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 
lol lets here it for unisex bathrooms! talk about awkward. wasnt that what alley mcbeal show was about? I personally hate waiting in those loooooooooooog lines at the bar and at concerts so, I dont mind making a whole room full of males very uncomfy. I cover my eyes and take my pee without a sound. SO WHAT. get over it most of us are adults or are pretending to be. As for ladies room rules, most chicks head straight for the roomy handycap at the very end of BR. Then the Middlemost stall. Finally the one closest to the door; which is the one nooo girl wants, if we poo everyone will findout that girls are full of sh*t. lolol
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“Its funny how readers fail to see that this entry is a direct rip-off of Maddox?s book ?The Alphabet of Manliness?.”
Because Maddox, who did write a very funny article, is obviously the only other person in the world EVER to talk about urinal etiquette. Yep. Nobody ever thought of or discussed such things before the book.
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Where would you stand in an infinite wall of urinals? What if the urinals are arranged around 3 walls?
What if it was a circular room which you entered via a circular staircase in the centre?
All these issues and more when you use non-standard urinal placement.
I should design a public toilet.
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I remember in high school, one of the most awkward things to do was to synch summer pool parties with the girls’ periods. In the end, one would always end up having hers and had to sit in the shade (with a supportive posse, of course, so she doens’t look like a fucking vampire) and just pretend to “not really feel like swimming.
A solution would be a nice challenge for the great Mr. Munroe, as well as help all the future generations’ girls’ geeky pool parties (even though I’m guessing there won’t be many..)
Also, GREAT work on this! 😀
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Hey guys…listen its a great study to read n see ppl commenting on it also in the same intellectual ways…
but i have a little doubt…u ppl always would have heard about india’s ppl tradition of pissing by the road…u would have said ewww…thts gross…
but let me put u in a condition…u waitin for ur bf/gf in some bus stop n may be s/he gets late due to some work…n u feel like peeing…n then u just see tht its dark enuf so tht ppl from far wont be noticing u much…n then u find a place close to the stop, near some tree…n then pee…omg i can’t tell u wht kinda relieved u feel…i m talkin real…u gotta try tht…
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You forget the coeficient X, where X to the power 3 or XXX, multipled by the inverse of the temperature of any beer drunk is proprtional to how likely it is the International Choice of Urinal Protocol is breached! Obviously this introduces some chaos, especially when X > than 3 or 4, then perhaps the problem of seeing double urinals might happen leading to the concept of sharing urinals [inadvertently]; this of course could lead to wet feet and puddles, but this becomes too complex a problem without the introduction of complex or imaginary numbers [ even more than likley if X is related to ingestion of other intoxicants such as hallucinogens, when the International Choice of Urinal Protocol may become very wobbly..]
However this is all great work.
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If someone comes in and uses the urinal next to yours when there was another obvious choice that would not create awkwardness try this: Look over and say “You know its really hard to pee with an erection.” They will move.
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I believe the protocol is itself a (periodic) function of time – in particular, the time since the urinals were last cleaned. This is because the first choice urinal when all are vacant is also influenced by the amount of piss splattered in and around the urinal.
To take the five urinal arrangement as an example, the most piss splattered are likely to be #1, #3 and #5. Therefore, over time, #2 and #4 will become increasingly attractive as a first choice when all are vacant.
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Interesting that you refer to ICUP as an ‘acronym’. The word ‘acronym’ used to be reserved specifically for abbreviations which were pronounced as words (for instance ‘NASA’ is an acronym, whereas ‘BBC’ isn’t).
ICUP, assuming you mean it to be pronounced ‘I see you pee’ (and I think that’s fair assumption, since it’s a good joke!) is an abbreviation – to be an acronym it would be pronounced ‘eyecup’. My son, who is possibly a bigger pedant even than me, assumed that, because you said ‘acronym’, you meant it the old-fashioned way, and that there was some sort of additional ‘n-girls-n-cups’ joke going on of which he was unaware.
The acronym/abbreviation distinction seems to be falling out of use nowadays, though. Increasingly the two words are used interchangeably to mean the same thing – abbreviation. I think that’s a shame…
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So True, So True
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You can exercise your understanding of this algorithm with this game:
http://www.onlinegames.net/games/1131/the-urinal-game.html
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And… how does this protocol work with drunk people? They usually see double the number of places to release the water.
Is there chance of a collision?
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This study is chewed grass. Preference evaporates with the degree of urgency and the venue.
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At our local pub there are three urinals. Two ordinary urinals next to each other then one urinal set slightly lower, I’m guessing for children. What should I do do when I walk to the bog and the first adult urinal is occupied? I’ve been wrestling with this dilemma for 6 months now. Help!
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For maximum awkwardness, no matter how many urinals are present, take the one next to the the guy peeing, unzip, sigh deeply and comment: “Nice dick!”
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Who cares? I’m just peeing. Don’t give a dang who is standing next to me. Why should anyone else?
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So I went to a concert once and instead of having urinals there was just this long trough.
what do you do in that situation?
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Our hostel’s urinal are the best case scenario. There are only three urinals out of which the middle’s one pipe is broken. So the only options left are 1st and the 3rd. 100% efficient .
To increase the efficiency in case of odd no. of urinals, remove the pipes of some urinals in such a manner that one removed pipe is in between the two perfect urinals..
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I’m printing this. It’s going on the door of my local pub’s bogs.
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Dang Brits’, what the heck is a bog?
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haha, good read.
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i always followed but never knew it was a protocol. I just don’t want to get splashed by some slob.
Doesn’t it stink when you follow protocol and the next guy doesn’t?
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See bathroom study by R.D. Middlemist in 1976 : http://www.psychologytoday.com/files/u47/middlemist_et_al.pdf
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I’ve never understood why people don’t just install some privacy screens between the urinals, therefore promoting 100% usage. Men. So inefficient.
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Works, but when you are in dire emergency and the urinals are arranged in 4 and the first and last are occupied, the best way to privacy, is to angle yourself against the person next to you, so your penis will not show. This gives us another variable, which can give another formula, and etc. Too lazy to figure it out.
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“At our local pub there are three urinals. Two ordinary urinals next to each other then one urinal set slightly lower, I?m guessing for children.”
Why does your pub have a urinal for children?
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“At our local pub there are three urinals. Two ordinary urinals next to each other then one urinal set slightly lower, I’m guessing for children.”
I prefer the lower one. I hate it when my unit dangles in the water…
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At least give some credit to Dave Barry, who was the first to tell this joke back in the 90’s.
http://www.angelfire.com/ma4/knickman11/r.htm
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What if you try to piss in the same urinal as another guy at the same time??
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What do you do when at the horse races and a female walks up stands next to you at the urinal, pulls up her dress and starts peeing.
It is awkward when she looks at you and says “Hi I see mine is bigger than yours”
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The same idea can be applied to people placement in an elevator or seating on the subway.
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hypotesis: No one likes akward situations.
in the 4 urinals senario, for example, the best replay to urinal 1, is urial 4, and no other.
The expectation of an akward scenario is 1/2 given that a 3rd guy will use an other urinal (lets hope he takes the 2nd and not the 3rd).
In general, the best replay given that a guy is pissing and you are 2nd in turn, is to take the oposite edge…the intuition is correct.
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I have one for women:
height of heels and proportionality to how long I can wear them for 🙂
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International Choice of Urinal Protocol
ICUP
Or maybe you don’t
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ICUP
Right over that rainbow
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“At least give some credit to Dave Barry, who was the first to tell this joke back in the 90’s.”
He was hardly the first. This observation is probably older than you are. Unless you are in your 60s, in which case you might be the same age.
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“Alignments of periods with sequences of holidays?”
This.
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How would this work with drunk people.
Just imagine after a few buds you would be seeing double!
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Well, im not a man, but I would hope it wouldn’t take too long for men to pee. So, there wouldn’t be that much of an awkward moment, maybe just awkward seconds.
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This is certainly not limited to urinals. Benches, for example.
Also, this doesn’t take into account that any man entering such a scenario has to weigh awkwardness against the desirabilities of each urinal — for example, cleanliness. The fact that different people weigh these differently often causes odd configurations and lowered efficiency.
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This protocol also misses a wrinkle, which is what happens in the toilets at gay bars. The accepted gay urinal protocol (GUP) is that the first person in takes the closest urinal to the door, and subsequent users take the one closest to any other current user. (Or, if it’s past about 1am and the night’s been going well, you can share!)
(captcha, oddly disturbingly, is ’21 98 raping’.)
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I noticed this difference in the gay urinal protocol as well while at a gay club. Another difference is that the “Eyes Forward at All Times” convention also does not seem to apply.
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I thought the One Urinal rule was the only one that mattered. Typically, if there are 5 urinals, there is one or two short (midget?) urinals, such that the first two guys take up the End (Regular) urinal, and the center Urinal, The third guy taking the short urinal, or waiting on one of the other regular sized ones to become available? So Long as there is at least one buffer urinal, I would disagree that the furthest from anyone else rule applies all the time, It just isn’t efficient!
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I do something Sort of Similar with Stalls… except I try to determine which stall is used lest, therefore likely to be cleaner. I havn’t found any satisfactory way to determine this yet. Any Hints/help?
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