Otherwise, after a variable delay time, you will see the elevator open its door while headed in the “wrong” direction.

The Ninth Floor is No Better

The probabilities are the same when you step into the lobby on the ninth floor. The elevator may have been on the tenth, or is just coming down from there; those are the 2 fortunate conditions out of 18.

Summary of the Simplified Elevator Puzzle

Regardless of whether you go from 2 to 9 or vice versa, in this ten-floor building with one relentless elevator, you only have 2 chances in 18 (or 0.111…) that the elevator will be going in the direction you want when it first opens its door.

In other words, you observe that you only have an 11% chance that the elevator will head in the right direction the first time you see it arrive. But it should be going up half the time, and down half the time.

The Observer Versus the Observation

In fact half of the times when it passes your floor it is going up.  If you were to stand quietly in the second-floor lobby for a full day and keep a log of the elevator’s activity, you observe it actually does have a 50% chance of being in state “2 : up” and “2 : down”. These are the only states you can observe from the second-floor lobby.

Because you normally take the elevator as soon as it is in the correct state, you do not take a proper statistical sampling of the elevator’s activity. Your 11% success ratio is due to the combination of where the elevator spends its time compared to your floor, and mainly because you cut short the “experiment” once your needs are met.

The Simplified General Single-Elevator Puzzle

Other Elevator Puzzles with 10 Floors. Image by Mike DeHaan

(Use this 10-floor image to help think about other situations.)

Rather than 10 floors, there are ‘n’ floors in a building with a single elevator that always stops at every floor. To make matters simpler, let’s just think about a single trip at any one time.

The elevator has “2*n – 2″ possible states. On the first floor, it can only be in the “1 : up” state. On the top floor, the only state is “n: down”. On every other ith floor, the elevator has two states: “i : up” or “i : down”.

Let us also stipulate “n > 1″. (A building with only one floor has no need of an elevator. A building with zero or fewer floor levels has trouble existing.)

Start Up from the First Floor

When you first enter the building, you are on the first floor and can only go up. Since the first floor is also the lowest level for the elevator, you will never be disappointed by seeing the elevator door open but indicate it will go down.

The elevator scores 100% satisfaction when you start from either extreme, because it can only go in the direction you want to go. (On the other hand, it is almost certain you have to wait for it to travel).

Based on the Second Floor

If the elevator is on the first floor, or just arriving at the second floor and heading up, you will be pleasantly surprised only two times out of the “2*n – 2″ states. That of course means you will see the elevator in state “2 : down” in “(2*n – 4)” cases out of “(2*n – 2)”.

However, if you are on the second floor but want to go down, your probability of seeing the “2 : down” elevator is “(2*n – 4)/(2*n – 2)”.

The Plot Thickens on the Third Floor

The for good states for the elevator are “2: down”, “1 : up”, “2 : up” and “3 : up” if you want to go higher. Therefore the probability of success is “4/(2*n – 2)”. Going down, however, you would succeed “(2*n – 6)/(2*n – 2)”% of the time.

The Generalized Elevator Puzzle for the ‘i’th Floor

We stipulate that “n > i > 1″ for this discussion, where ‘n’ is the number of floors and ‘i’ is the floor where you are currently. In the general case, you are on floor ‘i’, so there are “i – 1″ floors below and “n – i” floors above.

The elevator has “2n – 2″ possible states, pairing a floor and a direction. These states are {(1 : up), (2 : up),…,(n-1 : up), (n : down), (n-1 : down), (n-2 : down),…,(3 : down), (2 : down)}.

We want the “happy” state where elevator is heading in the direction you want, rather than going the other way.

Trivial Cases of the Elevator Puzzle

Let’s eliminate the trivial cases. When you are on the first floor, your only desire is to go up, and the elevator will never head down from the first floor. Likewise, on the nth floor, you can only want to go down, and the elevator will never go up from there. In both cases, you have a 100% guarantee of happiness.

As discussed in the ten-story building, when you start one floor away from the top or bottom, there are only two “Happy!” states, out of “2n – 2″ possibilities, if you are going toward the center of the building.

Generally, Things are Looking Up for the Elevator Puzzle

This section deals with going up.

In the general case, you are on floor ‘i’, so there are “i – 1″ floors below and “n – i” floors above. We restrict ‘i’ by stating “1 < i < (n-1)”.

When you want to go up, you would become “happy” in three situations as you step into the lobby. You hit the jackpot if the elevator immediately stops on your floor indicating “up”: this was the unique state “(i : up)”. If the elevator was in any state {(1 : up), (2 : up),…,(i-1 : up)}, you had to wait but never saw the elevator going down. Also, if the elevator was in state {(i-1 : down),…,(2 : down)}, you had a longer wait but, again, never saw the elevator on your floor but going down.

So the number of “happy” states when you head up from the ith floor are “1 + (i – 1) + (i – 2) = 2*i – 2″. The probability of being happy is “(2*i – 2)/(2*n – 2)”.

Therefore, when you want to go up, the probability that the elevator will first be on its way up is lower as you start on lower floors… with the exception of starting on the first floor.

Down is Just the Reverse of Up

Finally, note that the situations are the same as you head down from the top floors. The top floor will always be seen in the desirable “n : down” state. The (n-1)th floor has two desirable states when you want to go down, but “2*n – 2″ desirable for going up.

The “happy” states are: (i : down); {(n : down), (n-1, down),…,(i+1, down)}; and {(i+1, up), (i+2, up),…,(n-1, up)}. There are “1 + (n – i – 1) + (n – i – 2) = 2*n – 2*i – 2″ happy states out of “(2*n – 2)”.

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?

Who Were the Elevator Puzzle Physicists?

The two physicists were George Gamow and Otto Stern.

Who Was George Gamow?

George Gamow: Photo by Serge Lachinov

George Gamow was born in Odessa, Russia in 1904. His parents were teachers, but his mother died when he was nine years old.

Gamow studied at the University of Odessa and the University of St. Petersburg, but moved to the United States in 1936.

Among his professional interests were optics, nuclear physics and stellar mechanics. Gamow proposed a theory to explain radioactive nuclear decay, and aided research into DNA and the genetic code.

Gamow died at age 64, in 1968.

Who Was Otto Stern?

Photo of Otto Stern courtesy NNDB

Born in Sorau, Germany in 1888, Otto Stern earned his PhD in 1912 at the University of Breslau. He worked with Albert Einstein at the University of Prague and also the University of Zurich.

As his focus shifted from physical chemistry to theoretical physics, he also worked at several other European universities. He then moved to Pittsburgh in 1933 and joined the Carnegie Institute of Technology.

Stern was awarded the Nobel Prize in Physics in 1943, having contributed to: quantum theory; the molecular beam method for probing atoms; the verification of Maxwell’s laws of velocity distribution in gasses; and the wave nature of molecules.

Stern died in 1969.

Closing the Doors on the Elevator Puzzle

The elevator puzzle is a wonderful blend of science and mathematics. Gamow and Stern noticed it in the real world; anyone can list probabilities for a simplified model with limited floors; and anyone can capture data for themselves.

On the other hand, the elevator puzzle can challenge mathematicians and programmers when dealing with multiple elevators to solve the real-world goal of minimizing wait times and keeping building tenants happy.

Elevator Doors at the Waldorf. Image by Liz Henry

References:
Weisstein, Eric W. Elevator Paradox. MathWorld–A Wolfram Web Resource. Accessed Aug. 26, 2011.
George Gamow. EPS. Accessed Aug. 26, 2011.
Otto Stern – Biography. Nobelprize Org. [From Nobel Lectures, Physics 1942-1962, Elsevier Publishing Company, Amsterdam, 1964] (1943). Accessed Aug. 26, 2011.