When a guy goes into the bathroom, which urinal does he pick? Most guys are familiar with the International Choice of Urinal Protocol. It’s discussed at length elsewhere, but the basic premise is that the first guy picks an end urinal, and every subsequent guy chooses the urinal which puts him furthest from anyone else peeing. At least one buffer urinal is required between any two guys or Awkwardness ensues.
Let’s take a look at the efficiency of this protocol at slotting everyone into acceptable urinals. For some numbers of urinals, this protocol leads to efficient placement. If there are five urinals, they fill up like this:
The first two guys take the end and the third guy takes the middle one. At this point, the urinals are jammed — no further guys can pee without Awkwardness. But it’s pretty efficient; over 50% of the urinals are used.
On the other hand, if there are seven urinals, they don’t fill up so efficiently:
There should be room for four guys to pee without Awkwardness, but because the third guy followed the protocol and chose the middle urinal, there are no options left for the fourth guy (he presumably pees in a stall or the sink).
For eight urinals, the protocol works better:
So a row of eight urinals has a better packing efficiency than a row of seven, and a row of five is better than either.
This leads us to a question: what is the general formula for the number of guys who will fill in N urinals if they all come in one at a time and follow the urinal protocol? One could write a simple recursive program to solve it, placing one guy at a time, but there’s also a closed-form expression. If f(n) is the number of guys who can use n urinals, f(n) for n>2 is given by:
The protocol is vulnerable to producing inefficient results for some urinal counts. Some numbers of urinals encourage efficient packing, and others encourage sparse packing. If you graph the packing efficiency (f(n)/n), you get this:
This means that some large numbers of urinals will pack efficiently (50%) and some inefficiently (33%). The ‘best’ number of urinals, corresponding to the peaks of the graph, are of the form:
The worst, on the other hand, are given by:
So, if you want people to pack efficiently into your urinals, there should be 3, 5, 9, 17, or 33 of them, and if you want to take advantage of the protocol to maximize awkwardness, there should be 4, 7, 13, or 25 of them.
These calculations suggest a few other hacks. Guys: if you enter a bathroom with an awkward number of vacant urinals in a row, rather than taking one of the end ones, you can take one a third of the way down the line. This will break the awkward row into two optimal rows, turning a worst-case scenario into a best-case one. On the other hand, say you want to create awkwardness. If the bathroom has an unawkward number of urinals, you can pick one a third of the way in, transforming an optimal row into two awkward rows.
And, of course, if you want to make things really awkward, I suggest printing out this article and trying to explain it to the guy peeing next to you.
Discussion question: This is obviously a male-specific issue. Can you think of any female-specific experiences that could benefit from some mathematical analysis, experiences which — being a dude — I might be unfamiliar with? Alignments of periods with sequences of holidays? The patterns to those playground clapping rhymes? Whatever it is that goes on at slumber parties? Post your suggestions in the comments!
Edit: The protocol may not be international, but I’m calling it that anyway for acronym reasons.
xkcd 
For the girls, I think the period/holiday one is a good pick. Depends on the girl, since these things tend to slide a bit over time (although maybe you could take that into account?)…. but the fourth year in a row that you have cramps on christmas day or move-in weekend, you begin to notice a pattern. i suppose if you could reset the cycle using the pill, it’d be nice to know what the optimal cycle to aim for would be 🙂
btw, i’m teaching recurrence relations in my undergrad discrete math class later this semester. it’s gonna be really, really tempting to work the ICUP into the homework some way… after all, it’s a classic, real life problem.
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You’ve missed one part of the protocol, which generally only applies in situations where speed is a priority (I’ve seen it at sports events, high school class changes, and just before last call where there are enough urinals to prevent a line forming): If there is a free urinal which has people immediately either side (X-X) and you’re feeling bold or it’s already adjacent to a group of three (X-XXX), it may be used. This makes groups of 2^k+1 particularly efficient, since all it takes is one bold person to get from a little over 50% efficiency to 100%.
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I prefer to actually go directly to the middle urinals. This doubles my chances of a big wanker ending up at the urinal next to me. In this occurence, it’s customary to grab hold of said large wang and estimate weight, girth and length. Follow up with a few pumping squeezes to ensure working blood flow. It’s our duties to ensure our fellow urinators have a healthy working member.
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Nice article!
Now regarding to Women case. I think the case is completely different because they don’t follow the same arriving algorithm. They just arrive in groups of 4 or more, even if only one need to pee. However, all finally will pee making the problem a lot more complex.
Sounds impossible to solve…
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I like to poop in urinals.
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One female bathroom topic of interest might be figuring out whether it is in fact optimal for a guy to put the toilet seat down in a shared bathroom. I propose the measure of efficiency be total effort in managing seat state. Some assumptions and factors:
* It take more effort to lift the seat than to put it down (lifting requires bending over, putting the seat down usually requires just a flick of the wrist). The actual difference in effort is open to argument.
* One may wish to factor in the likelihood of a woman being “surprised” by an unlowered seat (which I’m told can happen on occasion) w/ some associated high cost of effort (“pain & suffering”)
* The ratio of men to women in the household as well as the average number of times the toilet is used and for which purpose would also be a factor
Now get to it. 😉
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That curve looks like an inverted version of the “new” Star Trek warp scale.
You made it up too, right?
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This fails to account for the lazy algorithm, where some dudes will take the closest available urinal with no adjoining used urinal. Now, obviously, if *everyone* followed this protocol you simply fill every other urinal left to right and the analysis is uninteresting. However, if only a certain percentage of urinal users are lazy, you could probably get some different statistics going.
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You did not take in account the possibility that the people may have consumed large amounts of alcohol. If that is the case, then the awkwardness levels tend to drop drastically, and so do the needed precautions.
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Your fill rate considerations are only on a very worst case scenario… Like all pooled resources, urinals get released at some point… Would be interesting to study the attrition rate and/or average probability of simultaneous arrival given the number of people…
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This article immediately reminded me of this flash game I saw years and years ago; http://gamescene.com/The_Urinal_Game_game.html
Perhaps the rules to the equation are more complex than stated above?
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I have a shy bladder, so I’ll just stand there and pretend to pee
if I can’t get the flow started. So, if I can start the flow before a
guy goes next to me, I’m fine … otherwise, I’m taking-up a urinal
without peeing. The time factor – how long it takes to pee must be
part of the equation.
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The one obvious flaw in the International Choice of Urinal Protocol is the one that everyone tiptoes around, but never has the guts to confront: What if one of the urinals is a children’s urinal?
It’s a usable urinal, but due to it being lower than the others, it’s statistically less probable that a urinal client will occupy it.
TEACH THE CONTRAVERSY
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Sigh.
CONTROVERSY.
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You forgot to factor in the recent trend of opaque partitions between urinals which eliminate awkwardness and are perhaps the best invention to overcome “stage fright” to date. They would maximize efficiency at minimal cost, regardless of number of urinals.
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>all it takes is one bold person to get from a little over 50% efficiency to 100%.
This is key when your in situations that have a high volume of people. once someone bridges this gap, the whole bathroom follows. It is useful when say you walk in with a friend, you its not just you being the outlier, but its already a trend and you not alone.
I’ve been at venues where they have a packed house and two bathrooms, on with girls lined up and one guy bathroom that is generally really fast. Its kinda of funny waiting for the bathroom(as a guy) and seeing a girl walk out. I commend you bold sister. In the end, when you’ve got a bladder full of used booze, its urgent that you just want to get the job done.
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“Sigh. CONTROVERSY” is my new way of mocking dumb arguments.
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Dammit, Jake, you beat me to the video!
1) Drunkenness or necessity does not ELIMINATE awkwardness, it merely overcomes it. You’ll notice that the above algorithm is not intending to predict urinal usage, merely describe urinal use configurations that become awkward.
2) Dividers may reduce awkwardness, but they do not eliminate it? Have you ever seen men, especially in america, being to use urinals sequentially just because they have dividers? No. They continue to follow the ICUP. Also, how many of you men LIKE standing in close proximity to other men that you don’t know, even fully clothed? As an American, I prefer about 6-12 inches, minimum, of space. Most urinals are not widely spaced enough to provide this, leading to awkwardness even when there are dividers. My ELBOW could be touching the next guys, for fucks sake! {shudder} 😉
3) The “gay” thing. I’m straight, and, having spent most of my life either in the midwest and in military environments, where people are as medieval as they can possibly be, I have had few gay friends. However, my guess is that gay men appreciate men peeing about as much as I appreciate women peeing. That is to say, not at all. Pee isn’t sexy for most of us. If it is, for you, I respectfully submit that you are projecting. If anyone feels like taking the time to google that, perhaps find the Kinsey study, feel free to correct me.
4) I wonder if there are any plumbing ramifications of the Stall/Urinal solution. Also, what are the space ramifications? I don’t have the math skills required, but it would be interesting to see how much extra space that took to achieve. My gut instinct is that it would require a fairly intense premium on floor space.
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And wow, I gotta remember to proofread. Minor Fail.
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Regarding awkwardness–yes, women have a bathroom issue. It deals with the amount of sinks, wastebaskets (and placement thereof) and paper towel dispensers. This factors in all the time when there is an unequal ration of papertowel dispenser to sink, but it compounds when menstration is involved. There is the awkwardness of needing to wash your hands, juggle the ‘wrapped’ parcel, wait for a sink, and get to a poorly placed wastebasket.
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The solution is to simply use a single urinal that extends along the entire length of the wall. Then user n+1 will simply find the position that makes his distance from other pee-ers as large as possible. This will continue until it is no longer possible to fit between two people without Physical Contact (which goes beyond the limits of the Awkwardness scale) taking place.
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@Rafael – Truth be told, there was a trough (really a tub) at my town’s carnival every year. At least when I was a kid. Come to think about it I -hope- that was the bathroom and not just someone’s tub.
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Or the solution can be stop being a shy wittle girl sucking of an oversized wowwy-pop with a big frilly pink ribbon in your hair and simply piss like a man.
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>> Most guys are familiar with the International Choice of Urinal Protocol. It’s discussed at length elsewhere […]<<
Can you provide a link to this? I assume its an RFC somewhere… I need to print it out and have all our employees sign that they understand proper urinal protocol as terms of their employment. xD
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one word… TROUGH
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Two things:
First, most of the bathrooms in my high school had five urinals. I’m not sure how it got started, but every guy in the school, teachers included, played by the MWF or Monday-Wednesday-Friday rule (assign days of the week to urinals from left to right, only ever use Monday Wednesday or Friday). This was observed at all times, and the Tuesday and Thursday urinals were noticeably cleaner than the other three. However, occasionally somebody would try to be a rebel and use Tuesday. This would cause backups and even tardiness to your next class. Nobody liked the kid who used Tuesday.
Second, the barrier solution is both a blessing and a curse. When barriers are involved, the urinals are usually placed closer together. So, even though line of sight is removed, you are still really close to your pee neighbor, increasing chance of strange sounds and smells, thereby increasing Awkwardness. Also, I was at a bathroom last week where they had barriers, but one was torn down. So there were three urinals, two of which were very very close to each other. I suppose that is a good thing if you are in there with a really close friend, or if you have two members, but otherwise, this is a highly Awkwardness causing situation. So the barrier solution has its own problems.
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This is pretty much identical to NYC subway seating protocol, and (in my experience) movie theater seating protocol. NYC subway seating protocol is a better analogy, however, because in both Randall’s protocol and subway seating protocol, there is a strong preference for those stations with only a single border, the edge stations. Movie theaters have the opposite preference.
I think his protocol is deficient, however, in noting that there are actually two values of awkwardness – one value, the lesser value, when both borders are occupied. Awkwardness goes up when only a single border will be occupied. For example, X-X becoming XXX is less awkward then X–X becoming either XX-X or X-XX.
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I would think that urinals in a convex arc would be a better arrangement. Decrease the exposure to the guys next to you and make everyone feel like they’re on an island.
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You should have pointed that an urinal is best used frontwards, and the fact you can pee from one position to a range of 6 or 7 urinals depending on your urinal skills…
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I never heard of that protocol, here in France, though I probably conform to it.
One could go further while studying the occupation ratio when you can use more than one wall to place the urinals.
For instance, 4 turns to be quite optimal if subdivided in 3 and 1. I don’t think it’s unreasonable in practice to make the assumption that at least 2 walls can be used.
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Finally, someone to write out that unwritten rule! Unfortunately, I don’t think everyone is born with this innate protocol. I once sat down on the end toilet (8 in the row, no stall dividers) in the military, and a guy comes in and sits on the one next to me… and we’re the only ones in there. At first I thought he was doing it on purpose, but after being forced into a conversation with him I figured out he was very simple-minded 😦
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Very interesting and true, but a friend brought to my attention that you left a variable without consideration, and that is the location of the door. Because, for instance in example one if there was a door next to the urinal on the right, the second subject would occupy the middle urinal instead (The one shown in the graph as occupied by the third guy).
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You know of course that the urinal rule is just another example of Hund’s rule of Maximum Multiplicity!
http://en.wikipedia.org/wiki/Hund%27s_rule_of_maximum_multiplicity
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There is also the issue of Friends, Awkward urinals and urinals with dividers.
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The trough urinals always produce 100% inefficiency when more than one person is involved.
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This issue is described in Dave Barry’s Guide to Guys, minus the math. He even drew a diagram just like yours.
Incidentally, if you haven’t read it, do: parts are juvenile, but it’s definitely hilarious.
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I presume the International Choice of Urinal Protocol described above is described in this movie?
http://www.youtube.com/watch?v=KnOaMC8KHA4&feature=player_embedded
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I think women need this equation: when to break into the men’s restroom because the women’s restroom line is too long. And do we need to worry about what the guys will think?
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In Sweden getting on the bus obeys the Pauli expultion principle. I don’t think most people know this, but it makes bus-riding for me all the more fun!
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lets monte carlo this mofo
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I think this description is seriously flawed. For a start it fails to take into account public toilets that have urinals along two walls, in an L-shape configuration.
Secondly, the people tending to go for an end urinal when all other urinals are unoccupied have an increased probability of Small Penis Syndrome, hence the end urinal position – less chance of someone seeing it and laughing! There’s no incentive otherwise to not go directly for a middle or off-centre position.
Finally, when the maximum number of patrons before Awkwardness is reached, the next user *can* use a stall, but risks shunning as “a big girl”. At this point, Awkwardness has to be accepted, with a loud clearing of throat to signal that your neighbours should assume the “eyes forward” protocol, if they haven’t already.
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@Tanya: You break into the men’s restroom…. never. Stay out of our restroom; you have your own.
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Should I enter as the first male, I prefer to use the furthest urinal as it limits the ‘pass’, which, I find, has the potential to be as creepy as the side-by-side.
Should I enter as the second male I prefer to take the next buffered location, thereby limiting the pass factor of newcomers.
I find it difficult when the restroom layout is formulated with three urinals between the sink and the stall. Where should one go in this scenario? Surely the middle is the least awkward when entering alone, but with the addition of others there is no escape.
One factor that you fail to mention however, is what to do when presented with a nicely spaced urinal usage [see 5 urinal model above], and one person finishes as you arrive. Do you take the recently pissed in spot, or proceed to an adjacent urinal creating an awkward presence.
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Apparently, you want to have your urinal cake and eat it too!
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