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 
To avoid Awkwardness, I pee in a stall. Judging by the number of times I find the seat up, other guys resort to the same tactic.
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“International Choice of Urinal Protocol”
Oh my, I laughed a little too hard at this.
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I have it on good authority (a female acquaintance who grew up on a ranch)
that women in the great outdoors often enjoy a communal pee, replete
with unbroken conversation, and separated by no more than the distance
needed to pass around a roll of toilet paper. This made me realize that
the ICUP protocol also does not apply to men in the wild. It’s often
the case that male camping buddies will enjoy a communal pee, accompanied
by expressions of a quality peeing experience, such as vocalizations
like, “AAAAHHHHHHH” or a forcefully expelled fart. This is especially
true if beer has been consumed. However, they usually pee radially outward
from the group. Parallel peeing is acceptable, such as when two guys pee on a
wall or off a bridge. However, every guy knows that even in the wild,
you must not break the “85-degree rule,” which says that the angle between
your pee and the next guy must never drop below 85 degrees. (Ninety
degrees is ideal, but there is a five-degree imprecision allowance.) For
instance, it would result in Extreme Awkwardness to start peeing
on a tree trunk that a friend is already using to mark his territory.
My rancher friend said she had heard rumors that guys sometimes write
their names in the snow. I fessed up about this, but pointed
out that this had to wait until we were taught cursive writing in
the third grade, as guys lack women’s guillotine-like ability to
cut off a pee in mid-stream.
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These findings (that urinals should be built in groups of 3, 5 or 9) should really be published in an engineering journal. As for the post immediately above mine: Ewww.
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Okay, my post was a little over the top. But did you seriously get all the way down here and think that this site was about serious engineering instead of bathroom humor? Also, my rancher friend bragged that if it weren’t for poor aim, writing in block letters would not be a problem for women. So I’m not just making this stuff up!
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I have a girl problem that is not exactly mathematical, but it does have a sort of protocol to it. Personally I don’t have any problem being next to other women in stalls, and ‘noises’ are not an issue. And if they are, it’s stupid.
BUT there is the protocol of the dirty toilet.
In general, when one looks into a stall to see if they can use it, we assess things like if there is pee on the seat, toilet paper on the floor, anything that would be disgusting to sit on. Also included in this is if the toilet has been used, but has not been flushed. If the latter is the case and nothing else is gross, very few women will actually just flush the toilet and then use it. ESPECIALLY if it is #2, but also if it’s just 1.
The procedure for finding a stall too gross to use is to look in and then just look out and either go to the next empty stall or to wait if there isn’t any other option.
This becomes a problem mostly when all other stalls are filled and there is a long line.
Here is where the awkwardness arises, if a woman who will NOT flush a clean, but unflushed toilet and instead elects to wait for another stall rejects the stall, no one else may use it. This is essentially a waste of a stall. If a woman who flushes is behind her or behind multiple other non-flushers in the line, it would be awkward to move up, flush the toilet and then use the stall because it would be usurping the first woman’s place in line.
It is also awkward because it indicates that the flusher has lower standards of cleanliness than the other women.
THUS A DILEMMA.
I think to figure out any kind of proof for this you would have to figure out the average percentage of women who are no-nonsense flushers, and women who are too icked out by it.
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It turns out that urinals exist for women also. Here’s a picture of one:
http://itsjustablogblog.myphotologs.com/?p=189
So, women, if there were a long row of these things, what would your protocol be?
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I was at school and there was a teacher there. Should there be a protocol about a significant age difference that would make things more awkward?
sinks? hand blow dryers? troughs at football games? public seating at movie theaters etc? locker room showers? The possibility of awkwardness is endless.
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Great article “Urinal Protocol”… hmmm, I wonder if this behavior is “ingrained” in our “DNA” from the start !! I’ve always followed most of these “protocols” it seems… but I never learned it from anything “written down”… I just did it automatically !! 🙂
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Ideal case:
Assuming 1 man takes 1 minute from the point he occupies a urinal until the point he leave, then the graph of decay will be a parabolic curve similar to the one in this pic. http://www.devmaster.net/articles/fast-sine-cosine/parabola.gif This is due to the fact that the speed and the quantity of the urine increases initially and as it reaches the peak, it (speed and quantity) decreases. Thus leading to the graph of a parabola.
Special Case:
The 1st part is the same till the speed and the quantity reaches the peak. In some exceptional cases if the quantity of the urine stored is more then, the speed of passing urine remains a constant for some period of time. And then decreases as a usual Parabola.
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I suppose this is only applicable when there are no stall dividers…otherwise you can attain maximum usage with almost absolutely no awkwardness (well, maybe no
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Summa,
Excellent questions, both.
I hypothesize that the women’s bathroom on the 3rd floor would be less crowded. No one wants to clim *up* stairs if they don’t have to, and many people would think, “I can just hit the 1st floor bathroom on the way out.”
On question number 2, when is the best time to go in a stadium, concert, etc. — right before the end of an ‘act,’ is good, if you can do it. But even if you go right at the beginning of the break, that’s good too. Most people get up, stretch, ask people, “So what did you think of the show?”, get a drink, and *then* go to the bathroom….so if you just go right away you’ll have only a short line.
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I’ve got a female equation for you to solve. How far away does the nearest boat have to be for a woman to attempt to pee off the side of a boat?
While fishing from behind a tree, I discovered that some women can pee extremely well while standing up. I’m sure that I would not have discovered this if I had been in plain sight. This begs the question: Just how far away from civilization does a woman have to get before she attempts to pee standing up?
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all I can say is… pure genious.
bah hah hah hah…
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Well, i dont really use this protocol, instead i cannot go to a urinal if there is a person 1 urinal away. EX. on the first diagram, if there is someone on the second, i go on the forth, if someones on first, i go on third or fifth. if someone is on 2 and on 4 then i go on 6. if there isnt a 6 then i’ll wait.
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I.C.U.P, Genius my friend, GENIUS!
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To anyone who may have suggested filling urinals 2k+1 where k=the order in which a man arrived in the restroom.
You obviously missed the point that the idea is to minimize any chance of dong rays entering your eyes by maximizing the distance between your eyes and another dong.
We’re not trying to fit as many dongs to urinals as possible, we’re trying to save our precious sight so it can be spent doing more productive things like watching videos of snatches on the internet.
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Ahahah! Simply great!
I love this useful/useless problem solved with math. Pleas post another one if you have.
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Or… You can space them out some more and section them off somewhat and get a 0% chance of awkwardness!
NB: Your post is very awesome!
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One problem, though. What about the short urinal? The one that you don’t go in unless the awkwardness potential is too high?
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I’ve spent quite some time considering a special case of this problem, and I was delighted to see the overall question analysed in such detail.
There is a row of three urinals in my place of residence which sees enough traffic to often be shared, and although the urinals are well spaced, awkwardness is still observably a consideration of many of the men who frequent them. In fact, one guy once told me off for choosing the middle urinal of the three. Perhaps he read this article, although I’m not sure if the dates match up.
However, I do not consider my choice a ‘hack’, or at least I do not believe it is quite that simple. There are several different preferences any urinal chooser might exhibit that would change their behaviour when presented with a urinal choice. As well as those objectives mentioned here (maximising or minimising awkwardness), a chooser might seek to appear unconcerned by choosing the closest urinal from the door, or they might seek to minimise awkwardness between themselves and pre-existing urinal users, with no concern for those that arrive mid-flow. And, in my case, I generally choose an end urinal if there is already an end urinal occupied, to follow protocol and be distant from the original urinator, but I choose the middle urinal if I am alone, not particularly to create discomfort in others, but to prevent anybody else having the satisfaction of forcing their uncomfortable proximity on me when they arrive.
To analyse this from a game theory perspective is particularly interesting – on the face of it, the first person arriving at a urinal can benefit themself by obeying protocol (if they care about proximity) or, if they do not care, then at no personal cost they can act unfavourably towards another player, either by causing them awkwardness, or by denying them the opportunity to deliberately create awkwardness. Therefore, for the person who, like me, does not care overly much about the proximity of other urinators, is the decision to take the centre urinal just griefing? I do not believe I aim specifically to cause discomfort in others (or “beat” the urinal protocol) when I take the middle of three urinals. Sometimes it’s just nice to be in control.
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An interesting game, in which the only winning move is not to play. Anyway, I like the soothing splash of water noise that peeing in the stall rewards me with.
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Is there a proof of this anywhere? I really want to see how the function works and I can’t quite work it out for myself
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One question, What if the urinals have adequate opaque dividers in between? say, from the floor to about chest height? That should help solve general awkwardness, unless you’re next to a tall guy who likes to peer over the dividers.
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Has anyone considered the fact that the distance between urinals that you are comfortable with is inversely proportional to how INTENSELY you have to go???
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I discovered that people do not follow ICUP protocol but behave according to the laws of thermodynamics.
There is a non-linear repulsion force between nearest neighbours. The probobability of chosing a given some urinal dpends on distance to occupied urinals and temperature. Only if the temperature is zero people follow ICUP.
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I disagree with one part of your premise, and that is that the man will always pick the urinal furthest from another urinal in use. I believe that the one-empty-urinal spacing is sufficient and more efficient, at least to the Awkward-Aware man.
The Awkward-Aware man will insure a distance of at least empty urinal between him and any other urinal in use, but unlike the 7-urinal example, the third man will pick urinal 3 or 5 instead of 4. This is because he is Awkward-Aware and realizes instinctively that by picking 4, he forces the next man into an Awkward situation, and that that there is a 50% chance that this next man will put him into an equally Awkward situation whereas either other man (each of whom should leave earlier) only has a 25% chance of Awkwardness. Thereby, by choosing 4, he puts himself at a higher risk for Awkwardness.
If he instead chooses 3 or 5, he leaves a urinal open for another man without Awkwardness arising. It is only if still another man chooses an open urinal will Awkwardness arise, and then in all cases, this will be a shared Awkwardness by three men, so it is less likely that the last man will enter into such an arrangement.
The assumption is that one empty urinal is sufficient space to prevent Awkwardness. Those men who do not find this sufficient usually go to a stall in all cases as any other presence at a urinal or the threat of such presence is sufficient to keep them from using the urinal at all.
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Would this complex senario and mathmatitical formula always workout the same if I had walked into the urinal with an AR15 rifle with me ? or would I find a urinal with vacant spaces either side quite easily ?
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This is an awesome animation of why this formula MUST be followed!! World peace DEPENDS on it!
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I was wondering why this was even an issue, being a guy and not minding peeing next to someone in a urinal, when I read the comments and realized you were talking about urinals without dividers. I seriously haven’t seen a row of urinals without a divider between them in several years. What kind of backwards places are you going out to where the urinals have nothing between them?
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I’m surprised Anonymous claims not to have seen a urinal without a divider in years. I don’t know what great country that he lives in, but in mine (Canada), many urinals don’t (I’d say less than half).
On one of my only two trips to the USA I went to the bathroom at a baseball stadium. There were not only no dividers, THEIR WASN’T EVEN URINALS, THERE WAS A BIG TROUGH! Not even kidding. One 50 foot long trough. Like those things animals eat out of. It didn’t look like anything that was meant for pissing into, it literally looked like cows or something should have been eating out of it.
I don’t know how I managed to piss into it. It was a pretty awkward experience, 30 people with their dicks out and pretty much no way of avoiding seeing each other.
That day my opinion of the USA went way down. Since then I’ve only been to big stadiums like that in Canada, and thankfully they were much better. I remember one with dividers and little tv screens infront of each urinal. Way more comfortable than just pulling your dick out in the open with a few dozen people in the room who can see.
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I cannot believe that men are that uptight about seeing other penises and having their own seen. Grow up!
I did enjoy reading the article though.
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lol, this was just linked to in Norways biggest newspaper in an article about standing in line xD
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Reminded me of http://www.radioactivepanda.com/comic/70 which might be Hat Guy’s solution as well…
Good work, though I’d say I’m an awkward-aware guy as Dupin describes — because of it, I usually choose one at the end if a row is unoccupied.
Still, great to see this analyzed — I wonder if there’s any way to get it to whoever designs men’s restrooms?
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I think I’m going to program this into my trusty TI-83 Plus and leave I.C.U.P. Efficiency Rating signs in all the bathrooms I visit now.
xP
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Something else to ponder, from a social perspective, is the pervasiveness of the protocol even in circumstances where there are dividers between urinals. I have personally noticed that even in the presence of dividers, many men still follow the ICUP rules (even to the extent that some others waited for an ICUP-compliant position), even though the possibility of awkwardness was eliminated by the presence of the dividers.
But I digress, this is beautifully written. I can finally explain this facet of male culture to my girlfriend in terms she’ll understand (she’s a math major 😛 ).
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this is the Pisot Sequence L(3,5): 3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073, 262145, 524289, 1048577, 2097153, 4194305, 8388609, 16777217, 33554433, 67108865, 134217729, 268435457, 536870913, 1073741825, 2147483649, 4294967297, 8589934593
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Um, this might sound slightly odd, but why exactly is there a need for the urinal in the first place? Why are your bathrooms not designed like the women’s side? Full stalls, with a fully functional toilet in each one. The invention of the urinal just never made much sense to me. I mean, it’s like somebody decided one day, “Hey, let’s make a toilet specifically for men that women can’t use and is only half functional!” Seriously, you wanna experience awkwardness? Try having the sudden urge to defecate while urinating…and you guys out there can’t possibly tell me that’s never happened to you…so now you have to look for an open stall, with your still dripping equipment dangling about freely in front of a room full of other men. Stalls are good, stalls are your pal. LOL
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Does this equation also work in men’s showers? I assume there may be an open room with shower heads. I was also under the impression that men do glance at each other to check for size comparison. Is this not the case?
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As a fundamentalist Christian white male, I have to say: this seems a little homophobic.
I mean, I like my personal space, too, but seriously, guys. Nobody’s going to steal your ding-dong.
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Just keep your eyes on the wall, and everything will be OK.
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A very unique use of math. However, I say ‘duck not causing awkewardness’. The only way to pee is to stand between two guys at a urinal and drop trow like you did in pre school!
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