Why do elevators usually run in the opposite direction of the floor you are going to?

In the preface to Puzzling Mathematics, authors George Gamow and Marvin Stern mentioned that they worked in Building A of Convair in the summer of 1956. Stern was an employee of the company, working on the 6th floor, while Gamow worked on the 2nd floor as a consultant to the company.

The latter often took the elevator to visit the former. Gradually, he found that on average, 5 out of 6 times, the first elevator that arrived went down rather than up. So Gamow asked Stern if Convair built the elevator on the top floor and then drove it down. Stern replied, "No way! Try taking the elevator down and see what happens."

After a while, Gamow said, "You are right. When I want to take the elevator down, only one out of six times does it go down. Are you building elevators in the basement and then sending them to the top of the building for airplanes to take away?" Stern replied, "Of course not! From your experience, it just proves that this building has 7 floors!" (I remember that there is no such thing as zero floor in the United States. Their first floor is the first floor.)

So what is going on?

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If we assume that there is only one elevator and that it stops at every floor, and that it takes 1 minute to get from one floor to another, then this problem is easy to understand. If a person is on the 1st floor at 10 o'clock, then at 10:01 he is on the 2nd floor, at 10:02 he is on the 3rd floor, and at 10:06 he is on the 7th floor. At this time, the elevator starts to descend, and at 10:07 he is on the 6th floor, at 10:08 he is on the 5th floor, and at 10:12 he is back on the 1st floor.

If Gamow arrived at the elevator room on the second floor between 10 and 10:01, the elevator was just ascending; if he arrived between 10:11 and 10:12, the elevator was descending. The situation was similar on the sixth floor.

The annoying thing is that in real life, if there are more floors above us than below us, the elevator will often be above us rather than below us. So here's the problem: In Gamow and Stern's book, they also came to the same conclusion when there is more than one elevator.

A few years later, Donald Knuth published an article that verified that even if the first elevator to arrive often comes from the direction with more floors, this probability will change, and the more elevators there are, the closer this probability is to 50%. So never trust your intuition too much.

Image source: Tuchong Creative

When we apply the mathematical knowledge of elevators to real life, we will find a very interesting phenomenon: when there is more than one elevator in a building, the software that manages the elevators is very smart and allows us to spend more time waiting for the elevator.

Suppose you are on the third floor of an underground parking lot. There are two elevators parked on the first and fourth floors. Then you press the elevator button. The elevator that comes down is the one parked on the fourth floor. It takes twice as long to come down. Why is that? Is the elevator parked on the first floor afraid of encountering dirt when it comes down and unwilling to come down? The reason is actually quite boring. Because more people come in from the first floor door, the elevator program sets the elevator to stop on the first floor more.

Who knows, a large group of people might suddenly come in from the door at any time. It is best not to let such a group of people wait for the elevator for too long. If it is just one person waiting for the elevator, it doesn't matter if they wait a little longer. Or suppose you are on the 6th floor and want to take the elevator down. There are two elevators, one on the 5th floor and the other on the 3rd floor. After you press the elevator button, the elevator on the 5th floor starts to go up, the elevator on the 3rd floor goes down, and the elevator stops in front of you and continues to go up. Then it is very likely that the person on the 2nd floor pressed the elevator to go up, and the person on the 8th floor pressed the elevator to go down.

In this way, the waiting time is the most reasonable for everyone, although some people may have to wait longer. (I'll tell you secretly, I think this is why some elevators only have floor indicators when they stop at the first floor, so that people waiting for the elevator won't get angry if they can't see the floor indicators.)

Finally, remember that while mathematics can help us reduce the time we wait, psychology can help us reduce the perceived time even more effectively. It is said that in a company, many people complained that the elevator was slow, so the company installed a mirror at the elevator entrance, and people began to tidy up their appearance in front of the mirror and stopped complaining about the long wait time for the elevator. What a smart move, right?