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The Urinal Problem

urinalToday 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.)
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