This article is from Proof Positive, our friendly newsletter that explores the joys and peculiarities of math. Sign up today for a weekly math essay and puzzle in your email inbox.

If you’ve ever spent any length of time in a tall building—either because you live or work in one—you probably know this feeling: The elevator's cars always seem to go in the wrong direction. If you want to go down, there’s an elevator going up, and vice versa.

On supporting science journalism

If you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.

The phenomenon intrigued physicists George Gamow and Marvin Stern in the mid-1950s. Gamow, who, among other things, published groundbreaking work on the theory of radioactivity, was working as a consultant for the company Convair in San Diego, Calif., in the summer of 1956. The company achieved worldwide prominence in the development and manufacture of military aircraft and space technology.

Gamow had an office on the second floor of the company’s six-story building while his friend and esteemed colleague Stern worked on the fifth floor. The two researchers frequently exchanged ideas and used the elevator to get from one’s office to the other’s.

At some point Gamow noticed that when he pressed the button, the first arriving car was usually on its way down, so he had to wait for it to come up again from the first floor. This pattern bothered him so much that, at some point, he began keeping records of it. He discovered that the elevator went down five out of six times when he pressed the button and that, only one out of six times, it would go in the desired direction, up.

Gamow and Stern made a joke of it, speculating that perhaps new cars were continuously being made on the building’s roof and sent down for storage in the basement. Stern, a scientist through and through, also began to keep records, and he, too, found that the elevator was traveling in the opposite direction of his desired travel more often than not: in five out of six cases, the car was on its way up when he wanted to go down.

The two physicists took a closer look at the mathematics of the problem—and came up with a plausible explanation. First, this strange tendency is not (or at least not only) a case of “Murphy’s Law,” where anything that can go wrong will. Nor is it the case that people are simply more likely to remember the long elevator waits. Instead it really is the case that elevators travel in the opposite direction than you intend more often than not, as Gamow and Stern’s statistics show.

If you are on the top floor of a building, then all elevators inevitably come up from the bottom and later descend. For the penultimate floor, then, a car goes up, and shortly afterward, it goes down again. The time interval between a car going up and a car going down is therefore extremely short. If you call the elevator at a random time, the probability is higher that you’ll catch an ascending car first. And that likelihood is noticeable—after all, with many floors below, there are good odds that you want to head down rather than up. That said, if you stay at the elevator bank and log all rides coming and going for hours or possibly days, you will eventually find that, on average, as many cars go up as go down.

The same logic applies in the reverse case for low floors. If a building has no basement, a car will always arrive at the first floor from above and then continue its journey upward. Thus, the time interval (if the car does not simply stay on the first floor) on the second floor between a descending and an ascending elevator is very small. For this reason, the probability of encountering a descending elevator first is higher.

To really think this through, take a larger building as an example. Imagine you work in a 30-story high-rise. Let’s imagine—in a case of truly terrible architectural planning—that there is only one extremely slow elevator, stopping at each floor and taking one minute per floor. To ensure that employees arrive on time at their respective floors, building management creates a timetable showing the departing elevators in each direction:

If you’re working on the second floor, don’t know about this schedule yet and walk over to the elevator bank, the first car you encounter will almost certainly be going down—unless you arrive at the exactly the hour or one minute after. So in 29 out of 30 cases, the elevator will go down first. The situation is similar with your colleagues on the 29th floor.

In more realistic scenarios, if a building has several elevators, or if one takes into account that people on low floors sometimes take the stairs, the corresponding calculations become much more complicated. Nevertheless, the trend remains the same: the elevator is going the “wrong” way most of the time. So if you find yourself standing in front of an elevator again, annoyed at having to wait, know that there is no great conspiracy against you—just simple mathematics.