Choose the best urinal in the men’s room — with math!

Men, we’ve all been there. A row of glistening white urinals appear before you. They’re all empty, thank goodness. You choose the cleanest one. That’s when you hear the door. Another person is walking into the bathroom. Don’t choose the stall next to me, is your first thought. He starts walking toward the urinals. Don’t do it. And closer. Don’t do it. Don’t...

Most men experience some level of social anxiety in the bathroom. We have an innate sense of personal space that we like to keep clear from strangers. When someone enters that personal space, we stiffen up.

This was demonstrated in a 1976 study. Researchers found the closer they stood in relationship to an unsuspecting participant in a men’s room, the longer it would take the participant to start urinating.

The researchers’ methods were deeply creepy. An observer sat in a closed-door stall next to the urinals and looked down at a hidden periscope on the floor. “An 11-inch (28-cm) space between the floor and the wall of the toilet stall provided a view, through the periscope, of the user’s lower torso and made possible direct visual sightings of the stream of urine,” the study reads. I assume if the participants knew they were being watched, they’d run out immediately.

The authors of that paper hypothesized that the intrusion into personal space provokes social anxiety, which in turn makes our muscles tense. Peeing involves the relaxation of sphincters around the urethra. If there are too many people around — if we’re too anxious — our urinary muscles can’t relax. We can’t go.

About 20 million Americans suffer from a more extreme form of this anxiety called paruresis, or shy-bladder syndrome. With paruresis, a person is unable to urinate in public. The UK Paruresis Trust suggests an evolutionary basis for the condition:

[Animals such as deer], when they sense a threat, freeze to be less noticeable; urination is disabled, because a flow would generate noise, smell and movement, all of which could attract the attention of a predator.

This condition makes attending lengthy public events — such as baseball games and movies — potentially awkward, uncomfortable affairs. Paruresis can be especially tricky for men, as urinals are not often designed with privacy in mind.

In 2010, researchers at Carleton University and Wesleyan published a paper that described the mathematically optimal strategy for choosing a urinal. The introduction to their paper reads like The Onion penned a question for the SAT:

A man walks into a men’s room and observes n empty urinals. Which urinal should he pick so as to maximize his chances of maintaining privacy, i.e., minimize the chance that someone will occupy a urinal beside him?

The key is figuring out when an array of stalls will reach their “saturation point” — the moment at which every other stall is occupied and the next person who enters the bathroom will be forced to stand next to someone else. If there are five stalls, the saturation point looks like this (the o’s are open stalls and the x’s are occupied stalls):

X O X O X

In this five-stall scenario, it’s best to choose an odd-numbered stall if you’re the first person to enter the bathroom. If you start on an even-numbered stall, the bathroom will become saturated with fewer occupants, upping your chances of a stall neighbor:

O X O X O

The researchers tested a number of hypothetical scenarios — what if the next person always chooses the closest urinal to the door out of laziness, what if people disregard personal space and choose at random, and so on. Danny Krizanc, the Wesleyan University mathematician who co-authored the paper, notes in an email: “We derived our assumptions from introspection and not from real world data or sociological studies. ... Our hope was they might be a reasonable first approximation.”

Across all the conditions tested, they found the same general rule holds true. “In order to maximize your privacy, you should probably choose the one furthest from the door if it is available and the one next to it is unoccupied,” they conclude.

The general framework holds up even in a larger array of stalls. Let’s consider a seven-stall scenario. “This is where it starts to get interesting,” Krizanc says. If you choose the stall farthest from the door, the array becomes saturated with three occupants:

X O O X O O X

It may seem better, then, in this seven-stall scenario, to take the third urinal from the end, which would result in a saturation point of four.

X O X O X O X

But even in this scenario, “the end is a slightly better choice then the third position,” Krizanc says. Why? Because the end only has one neighbor, so your chances of peeing next to someone are still reduced, even if your choice greatly increases the likelihood that the stalls become saturated.

The web comic illustrator and physics blogger Randall Munroe, who runs XKCD, looked into this problem in a post from 2009. His advice was for bathroom designers. According to Munroe, a bathroom should have three, five, nine, or 17 stalls for maximum saturation efficiency. Bathrooms with four, seven, or 13 stalls are begging for awkward interactions.

“If you want to make things really awkward,” Munroe wrote, “I suggest printing out this article and trying to explain it to the guy peeing next to you.”

Read more: It’s possible, through cognitive behavioral therapy, to learn how to relax the urinary sphincter muscles in tense situations.