The Urinal Problem

Today we study a particular variation of the classic so-called Urinal Problem. For millennia great thinkers like Socrates, Plato, Leonardo da Vinci, Bill Gates and others have pondered the great mysteries of gentlemen’s restroom etiquette. Now it’s my turn to take the problem out for a spin.

The classic definition of the problem, of course, involves an infinite number of monkeys and an infinite number of urinals. It’s easy to see how a problem like that could humble even the greats. In a flight of hubris, even I once made the attempt, and was left humbled and feeling flushed.

For simplicity, we will closely examine a three-urinal subset of n and attempt to fully solve the problem variation.

Abstract. A man walks into a men’s room and observes n empty urinals. Which urinal should he pick so as to minimize his chances of maintaining privacy, i.e., minimize the chance that someone will occupy a urinal beside him? In this paper, we attempt to answer this question under a variety of models for standard men’s room behavior. Our results suggest that for the most part one should probably choose the urinal furthest from the door (with some interesting exceptions). We also suggest a number of variations on the problem that lead to many open problems.

Source: Springer Link – The Urinal Problem. The complete paper is available for purchase.

It was easy to theorize a solution for the three urinal-subset based on the process of elimination (no pun intended). This is also known as The Vizzini Gambit. (See: The Princess Bride.)

Clearly you should not choose the urinal in the center as the next visitor must choose one of the adjacent urinals. Thus, it is obvious that the solution must be one of the end urinals. But which one? Elimination only gets us so far.

As is often the case, field research is required to test theoretical constructs. And that’s where the shit hit the fan. (The results of that experiment are beyond the scope of this article.)

For the purpose of experimentation using actual urinals would have been problematic on several fronts. Thus we used a model that we hoped would approximate similar results.

The setting was a small Italian coffee house and bakery. To stay under cover we ordered fancy Italian coffees and shared a chicken sausage frittata. We positioned ourselves at the end of a series of three window tables. Yes, we could have also performed the experiment in a movie theater, but limiting n to only three is difficult and besides, we’ve already performed that experiment innumerable times. The Highway Problem we had also beaten to death.

So, the experiment was under way!

Our blind group entered the restaurant. It was a pair of older women. Eagerly we awaited to see where they would choose to sit. The entire restaurant was available to them, but only three window tables, all in row, were available, and we occupied the one furthest from the door. We designated this as table #1. In the middle was table #2 and closest to the door was table #3.

They could have selected any table in the restaurant which was completely empty and wide open to them. But if they choose table #2 or table #3 we would gain valuable empirical data.

We were settled in for a long wait, since logic dictated they would likely choose away from the window to maintain distance from us. And, even if they did choose the window, they would likely choose table #3 and we would learn very little, since it was the only remaining logical option.

Eureka! They selected table #2 and settled in mere centimeters from us.

Isn’t science exciting? We had defied the odds and obtained a significant result on our very first try. I thought science was supposed to be harder than this.

I attempted to remain calm but communicated much emotion to my partner using only my eyes. Only later would I discover that my meaning was misconstrued as animalistic rage and primal physical confrontation urges. I also had this sudden urge to pee.

What was the meaning of their mysterious decision to avoid an entire restaurant of empty tables and sit as closely as possible to us? Direct observation was required. I had to learn more! And that’s where things became even more strange.

The women were staring at us. This almost caused my partner to violate the conditions of the experiment. They didn’t speak to each other very much. They looked funny. There was a dead silence in the room and with each passing moment the awkwardness was exponentially increasing. When my partner and I did attempt to communicate we leaned away from table #2 and whispered in hushed tones. Awkward.

Eventually our test group did emit some vocalizations, but those were limited to statements involving the acquisition of a piece of pumpkin cheesecake and comments pertaining to a stranded dog visible out the window and imprisoned in a vehicle. They didn’t offer much illumination.

Sadly, in the end, we didn’t actually learn very much. The reasons behind the actions of those two idiots still remains very much a mystery. The effect was that our morning was pretty much shat upon but we still do not understand the cause.

We publish our findings here today in the hopes that others may benefit from our research. Perhaps the next time you choose a space in an empty campgrounds and the only other guest chooses the space ten feet away (out of the hundreds available) this will give you something to think about as they bust out their boombox and put their tunes on MAX. So much for the bird watching, eh?

This is a very interesting field of study.